# EBook Problems

(Difference between revisions)
 Revision as of 21:25, 16 November 2008 (view source)JayZzz (Talk | contribs)← Older edit Current revision as of 06:31, 14 December 2010 (view source)IvoDinov (Talk | contribs) (→V. Normal Probability Distribution: added Multivariate Normal Problems) (48 intermediate revisions not shown) Line 4: Line 4: =I. Introduction to Statistics= =I. Introduction to Statistics= - ==The Nature of Data and Variation== + ==[[AP_Statistics_Curriculum_2007_IntroVar | The Nature of Data and Variation]]== - ==Uses and Abuses of Statistics== + Although natural phenomena in real life are unpredictable, the designs of experiments are bound to generate data that varies because of intrinsic (internal to the system) or extrinsic (due to the ambient environment) effects. - ==Design of Experiments== + How many natural processes or phenomena in real life can we describe that have an exact mathematical closed-form description and are completely deterministic? How do we model the rest of the processes that are unpredictable and have random characteristics? - ==Statistics with Tools (Calculators and Computers)== + ===[[EBook_Problems_EDA_IntroVar | Problems]]=== - =II. Describing, Exploring, and Comparing Data= + ==[[AP_Statistics_Curriculum_2007_IntroUses |Uses and Abuses of Statistics]]== - ==Types of Data== + Statistics is the science of variation, randomness and chance. As such, statistics is different from other sciences, where the processes being studied obey exact deterministic mathematical laws. Statistics provides quantitative inference represented as long-time probability values, confidence or prediction intervals, odds, chances, etc., which may ultimately be subjected to varying interpretations. The phrase ''Uses and Abuses of Statistics'' refers to the notion that in some cases statistical results may be used as evidence to seemingly opposite theses. However, most of the time, common [http://en.wikipedia.org/wiki/Logic principles of logic] allow us to disambiguate the obtained statistical inference. - ==Summarizing Data with Frequency Tables== + ===[[EBook_Problems_EDA_IntroUses | Problems]]=== - ==[[AP_Statistics_Curriculum_2007_EDA_Pics | Pictures of Data]]== + - There are many different ways to display and graphically visualize data. These graphical techniques facilitate the understanding of the dataset and enable the selection of an appropriate statistical methodology for the analysis of the data. + - ===[[EBook_Problems_EDA_Pics | Problems]]=== + - ==Measures of Central Tendency== + ==[[AP_Statistics_Curriculum_2007_IntroDesign | Design of Experiments]]== - '''1. Suppose that in a certain country, the average yearly income for 75% of the population is below average, what would you use as the measure of center and spread? + Design of experiments is the blueprint for planning a study or experiment, performing the data collection protocol and controlling the study parameters for accuracy and consistency. Data, or information, is typically collected in regard to a specific process or phenomenon being studied to investigate the effects of some controlled variables (independent variables or predictors) on other observed measurements (responses or dependent variables). Both types of variables are associated with specific observational units (living beings, components, objects, materials, etc.) + ===[[EBook_Problems_EDA_IntroDesign | Problems]]=== - '''Choose one answer. + ==[[AP_Statistics_Curriculum_2007_IntroTools |Statistics with Tools (Calculators and Computers)]]== + All methods for data analysis, understanding or visualizing are based on models that often have compact analytical representations (e.g., formulas, symbolic equations, etc.) Models are used to study processes theoretically. Empirical validations of the utility of models are achieved by inputting data and executing tests of the models. This validation step may be done manually, by computing the model prediction or model inference from recorded measurements. This process may be possible by hand, but only for small numbers of observations (<10). In practice, we write (or use existent) algorithms and computer programs that automate these calculations for better efficiency, accuracy and consistency in applying models to larger datasets. + ===[[EBook_Problems_EDA_IntroTools | Problems]]=== - :''(a) Mean and interquartile range + =II. Describing, Exploring, and Comparing Data= + ==[[AP_Statistics_Curriculum_2007_EDA_DataTypes |Types of Data ]]== + There are two important concepts in any data analysis - '''Population''' and '''Sample'''. + Each of these may generate data of two major types - '''Quantitative''' or '''Qualitative''' measurements. + ===[[EBook_Problems_EDA_DataTypes | Problems]]=== - :''(b) Mean and standard deviation + ==[[AP_Statistics_Curriculum_2007_EDA_Freq |Summarizing Data with Frequency Tables ]]== + There are two important ways to describe a data set (sample from a population) - '''Graphs''' or '''Tables'''. + ===[[EBook_Problems_EDA_Freq | Problems]]=== - :''(c) Median and interquartile range + ==[[AP_Statistics_Curriculum_2007_EDA_Pics | Pictures of Data]]== + There are many different ways to display and graphically visualize data. These graphical techniques facilitate the understanding of the dataset and enable the selection of an appropriate statistical methodology for the analysis of the data. + ===[[EBook_Problems_EDA_Pics | Problems]]=== - :''(d) Mean and standard deviation + ==[[AP_Statistics_Curriculum_2007_EDA_Center | Measures of Central Tendency]]== + There are three main features of populations (or sample data) that are always critical in understanding and interpreting their distributions - Center, Spread and Shape. The main measures of centrality are Mean, Median and Mode(s). + ===[[EBook_Problems_EDA_Center | Problems]]=== - '''2. According to a story in the Guardian newspaper in the U.K., the mean wage for a Premiership player in 2001-2002 in the U.K. was 600,000 pounds. Which of the following is most likely true?''' + ==[[AP_Statistics_Curriculum_2007_EDA_Var | Measures of Variation]]== + There are many measures of (population or sample) spread, e.g., the range, the variance, the standard deviation, mean absolute deviation, etc. These are used to assess the dispersion or variation in the population. + ===[[EBook_Problems_EDA_Var | Problems]]=== - '''Choose one answer.''' + ==[[AP_Statistics_Curriculum_2007_EDA_Shape | Measures of Shape]]== + The shape of a distribution can usually be determined by looking at a histogram of a (representative) sample from that population; Frequency Plots, Dot Plots or Stem and Leaf Displays may be helpful. + ===[[EBook_Problems_EDA_Shape | Problems]]=== - :''(a) About as many Premiership players make more than 600,000 pounds as make less.'' + ==[[AP_Statistics_Curriculum_2007_EDA_Statistics | Statistics]]== + Variables can be summarized using statistics - functions of data samples. + ===[[EBook_Problems_EDA_Statistics | Problems]]=== - :''(b) Most Premiership players make close to 600,000 pounds.'' + ==[[AP_Statistics_Curriculum_2007_EDA_Plots | Graphs and Exploratory Data Analysis]] == - + Graphical visualization and interrogation of data are critical components of any reliable method for statistical modeling, analysis and interpretation of data. - :''(c) Most Premiership players make less than 600,000 pounds.'' + ===[[EBook_Problems_EDA_Plots | Problems]]=== - + - :''(d) Most Premiership players make more than 600,000 pounds.'' + - + - '''3. If the observations larger than 40 are deleted in the following histogram, what will happen to the value of the mean?''' + - + - '''Choose one answer.''' + - + - :''(a) It will become larger.'' + - + - :''(b) It will become smaller.'' + - + - :''(c) Need more information to answer this question.'' + - + - :''(d) It will not change.'' + - + - ==Measures of Variation== + - '''1. The number of flaws of an electroplated automobile grill is known to have the following probability distribution: + - + - {| border="1" + - |- + - | X || 0 || 1 || 2 || 3 + - |- + - | P(X) || 0.8 || 0.1 || 0.05 || 0.05 + - |- + - |} + - + - '''What would be the standard deviation of the sample means if we took 100 samples, each sample with 200 grills, and computed their sample means? + - + - '''Choose One Answer. + - + - :''(a) 0.6275 + - + - :''(b) 0.0560 + - + - :''(c) None of the Above + - + - :''(d) 0.89269 + - + - '''2. Suppose that in a certain country, the average yearly income for 75% of the population is below average, what would you use as the measure of center and spread? + - + - '''Choose one answer. + - + - :''(a) Mean and interquartile range + - + - :''(b) Mean and standard deviation + - + - :''(c) Median and interquartile range + - + - :''(d) Mean and standard deviation + - + - + - + - ==Measures of Shape== + - ==Statistics== + - '''1. A recent Gallup Poll found that 23% of senior citizens exercise at least 3 times a week. The number 23% is:''' + - + - '''Choose one answer.''' + - + - :''(a) A sample'' + - + - :''(b) An estimate of the percentage of all senior citizens who exercise in the population'' + - + - :''(c) The percentage of all senior citizens who exercise in the population'' + - + - :''(d) A parameter'' + - + - '''2. A student said his SAT Math score was at the 90th percentile. This means that:''' + - + - '''Choose one answer.''' + - + - :''(a) The student got 90% of the questions wrong'' + - + - :''(b) 90% of the class had a lower score than the student'' + - + - :''(c) The student got 90% of the questions right'' + - + - :''(d) 90% of the class had a higher score than the student'' + - + - '''3. A random sample of 1000 US adults were interviewed and it was found that 2 of them had a rare disease known as diseaseA. Which of the following is true?''' + - + - '''Choose one answer.''' + - + - :''(a) The standard error of the sample proportion is 5%'' + - + - :''(b) 1000 is not a large enough sample to be able to construct a 99.7% confidence interval'' + - + - :''(c) There is no way we can figure out whether the sample is too large or too small to construct an interval'' + - + - :''(d) 2% of people in the sample have diseaseA'' + - + - '''4. The Caldwells want to buy a new car, and they have narrowed their choices to a Buick or an Oldsmobile. They first consulted an issue of Consumer Reports, which compared rates of repairs for various cars. Records of repairs done on 400 cars of each type showed somewhat fewer mechanical problems with the Buick than with the Oldsmobile. The Caldwells then talked to three friends, two Oldsmobile owners and one former Buick owner. Both Oldsmobile owners reported having a few mechanical problems, but nothing major. The Buick owner, however, exploded when asked how he liked his car: first, the fuel injection went out, which cost $250 to fix. Next, he started having trouble with the rear end and had to replace it. He finally decided to sell it after the transmission went. He says he'd never buy another Buick. The Caldwells want to buy the car that is less likely to require repairs. Given what they currently know, which car would you recommend that they buy?''' + - + - '''Choose one answer.''' + - + - :''(a) I would recommend that they buy the Buick despite their friend's bad experience. He is just one case, while the information reported in Consumer Reports is based on many cases. According to that data, the Buick is somewhat less likely to require repair.'' + - + - :''(b) I would recommend that they buy the Oldsmobile, primarily because of all the trouble their friend had with his Buick. Since they haven't heard similar horror stories about the Oldsmobile, they should go with it.'' + - + - :''(c) I would tell them that it does not matter which car they bought. Even though one of the models might be more likely than the other to require repairs, they could still, just by chance, get stuck with a particular car that would need a lot of repairs.'' + - + - + - ==Graphs and Exploratory Data Analysis== + =III. Probability= =III. Probability= - ==Fundamentals== + Probability is important in many studies and disciplines because measurements, observations and findings are often influenced by variation. In addition, probability theory provides the theoretical groundwork for statistical inference. - '''1. In a large midwestern university with 30 different departments, the university is considering eliminating standardized scores from their admission requirements. The university wants to find out whether the students agree with this plan. They decide to randomly select 100 students from each department, send them a survey, and follow up with a phone call if they do not return the survey within a week. What kind of sampling plan did they use?''' + - '''Choose one answer.''' + ==[[AP_Statistics_Curriculum_2007_Prob_Basics |Fundamentals]]== + Some fundamental concepts of probability theory include random events, sampling, types of probabilities, event manipulations and axioms of probability. + ===[[EBook_Problems_Prob_Basics | Problems]]=== - :''(a) Stratified random sampling'' + ==[[AP_Statistics_Curriculum_2007_Prob_Rules | Rules for Computing Probabilities]]== + There are many important rules for computing probabilities of composite events. These include conditional probability, statistical independence, multiplication and addition rules, the law of total probability and the Bayesian rule. + ===[[EBook_Problems_Prob_Rules| Problems]]=== - :''(b) Simple random sampling'' + ==[[AP_Statistics_Curriculum_2007_Prob_Simul |Probabilities Through Simulations]]== + Many experimental setting require probability computations of complex events. Such calculations may be carried out exactly, using theoretical models, or approximately, using estimation or simulations. + ===[[EBook_Problems_Prob_Simul | Problems]]=== - :''(c) Multi-stage sampling'' + ==[[AP_Statistics_Curriculum_2007_Prob_Count |Counting]]== - + There are many useful counting principles (including permutations and combinations) to compute the number of ways that certain arrangements of objects can be formed. This allows counting-based estimation of probabilities of complex events. - :''(d) Cluster sampling'' + ===[[EBook_Problems_Prob_Count | Problems]]=== - + - '''2. It is believed that 5% of elementary school children have some kind of ADD (Attention Deficit Disorder). Researchers are hoping to track 60 or more of these students for several years. They decide to test 1500 first graders for this problem. What is the probability that they will find enough subjects for their study?''' + - + - '''Choose one answer.''' + - + - :''(a) Cannot be calculated with the given data'' + - + - :''(b) More than 95%'' + - + - :''(c) Less than 5%'' + - + - :''(d) Between 70% to 80%'' + - + - '''3. A box contains 6 balls, where 2 are red, 2 are white, and 2 are blue. Four balls are picked at random, one at a time. Each time a ball is picked, the color is recorded, and the ball is put back in the box. If the first 3 balls are red, what color is the fourth ball most likely to be?''' + - + - '''Choose one answer.''' + - + - :''(a) Red'' + - + - :''(b) White'' + - + - :''(c) Blue'' + - + - :''(d) Blue and white are equally likely and more likely than red.'' + - + - :''(e) Red, blue, and white are all equally likely.'' + - + - '''4. A coin is tossed 400 times and 170 heads are observed. This coin is''' + - + - '''Choose one answer.''' + - + - :''(a) fair, because the probability of seeing that amount of heads or less is approximately 0.0013'' + - + - :''(b) neither fair or unfair. There is not enough information to determine that.'' + - + - :''(c) fair, because the probability of seeing that amount of heads or less is approximately 0.5'' + - + - :''(d) not fair, because the probability of seeing that amount of heads or less is close to 0.'' + - + - + - ==Rules for Computing Probabilities== + - '''1. A professor who teaches 500 students in an introductory psychology course reports that 250 of the students have taken at least one introductory statistics course, and the other 250 have not taken any statistics courses. 200 of the students were freshmen, and the other 300 students were not freshmen. Exactly 50 of the students were freshmen who had taken at least one introductory statistics course.''' + - + - '''If you select one of these psychology students at random, what is the probability that the student is not a freshman and has never taken a statistics course?''' + - + - :''(a) 30%'' + - + - :''(b) 40%'' + - + - :''(c) 50%'' + - + - :''(d) 60%'' + - + - :''(e) 20%'' + - + - '''2. A box contains 30 pens, where 5 are red, 14 are black, and 11 are blue. If you pick three pens from the box at random without replacement, what is the probability that these three pens will all be black?''' + - + - '''Choose one answer.''' + - + - :''(a) 14/30 + 14/30 + 14/30'' + - + - :''(b) 14/30 + 13/29 + 12/28'' + - + - :''(c) 14/30 x 13/29 x 12/28'' + - + - :''(d) 1 - (14/30 x 13/29 x 12/28)'' + - + - '''3. When three fair dice are simultaneously thrown, which of these three results is least likely to be obtained?''' + - + - '''Choose one answer.''' + - + - :''(a) All three results are equally unlikely.'' + - + - :''(b) Two fives and a 3 in any order.'' + - + - :''(c) A 5, a 3 and a 6 in any order.'' + - + - :''(d) Three 5's.'' + - + - '''4. Suppose that you take a three question "true/false" quiz for which you are completely unprepared. You have to guess the correct answer for each question. What is the probability of answering at least one question correctly?''' + - + - '''Choose one answer.''' + - + - :''(a) 4/8'' + - + - :''(b) 5/8'' + - + - :''(c) 7/8'' + - + - :''(d) 1/8'' + - + - :''(e) 3/8'' + - + - '''5. Records show that in an introductory chemistry course in a college, 20% of the students get an A, 30% get a B, 40% get a C, and 10% get a D. If you pick three students at random, what is the probability that all three will get an A?''' + - + - '''Choose one answer.''' + - + - :''(a) 0.8*0.8*0.8'' + - + - :''(b) 0.2*0.2*0.2'' + - + - :''(c) 200*0.2*0.2*0.2'' + - + - :''(d) 0.2*3'' + - + - '''6.A newly born child is equally likely to be a boy or a girl. What is the probability that in a family of three children there are less than 3 boys?''' + - + - :''(a) 0.125'' + - + - :''(b) 0.75'' + - + - :''(c) 0.875'' + - + - :''(d) 0.5'' + - + - '''7.A professor who teaches 300 students in an introductory psychology course reports that 135 of the students have taken exactly one introductory statistics course, 60 have taken two or more introductory statistics courses, and the other 105 have not taken any statistics courses. If you select one of these psychology students at random, what is the probability that the student has taken at least one statistics class?''' + - + - '''Choose one answer.''' + - + - :''(a) 0.20'' + - + - :''(b) 0.45'' + - + - :''(c) 0.65'' + - + - :''(d) 0.35'' + - + - '''8. Three fair coins are flipped. Find the probability that at least one comes up heads.''' + - + - '''Choose one answer.''' + - + - :''(a) 7/8'' + - + - :''(b) 4/8'' + - + - :''(c) 6/8'' + - + - :''(d) 3/8'' + - + - :''(e) 5/8'' + - + - '''9. Two fair coins are flipped. The probability that both are heads is:''' + - + - '''Choose one answer.''' + - + - :''(a) About 33%'' + - + - :''(b) Exactly 25%'' + - + - :''(c) Exactly 12.5%'' + - + - :''(d) Exactly 50%'' + - + - :''(e) Exactly 75%'' + - + - '''10. Two fair coins are flipped. The probability that the second coin is a head, given that the first was a head, is:''' + - + - '''Choose one answer.''' + - + - :''(a) Exactly 50%'' + - + - :''(b) Exactly 25%'' + - + - :''(c) Exactly 75%'' + - + - :''(d) Exactly 12.5%'' + - + - :''(e) About 33%'' + - + - '''10. Three dice are rolled. The probability that at least one is a 5 is:''' + - + - '''Choose one answer.''' + - + - :''(a) 1/6 + 1/6 + 1/6'' + - + - :''(b) 1/6 x 1/6 x 1/6'' + - + - :''(c) 1 - (5/6 x 5/6 x 5/6)'' + - + - :''(d) 5/6 + 4/6 + 3/6'' + - + - :''(e) 5/6 x 4/6 x 3/6'' + - ==Probabilities Through Simulations== + - '''1. A certain soft drink company was having a promotional contest in which they claimed that 1 in 3 bottles contained a free download from an mp3 server. A professor noticed that one machine in the Math Sciences building gave him 8 free downloads in 11 purchases. If the company's claim is true, what is the probability of getting 8 or more free downloads in 11 purchases? We can design a simulation to find out. + - + - The first step in a simulation is to identify the component to be repeated. Which of the choices below would be the best choice for the component to be repeated?''' + - + - '''Choose one answer.''' + - + - :''(a) The selection of 11 bottles of the soft drink.'' + - + - :''(b) The selection of a bottle of the soft drink.'' + - + - :''(c) The selection of 3 bottles of the soft drink.'' + - + - :''(d) the selection of 8 bottles of the soft drink.'' + - ==Counting== + =IV. Probability Distributions= =IV. Probability Distributions= - ==Random Variables== + There are two basic types of processes that we observe in nature - '''Discrete''' and '''Continuous'''. We begin by discussing several important discrete random processes, emphasizing the different distributions, expectations, variances and applications. In the [[AP_Statistics_Curriculum_2007#Chapter_V:_Normal_Probability_Distribution | next chapter]], we will discuss their continuous counterparts. The complete list of all [[About_pages_for_SOCR_Distributions |SOCR Distributions is available here]]. - ==Expectation(Mean) and Variance)== + - '''1. Ming’s Seafood Shop stocks live lobsters. Ming pays$6.00 for each lobster and sells each one for $12.00. The demand X for these lobsters in a given day has the following probability mass function.''' + - + - {| border="1" + - |- + - | X || 0 || 1 || 2 || 3 || 4 || 5 + - |- + - | P(x) || 0.05 || 0.15 || 0.30 || 0.20 || 0.20 || 0.1 + - |} + - + - '''What is the Expected Demand?''' + - + - '''Choose one answer.''' + - + - :''(a) 13.5'' + - + - :''(b) 3.1'' + - + - :''(c) 2.65'' + - + - :''(d) 5.2'' + - '''2. If sampling distributions of sample means are examined for samples of size 1, 5, 10, 16 and 50, you will notice that as sample size increases, the shape of the sampling distribution appears more like that of the:''' - '''Choose one answer.''' + ==[[AP_Statistics_Curriculum_2007_Distrib_RV | Random Variables]]== + To simplify the calculations of probabilities, we will define the concept of a '''random variable''' which will allow us to study uniformly various processes with the same mathematical and computational techniques. + ===[[EBook_Problems_Distrib_RV | Problems]]=== - :''(a) normal distribution'' + ==[[AP_Statistics_Curriculum_2007_Distrib_MeanVar | Expectation (Mean) and Variance]]== + The expectation and the variance for any discrete random variable or process are important measures of [[AP_Statistics_Curriculum_2007#Measures_of_Central_Tendency | Centrality]] and [[AP_Statistics_Curriculum_2007#Measures_of_Variation |Dispersion]]. This section also presents the definitions of some common population- or sample-based moments. + ===[[EBook_Problems_Distrib_MeanVar | Problems]]=== - :''(b) uniform distribution'' + ==[[AP_Statistics_Curriculum_2007_Distrib_Binomial |Bernoulli and Binomial Experiments]]== + The '''Bernoulli''' and '''Binomial''' processes provide the simplest models for discrete random experiments. + ===[[EBook_Problems_Distrib_Binomial | Problems]]=== - :''(c) population distribution'' + ==[[AP_Statistics_Curriculum_2007_Distrib_Multinomial |Multinomial Experiments]]== + '''Multinomial processes''' extend the [[AP_Statistics_Curriculum_2007_Distrib_Binomial |Binomial experiments]] for the situation of multiple possible outcomes. + ===[[EBook_Problems_Distrib_Multinomial | Problems]]=== + ==[[AP_Statistics_Curriculum_2007_Distrib_Dists |Geometric, Hypergeometric and Negative Binomial]]== + The '''Geometric, Hypergeometric and Negative Binomial distributions''' provide computational models for calculating probabilities for a large number of experiment and random variables. This section presents the theoretical foundations and the applications of each of these discrete distributions. + ===[[EBook_Problems_Distrib_Dists | Problems]]=== - :''(d) binomial distribution'' + ==[[AP_Statistics_Curriculum_2007_Distrib_Poisson |Poisson Distribution]]== - ==Bernoulli and Binomial Experiments== + The '''Poisson distribution''' models many different discrete processes where the probability of the observed phenomenon is constant in time or space. Poisson distribution may be used as an approximation to the Binomial distribution. - ==Multinomial Experiments== + ===[[EBook_Problems_Distrib_Poisson | Problems]]=== - ==Geometric, Hypergeometric, and Negative Binomial== + - ==Poisson Distribution== + =V. Normal Probability Distribution= =V. Normal Probability Distribution= - ==The Standard Normal Distribution== + The Normal Distribution is perhaps the most important model for studying quantitative phenomena in the natural and behavioral sciences - this is due to the [[AP_Statistics_Curriculum_2007_Limits_CLT | Central Limit Theorem]]. Many numerical measurements (e.g., weight, time, etc.) can be well approximated by the normal distribution. - '''1. Weight is a measure that tends to be normally distributed. Suppose the mean weight of all women at a large university is 135 pounds, with a standard deviation of 12 pounds. If you were to randomly sample 9 women at the university, there would be a 68% chance that the sample mean weight would be between:''' + - '''Choose one answer.''' + ==[[AP_Statistics_Curriculum_2007_Normal_Std |The Standard Normal Distribution]]== + The Standard Normal Distribution is the simplest version (zero-mean, unit-standard-deviation) of the (General) Normal Distribution. Yet, it is perhaps the most frequently used version because many tables and computational resources are explicitly available for calculating probabilities. + ===[[EBook_Problems_Normal_Std | Problems]]=== - :''(a) 131 and 139 pounds.'' + ==[[AP_Statistics_Curriculum_2007_Normal_Prob |Nonstandard Normal Distribution: Finding Probabilities]]== + In practice, the mechanisms underlying natural phenomena may be unknown, yet the use of the normal model can be theoretically justified in many situations to compute critical and probability values for various processes. + ===[[EBook_Problems_Normal_Prob | Problems]]=== - :''(b) 133 and 137 pounds.'' + ==[[AP_Statistics_Curriculum_2007_Normal_Critical |Nonstandard Normal Distribution: Finding Scores (Critical Values)]]== + In addition to being able to compute probability (p) values, we often need to estimate the critical values of the Normal Distribution for a given p-value. + ===[[EBook_Problems_Normal_Critical | Problems]]=== - :''(c) 119 and 151 pounds'' + ==[[AP_Statistics_Curriculum_2007_MultivariateNormal |Multivariate Normal Distribution]]== - + The multivariate normal distribution (also known as multivariate Gaussian distribution) is a generalization of the [[AP_Statistics_Curriculum_2007_Normal_Prob|univariate (one-dimensional) normal distribution]] to higher dimensions (2D, 3D, etc.) The multivariate normal distribution is useful in studies of correlated real-valued random variables. - :''(d) 125 and 145 pounds.'' + ===[[EBook_Problems_MultivariateNormal | Problems]]=== - + - :''(e) 123 and 147 pounds.'' + - + - '''2. The amount of money college students spend each semester on textbooks is normally distributed with a mean of$195 and a standard deviation of $20. Suppose you take a random sample of 100 college students from this population. There is a 68% chance that the sample mean amount spent on textbooks is between:''' + - + - '''Choose one answer.''' + - + - :''(a)$193 and $197.'' + - + - :''(b)$155 and $235.'' + - + - :''(c)$191 and $199.'' + - + - :''(d)$175 and $215.'' + - + - '''3. A researcher converts 100 lung capacity measurements to z-scores. The lung capacity measurements do not follow a normal distribution. What can we say about the standard deviation of the 100 z-scores?''' + - + - '''Choose one answer.''' + - + - :''(a) It depends on the standard deviation of the raw scores'' + - + - :''(b) It equals 1'' + - + - :''(c) It equals 100'' + - + - :''(d) It must always be less than the standard deviation of the raw scores'' + - + - :''(e) It depends on the shape of the raw score distribution'' + - + - '''4. The weights of packets of cookies produced by a certain manufacturer have a normal distribution with a mean of 202 grams and a standard deviation of 3 grams. What is the weight that should be stamped on the packet so that only 0.99% of packets are underweight?''' + - + - '''Choose one answer.''' + - + - :''(a) 200'' + - + - :''(b) 195'' + - + - :''(c) 190'' + - + - :''(d) 205'' + - + - '''5. GSP Inc. is trying two different marketing techniques for its toothpaste. In 20 test cities, it is using family branding. This sells toothpaste with a mean of 2,250 units per week and a standard deviation of 250 units per week. In 20 other test cities, GSP is using individual branding. This sells toothpaste with a mean of 2,250 units per week and a standard deviation of 500 units per week. GSP wants to select the marketing technique that sells at least 2,350 units per week more often. If the number of units sold per week follows a normal distribution, which marketing technique should GSP choose?''' + - + - '''Choose one answer.''' + - + - :''(a) Individual Branding'' + - + - :''(b) Can't be answered with the information given'' + - + - :''(c) Family Branding'' + - + - :''(d) They each get the same result'' + - + - '''6. Among first year students at a certain university, scores on the verbal SAT follow the normal curve. The average is around 500 and the SD is about 100. Tatiana took the SAT, and placed at the 85% percentile. What was her verbal SAT score?''' + - + - '''Choose one answer.''' + - + - :''(a) 604'' + - + - :''(b) 560'' + - + - :''(c) 90'' + - + - :''(d) 403'' + - + - ==Nonstandard Normal Distribution: Finding Probabilities== + - + - ==Nonstandard Normal Distribution: Finding Scores(Critical Values)== + =VI. Relations Between Distributions= =VI. Relations Between Distributions= - ==The Central Limit Theorem== + In this chapter, we will explore the relations between different distributions. This knowledge will help us to compute difficult probabilities using reasonable approximations and identify appropriate probability models, graphical and statistical analysis tools for data interpretation. - '''1. Which of the following would make the sampling distribution of the sample mean narrower? Check all answers that apply.''' + The complete list of all [[About_pages_for_SOCR_Distributions |SOCR Distributions is available here]] and the [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome applet] provides an interactive graphical interface for exploring the relations between different distributions. - '''Choose at least one answer.''' + ==[[AP_Statistics_Curriculum_2007_Limits_CLT |The Central Limit Theorem]]== + The exploration of the relation between different distributions begins with the study of the '''sampling distribution of the sample average'''. This will demonstrate the universally important role of normal distribution. + ===[[EBook_Problems_Limits_CLT | Problems]]=== - :''(a) A smaller population standard deviation'' + ==[[AP_Statistics_Curriculum_2007_Limits_LLN |Law of Large Numbers]]== + Suppose the relative frequency of occurrence of one event whose probability to be observed at each experiment is ''p''. If we repeat the same experiment over and over, the ratio of the observed frequency of that event to the total number of repetitions converges towards ''p'' as the number of experiments increases. Why is that and why is this important? + ===[[EBook_Problems_Limits_LLN | Problems]]=== - :''(b) A smaller sample size'' + ==[[AP_Statistics_Curriculum_2007_Limits_Norm2Bin |Normal Distribution as Approximation to Binomial Distribution]]== + Normal Distribution provides a valuable approximation to Binomial when the sample sizes are large and the probability of successes and failures are not close to zero. + ===[[EBook_Problems_Limits_Norm2Bin | Problems]]=== - :''(c) A larger standard error'' + ==[[AP_Statistics_Curriculum_2007_Limits_Poisson2Bin |Poisson Approximation to Binomial Distribution]]== + Poisson provides an approximation to Binomial Distribution when the sample sizes are large and the probability of successes or failures is close to zero. + ===[[EBook_Problems_Limits_Poisson2Bin | Problems]]=== - :''(d) A larger sample size'' + ==[[AP_Statistics_Curriculum_2007_Limits_Bin2HyperG |Binomial Approximation to Hypergeometric]]== + Binomial Distribution is much simpler to compute, compared to Hypergeometric, and can be used as an approximation when the population sizes are large (relative to the sample size) and the probability of successes is not close to zero. + ===[[EBook_Problems_Limits_Bin2HyperG | Problems]]=== - :''(e) A larger population standard deviation'' + ==[[AP_Statistics_Curriculum_2007_Limits_Norm2Poisson |Normal Approximation to Poisson]]== - ==Law of Large Numbers== + The Poisson can be approximated fairly well by Normal Distribution when λ is large. - ==Normal Distribution as Approximation to Binomial Distribution== + ===[[EBook_Problems_Limits_Norm2Poisson | Problems]]=== - '''1. Under what condition will the approximation to the binomial distribution using the normal curve be most accurate?''' + - + - '''Choose one answer.''' + - + - :''(a) np>10 and n(1-p)>10'' + - + - :''(b) Bernoulli trials for each member of the sample'' + - + - :''(c) Dependence of the members of the sample.'' + - + - :''(d) np>10 and n(1-p)<10'' + - ==Poisson Approximation to Binomial Distribution== + - ==Binomial Approximation to Hypergeometric== + - ==Normal Approximation to Poisson== + =VII. Point and Interval Estimates= =VII. Point and Interval Estimates= - ==Method of Moments and Maximum Likelihood Estimation== + Estimation of population parameters is critical in many applications. Estimation is most frequently carried in terms of point-estimates or interval (range) estimates for population parameters that are of interest. - ==Estimating a Population Mean: Large Samples== + - + - ==Estimating a Population Mean: Small Samples== + - ==Student's T Distribution== + - ==Estimating a Population Proportion== + - '''1. A 1996 poll of 1,200 African American adults found that 708 think that the American dream has become impossible to achieve. The New Yorker magazine editors want to estimate the proportion of all African American adults who feel this way. Which of the following is an approximate 90% confidence interval for the proportion of all African American adults who feel this way?''' + - + - '''Choose one answer.''' + - + - :''(a) (.56, .62)'' + - + - :''(b) (.57, .61)'' + - + - :''(c) Can't be calculated because the population size is too small.'' + - + - :''(d) Can't be calculated because the sample size is too small.'' + - + - '''2. True or False: In a well-designed sample survey like the Current Population Survey, the observed sample percentage (e.g, percentage unemployed) is equal to the population percentage. Thus, it is appropriate to just report the sample percentage, without any measure of accuracy (i.e. without the margin of error).''' + - + - '''Choose one answer.''' + - + - :''(a) True'' + - :''(b) False'' + ==[[AP_Statistics_Curriculum_2007_Estim_MOM_MLE |Method of Moments and Maximum Likelihood Estimation]]== + There are many ways to obtain point (value) estimates of various population parameters of interest, using observed data from the specific process we study. The '''method of moments''' and the '''maximum likelihood estimation''' are among the most popular ones frequently used in practice. + ===[[EBook_Problems_Estim_MOM_MLE | Problems]]=== - '''3. The BBC news does a story and at one point the reporter says: A polling agency reports that the percentage of the American public who agree we should spend more money on the mental health of the war veterans is 42% +/- 3%''' + ==[[AP_Statistics_Curriculum_2007_Estim_L_Mean |Estimating a Population Mean: Large Samples]]== + This section discusses how to find point and interval estimates when the sample-sizes are large. + ===[[EBook_Problems_Estim_L_Mean | Problems]]=== - '''Choose one answer.''' + ==[[AP_Statistics_Curriculum_2007_Estim_S_Mean |Estimating a Population Mean: Small Samples]]== + Next, we discuss point and interval estimates when the sample-sizes are small. Naturally, the point estimates are less precise and the interval estimates produce wider intervals, compared to the case of large-samples. + ===[[EBook_Problems_Estim_S_Mean | Problems]]=== - :''(a) The probability that the American public agree that we should spend more money on the mental health of the war veterans is between 39% to 42%.'' + ==[[AP_Statistics_Curriculum_2007_StudentsT |Student's T distribution]]== + The '''Student's T-Distribution''' arises in the problem of estimating the mean of a normally distributed population when the sample size is small and the population variance is unknown. + ===[[EBook_Problems_StudentsT | Problems]]=== - :''(b) The percentage of the American public who agree that we should spend more money on the mental health of the war veterans is between 39% to 45%.'' + ==[[AP_Statistics_Curriculum_2007_Estim_Proportion |Estimating a Population Proportion]]== + '''Normal Distribution''' is appropriate model for proportions, when the sample size is large enough. In this section, we demonstrate how to obtain point and interval estimates for population proportion. - :''(c) We are 95% confident that the percentage of the American public who agree that we should spend more money on the mental health of the war veterans is between 39% to 45%.'' + ===[[EBook_Problems_Estim_Proportion | Problems]]=== - :''(d) The percentage of the American public who agree that we should spend more money on the mental health of the war veterans is 42%.'' + ==[[AP_Statistics_Curriculum_2007_Estim_Var |Estimating a Population Variance]]== - ==Estimating a Population Variance== + In many processes and experiments, controlling the amount of variance is of critical importance. Thus the ability to assess variation, using point and interval estimates, facilitates our ability to make inference, revise manufacturing protocols, improve clinical trials, etc. + ===[[EBook_Problems_Estim_Var | Problems]]=== =VIII. Hypothesis Testing= =VIII. Hypothesis Testing= - ==Fundamentals of Hypothesis Testing== + '''Hypothesis Testing''' is a statistical technique for decision making regarding populations or processes based on experimental data. It quantitatively answers the possibility that chance alone might be responsible for the observed discrepancy between a theoretical model and the empirical observations. - '''1. Suppose you were hired to conduct a study to find out which of two brands of soda college students think taste better. In your study, students are given a blind taste test. They rate one brand and then rated the other, in random order. The ratings are given on a scale of 1 (awful) to 5 (delicious). Which type of test would be the best to compare these ratings?''' + - :''(a) One-Sample t'' + ==[[AP_Statistics_Curriculum_2007_Hypothesis_Basics |Fundamentals of Hypothesis Testing]]== + In this section, we define the core terminology necessary to discuss Hypothesis Testing (Null and Alternative Hypotheses, Type I and II errors, Sensitivity, Specificity, Statistical Power, etc.) + ===[[EBook_Problems_Hypothesis_Basics | Problems]]=== - :''(b) Chi-Square'' + ==[[AP_Statistics_Curriculum_2007_Hypothesis_L_Mean |Testing a Claim about a Mean: Large Samples]]== + As we already saw how to construct point and interval estimates for the population mean in the large sample case, we now show how to do hypothesis testing in the same situation. + ===[[EBook_Problems_Hypothesis_L_Mean | Problems]]=== - :''(c) Paired Difference t'' + ==[[AP_Statistics_Curriculum_2007_Hypothesis_S_Mean |Testing a Claim about a Mean: Small Samples]]== + We continue with the discussion on inference for the population mean for small samples. + ===[[EBook_Problems_Hypothesis_S_Mean | Problems]]=== - :''(d) Two-Sample t'' + ==[[AP_Statistics_Curriculum_2007_Hypothesis_Proportion |Testing a Claim about a Proportion]]== + When the sample size is large, the sampling distribution of the sample proportion $\hat{p}$ is approximately Normal, by [[AP_Statistics_Curriculum_2007_Limits_CLT | CLT]]. This helps us formulate hypothesis testing protocols and compute the appropriate statistics and p-values to assess significance. + ===[[EBook_Problems_Hypothesis_Proportion | Problems]]=== - '''2. USA Today's AD Track examined the effectiveness of the new ads involving the Pets.com Sock Puppet (which is now extinct). In particular, they conducted a nationwide poll of 428 adults who had seen the Pets.com ads and asked for their opinions. They found that 36% of the respondents said they liked the ads. Suppose you increased the sample size for this poll to 1000, but you had the same sample percentage who like the ads (36%). How would this change the p-value of the hypothesis test you want to conduct? + ==[[AP_Statistics_Curriculum_2007_Hypothesis_Var |Testing a Claim about a Standard Deviation or Variance]]== + The significance testing for the variation or the standard deviation of a process, a natural phenomenon or an experiment is of paramount importance in many fields. This chapter provides the details for formulating testable hypotheses, computation, and inference on assessing variation. + ===[[EBook_Problems_Hypothesis_Var | Problems]]=== - '''Choose One Answer. + =IX. Inferences from Two Samples= + In this chapter, we continue our pursuit and study of significance testing in the case of having two populations. This expands the possible applications of one-sample hypothesis testing we saw in the [[EBook#Chapter_VIII:_Hypothesis_Testing | previous chapter]]. - :''(a) No way to tell + ==[[AP_Statistics_Curriculum_2007_Infer_2Means_Dep |Inferences About Two Means: Dependent Samples]]== + We need to clearly identify whether samples we compare are '''Dependent''' or '''Independent''' in all study designs. In this section, we discuss one specific dependent-samples case - '''Paired Samples'''. + ===[[EBook_Problems_Infer_2Means_Dep | Problems]]=== - :''(b) The new p-value would be the same as before + ==[[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |Inferences About Two Means: Independent Samples]]== + '''Independent''' Samples designs refer to experiments or observations where all measurements are individually independent from each other within their groups and the groups are independent. In this section, we discuss inference based on independent samples. + ===[[EBook_Problems_Infer_2Means_Indep | Problems]]=== - :''(c) The new p-value would be smaller than before + ==[[AP_Statistics_Curriculum_2007_Infer_BiVar |Comparing Two Variances]]== + In this section, we compare '''variances (or standard deviations)''' of two populations using randomly sampled data. + ===[[EBook_Problems_Infer_BiVar | Problems]]=== - :''(d) The new p-value would be larger than before + ==[[AP_Statistics_Curriculum_2007_Infer_2Proportions |Inferences about Two Proportions]]== - + This section presents the '''significance testing''' and '''inference on equality''' of proportions from two independent populations. - '''3. A marketing director for a radio station collects a random sample of three hundred 18 to 25 year-olds and two hundred and fifty 25 to 40 year-olds. She records the percent of each group that had purchased music online in the last 30 days. She performs a hypothesis test, and the p-value of her test turns out to be 0.15. From this she should conclude:''' + ===[[EBook_Problems_Infer_2Proportions | Problems]]=== - + - '''Choose one answer.''' + - + - :''(a) that about 15% more people purchased on-line music in the younger group than in the older group.'' + - + - :''(b) there is insufficient evidence to conclude that there is a difference in the proportion of on-line music purchases in the younger and older group.'' + - + - :''(c) the proportion of on-line music purchasers is the same in the under-25 year-old group as in the older group.'' + - + - :''(d) the probability of getting the same results again is 0.15.'' + - + - '''4. If we want to estimate the mean difference in scores on a pre-test and post-test for a sample of students, how should we proceed?''' + - + - '''Choose one answer.''' + - + - :''(a) We should construct a confidence interval or conduct a hypothesis test'' + - + - :''(b) We should collect one sample, two samples, or conduct a paired data procedure'' + - + - :''(c) We should calculate a z or a t statistic'' + - + - '''5. The paint used to make lines on roads must reflect enough light to be clearly visible at night. Let mu denote the true average reflectometer reading for a new type of paint under consideration. A test of the null hypothesis that mu = 20 versus the alternative hypothesis that mu > 20 will be based on a random sample of size n from a normal population distribution. In which of the following scenarios is there significant evidence that mu is larger than 20?''' + - + - '''(i) n=15, t=3.2, alpha=0.05''' + - + - '''(ii) n=9, t=1.8, alpha=0.01''' + - + - '''(iii) n=24, t=-0.2, alpha=0.01''' + - + - '''Choose one answer.''' + - + - :''(a) (ii) and (iii)'' + - + - :''(b) (i)'' + - + - :''(c) (iii)'' + - + - :''(d) (ii)'' + - + - '''6. The average length of time required to complete a certain aptitude test is claimed to be 80 minutes. A random sample of 25 students yielded an average of 86.5 minutes and a standard deviation of 15.4 minutes. If we assume normality of the population distribution, is there evidence to reject the claim? (Select all that applies).''' + - + - '''Choose at least one answer.''' + - + - :''(a) No, because the probability that the null is true is > 0.05'' + - + - :''(b) Yes, because the observed 86.5 did not happen by chance'' + - + - :''(c) Yes, because the t-test statistic is 2.11'' + - + - :''(d) Yes, because the observed 86.5 happened by chance'' + - + - '''7. We observe the math self-esteem scores from a random sample of 25 female students. How should we determine the probable values of the population mean score for this group?''' + - + - '''Choose one answer.''' + - + - :''(a) Test the difference in means between two paired or dependent samples.'' + - + - :''(b) Test that a correlation coefficient is not equal to 0 (correlation analysis).'' + - + - :''(c) Test the difference between two means (independent samples).'' + - + - :''(d) Test for a difference in more than two means (one way ANOVA).'' + - + - :''(e) Construct a confidence interval.'' + - + - :''(f) Test one mean against a hypothesized constant.'' + - + - :''(g) Use a chi-squared test of association.'' + - + - '''8. Food inspectors inspect samples of food products to see if they are safe. This can be thought of as a hypothesis test where H0: the food is safe, and H1: the food is not A. If you are a consumer, which type of error would be the worst one for the inspector to make, the type I or type II error?''' + - + - '''Choose one answer.''' + - + - :''(a) Type I'' + - + - :''(b) Type II'' + - + - '''9. A college admissions officer is concerned that their admission criteria might not treat men and women with equal weight. To test this, the college took a random sample of male and female high school seniors from a very large local school district and determined the percent of males and females who would be eligible for admission at the college. Which of the following is a suitable null hypothesis for this test?''' + - + - '''Choose one answer.''' + - + - :''(a) p = 0.5'' + - + - :''(b) The proportion of all eligible men in the district will not equal the proportion of all eligible women in the district.'' + - + - :''(c) The proportion of all eligible men in the school district should be equal to the proportion of all eligible women in the school district.'' + - + - :''(d) The proportion of eligible men sampled should equal the propotion of eligible women sampled.'' + - ==Testing a Claim About a Mean: Large Samples== + - '''1. Hong is a pharmacist studying the effect of an anti-depressant drug. She organizes a simple random sample of 100 patients, and then collect their anxiety test scores before and after administering the anti-depressant drug. Hong wants to estimate the mean difference between the pre-drug and post-drug test scores. How should she proceed?''' + - + - '''Choose one answer.''' + - + - :''(a) She should compute a confidence interval or conduct a hypothesis test'' + - + - :''(b) She should calculate the z or the t statistics'' + - + - :''(c) She should compute the correlation between the two samples'' + - + - :''(d) Not enough information to tell'' + - ==Testing a Claim About a Mean: Small Samples== + - '''1. To test the claim that the average home in a certain town is within 5.5 miles of the nearest fire station, and insurance company measured the distances from 25 randomly selected homes to the nearest fire station and found x-bar = 5.8 miles and sd = 2.4 miles. Determine what the insurance company found out with a test of significance. Check all that apply.''' + - + - '''Choose at least one answer.''' + - + - :''(a) There is no evidence in the data to conclude that the distance is different from 5.5.'' + - + - :''(b) The average of 5.8 miles observed is by chance.'' + - + - :''(c) We cannot reject the null.'' + - + - :''(d) There is evidence in the data to conclude that the distance is 5.5.'' + - ==Testing a Claim About a Proportion== + - ==Testing a Claim About a Standard Deviation or Variance== + - + - =IX. Inferences from Two Samples= + - ==Inferences About Two Means: Dependent Samples== + - ==Inferences About Two Means: Independent Samples== + - ==Comparing Two Variances== + - ==Inferences About Two Proportions== + =X. Correlation and regression= =X. Correlation and regression= - ==Correlation== + Many scientific applications involve the analysis of relationships between two or more variables involved in a process of interest. We begin with the simplest of all situations where '''Bivariate Data''' (X and Y) are measured for a process and we are interested on determining the association, relation or an appropriate model for these observations (e.g., fitting a straight line to the pairs of (X,Y) data). - '''1. A positive correlation between two variables X and Y means that if X increases, this will cause the value of Y to increase.''' + - :''(a) This is always true.'' + ==[[AP_Statistics_Curriculum_2007_GLM_Corr |Correlation]]== + The '''Correlation''' between X and Y represents the first bivariate model of association which may be used to make predictions. + ===[[EBook_Problems_GLM_Corr | Problems]]=== - :''(b) This is sometimes true.'' + ==[[AP_Statistics_Curriculum_2007_GLM_Regress |Regression]]== + We are now ready to discuss the modeling of linear relations between two variables using '''Regression Analysis'''. This section demonstrates this methodology for the SOCR California Earthquake dataset. + ===[[EBook_Problems_GLM_Regress | Problems]]=== - :''(c) This is never true.'' + ==[[AP_Statistics_Curriculum_2007_GLM_Predict |Variation and Prediction Intervals]]== - {{hidden|Answer|''(c)''}} + In this section, we discuss point and interval estimates about the slope of linear models. + ===[[EBook_Problems_GLM_Predict | Problems]]=== - '''2. The correlation between high school algebra and geometry scores was found to be + 0.8. Which of the following statements is not true?''' + ==[[AP_Statistics_Curriculum_2007_GLM_MultLin |Multiple Regression]]== + Now, we are interested in determining linear regressions and multilinear models of the relationships between one dependent variable Y and many independent variables $X_i$. + ===[[EBook_Problems_GLM_MultLin | Problems]]=== - :''(a) Most of the students who have above average scores in algebra also have above average scores in geometry. '' + =XI. Analysis of Variance (ANOVA)= - :''(b) Most people who have above average scores in algebra will have below average scores in geometry '' + ==[[AP_Statistics_Curriculum_2007_ANOVA_1Way | One-Way ANOVA]]== + We now expand our inference methods to study and compare ''k'' '''independent''' samples. In this case, we will be decomposing the entire variation in the data into independent components. + ===[[EBook_Problems_ANOVA_1Way | Problems]]=== - :''(c) If we increase a student's score in algebra (ie. with extra tutoring in algebra), then the student's geometry scores will always increase accordingly.'' + ==[[AP_Statistics_Curriculum_2007_ANOVA_2Way | Two-Way ANOVA]]== + Now we focus on decomposing the variance of a dataset into (independent/orthogonal) components when we have two (grouping) factors. This procedure called '''Two-Way Analysis of Variance'''. + ===[[Ebook_Problems_ANOVA_2Way | Problems]]=== - :''(d) Most students who have below average scores in algebra also have below average scores in geometry. '' + =XII. Non-Parametric Inference= - {{hidden|Answer|''(c)''}} + To be valid, many statistical methods impose (parametric) requirements about the format, parameters and distributions of the data to be analyzed. For instance, the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep | Independent T-Test]] requires the distributions of the two samples to be Normal, whereas Non-Parametric (distribution-free) statistical methods are often useful in practice, and are [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | less-powerful]]. + ==[[AP_Statistics_Curriculum_2007_NonParam_2MedianPair | Differences of Medians (Centers) of Two Paired Samples]]== + The '''Sign Test''' and the '''Wilcoxon Signed Rank Test''' are the simplest non-parametric tests which are also alternatives to the [[AP_Statistics_Curriculum_2007_Infer_2Means_Dep | One-Sample and Paired T-Test]]. These tests are applicable for paired designs where the data is not required to be normally distributed. + ===[[EBook_Problems_NonParam_2MedianPair | Problems]]=== - '''3. Researchers discover that the correlation between miles ran per week and cardiovascular endurance is +0.75. They also discover that the correlation between hours spent watching television per week and cardiovascular endurance is -0.75. What is the conclusion that best characterizes the result of this study?''' + ==[[AP_Statistics_Curriculum_2007_NonParam_2MedianIndep | Differences of Medians (Centers) of Two Independent Samples]]== + The '''Wilcoxon-Mann-Whitney (WMW) Test''' (also known as Mann-Whitney U Test, Mann-Whitney-Wilcoxon Test, or Wilcoxon rank-sum Test) is a ''non-parametric'' test for assessing whether two samples come from the same distribution. + ===[[EBook_Problems_NonParam_2MedianIndp | Problems]]=== - '''Choose one answer.''' + ==[[AP_Statistics_Curriculum_2007_NonParam_2PropIndep | Differences of Proportions of Two Samples]]== + Depending upon whether the samples are dependent or independent, we use different statistical tests. + ===[[EBook_Problems_NonParam_2PropIndep | Problems]]=== - :''(a) Most people who spend a lot of hours watching television have low cardiovascular endurance.'' + ==[[AP_Statistics_Curriculum_2007_NonParam_ANOVA | Differences of Means of Several Independent Samples]]== + We now extend the [[EBook#Chapter_XI:_Analysis_of_Variance_.28ANOVA.29 | multi-sample inference which we discussed in the ANOVA section]], to the situation where the [[AP_Statistics_Curriculum_2007_ANOVA_1Way#ANOVA_Conditions| ANOVA assumptions]] are invalid. + ===[[EBook_Problems_NonParam_ANOVA | Problems]]=== - :''(b) Most people who have good cardiovascular endurance spend a lot of time running and little time watching television.'' + ==[[AP_Statistics_Curriculum_2007_NonParam_VarIndep | Differences of Variances of Independent Samples (Variance Homogeneity)]]== + There are several tests for variance equality in ''k'' samples. These tests are commonly known as tests for '''Homogeneity of Variances'''. + ===[[EBook_Problems_NonParam_VarIndep | Problems]]=== - :''(c) Based on the correlation, if you increase your running hours per week, your cardiovascular endurance will decrease.'' + =XIII. Multinomial Experiments and Contingency Tables= - :''(d) Based on the correlation, if you increases your television watching time, your cardiovascular endurance will decrease.'' + ==[[AP_Statistics_Curriculum_2007_Contingency_Fit |Multinomial Experiments: Goodness-of-Fit]]== + The '''Chi-Square Test''' is used to test if a data sample comes from a population with specific characteristics. + ===[[EBook_Problems_Contingency_Fit | Problems]]=== - :''(e) Most people with a lot of miles ran per week have high cardiovascular endurance.'' + ==[[AP_Statistics_Curriculum_2007_Contingency_Indep |Contingency Tables: Independence and Homogeneity]]== - + The '''Chi-Square Test''' may also be used to test for independence (or association) between two variables. - '''4. The correlation between working out and body fat was found to be exactly -1.0. Which of the following would not be true about the corresponding scatterplot?''' + ===[[EBook_Problems_Contingency_Indep | Problems]]=== - + - '''Choose one answer.''' + - + - :''(a) The slope of the best line of fit should be -1.0.'' + - + - :''(b) All the points would lie along a perfect straight line, with no deviation at all.'' + - + - :''(c) The best fitting line would have a downhill (negative) slope.'' + - + - :''(d) 100% of the variance in body fat can be predicted from workout.'' + - + - '''5. Suppose that the correlation between working out and body fat was found to be exactly -1.0. Which of the following would NOT be true, about the corresponding scatterplot?''' + - + - '''Choose one answer.''' + - + - :''(a) All points would lie along a straight line, with no deviation at all.'' + - + - :''(b) 100% of the variance in body fat can be predicted from the workout.'' + - + - :''(c) The slope of the linear model is -1.0.'' + - + - :''(d) The best fitting line would have a negative slope.'' + - + - '''6. A recent article in an educational research journal reports a correlation of +0.8 between math achievement and overall math aptitude. It also reports a correlation of -0.8 between math achievement and a math anxiety test. Which of the following interpretations is the most correct?''' + - + - '''Choose one answer''' + - + - :''(a) You cannot compare a positive and a negative correlation.'' + - + - :''(b) The correlation of +0.8 indicates a stronger relationship than the correlation of -0.8.'' + - + - :''(c) The correlation of +0.8 is just as strong as the correlation of -0.8.'' + - + - :''(d) It is impossible to tell which correlation is stronger.'' + - {{hidden|Answer|''(c)''}} + - + - '''7. Psychologists have shown that there is a relationship between stress levels and productivity. As stress levels increase, productivity also increases up to a certain point, and after that productivity decreases as stress levels increase. Suppose you were given this data for a random sample of 200 adults. If you calculated the Pearson coefficient of correlation, what would you expect to find?''' + - + - '''Choose one answer.''' + - + - :''(a) I would expect r to be between -0.50 to -0.70.'' + - + - :''(b) I would expect r to be -1.'' + - + - :''(c) I would expect r to be between 0.50 and 0.70.'' + - + - :''(d) I would expect r to be +1.'' + - + - :''(e) I would expect r to be zero.'' + - ==Regression== + - '''1. Use the information from the Heights of Fathers and Sons to write the linear model that best predicts the height of the son from the height of the father.''' + - + - '''Choose one answer.''' + - + - :''(a) Son's height = 35 + 0.5*Father's height''' + - + - :''(b) Son's height = 1.00 + 1.00* Father's height'' + - + - :''(c) The model cannot be determined without the actual data'' + - + - :''(d) Son's height = 0.5 + 35*Father's height'' + - + - '''2. A congressional report investigates the relationship between income of parents and educational attainment of their daughters. Data are from a sample of families with daughters age 18-24. Average parental income is$29,300, average educational attainment of the daughters is 13.1 years of schooling completed, and the correlation is 0.37. + - + - The regression line for predicting daughter’s education from parental income is reported as: Predicted education = 0.000617*(income) + 8.1 + - + - Is the following statement true or false? "The above line is the regression line to predict education from income."''' + - + - :''(a)True.'' + - + - :''(b)False.'' + - + - '''3. Heights of Fathers and Sons''' + - + - '''In the early 1900's when Francis Galton and Karl Pearson measured 1078 pairs of fathers and their grown-up sons, they calculated that the mean height for fathers was about 68 inches with deviation of 3 inches. For their sons, the mean height was 69 inches with deviation of 3 inches. (The actual numbers are slightly smaller, but we will work with these values to keep the calculations simple.) The correlation coefficient was 0.50. Use the information to calculate the slope of the linear model that predicts the height of the son from the height of the father.''' + - + - '''Choose one answer.''' + - + - :''(a) 0.50'' + - + - :''(b) The slope cannot be determined without the actual data'' + - + - :''(c) 35.00'' + - + - :''(d) 3/3 = 1.00'' + - + - '''4. The National Highway Safety Administration is interested in the effect of seat belt use on saving lives. One study reported statistics on children under the age of 5 who were involved in motor vehicles accidents in which at least one fatality occurred. 7,060 such accidents between 1985 and 1989 were studied. Of those who survived, 1129 weren't wearing a seat belt, 432 were wearing an adult seat belt and 733 had a children's carseat belt. Of those with fatalities, 509 had no belt, 73 had an adult seat belt, and 139 had a children's carseat belt.''' + - + - '''Are seat belt status and the outcome of the accidents independent?''' + - + - '''Choose one answer.''' + - + - :''(a) Yes'' + - + - :''(b) No'' + - + - :''(c) Can't tell with the information provided'' + - + - ==Variation and Prediction Intervals== + - '''1. Two researchers are going to take a sample of data from the same population of physics students. Researcher A will select a random sample of students from among all students taking physics. Researcher B's sample will consist only of the students in her class. Both researchers will construct a 95% confidence interval for the mean score on the physics final exam using their own sample data. Which researcher's method has a 95% chance of capturing the true mean of the population of all students taking physics?''' + - + - '''Choose one answer.''' + - + - :''(a) Research B'' + - + - :''(b) Researcher A'' + - + - :''(c) Both methods have a 95% chance of capturing the true mean'' + - + - :''(d) Neither'' + - + - '''2. A random sample of 150 UCLA students found that 35% of the respondants wanted a elevator to replace Bruin Walk. A 95% confidence interval for the percentage of all UCLA students who feel this way is approximately:''' + - + - '''Choose one answer.''' + - + - :''(a) (24%, 46%)'' + - + - :''(b) (32%, 38%)'' + - + - :''(c) The sample size is too small to compute a confidence interval.'' + - + - :''(d) (27%, 43%)'' + - + - '''3. According to Terry Prachett, the short unit of time in the multiverse is the New York second, defined as the time interval between the light turning green and the cab behind you honking. A magazine took a poll of 100 New Yorkers and found that 90 people agree with that statement wholeheartedly. Which of the following is a 90% confidence interval for the proportion of people who agree with that statement?''' + - + - '''Choose one answer.''' + - + - :''(a) 0.9 +\- 0.50'' + - + - :''(b) 0.9 +\- .05'' + - + - :''(c) 0.9 +\- .03'' + - + - :''(d) 0.9 +\- .06'' + - + - '''4. A national poll found that 62% of all Americans agreed that more attention should be paid to mental health of war veterans. If a simple random sample of 326 people was used to make a 95% confidence interval of (0.57,0.67), what is the margin of error?''' + - + - '''Choose one answer.''' + - + - :''(a) 0.03'' + - + - :''(b) 0.05'' + - + - :''(c) 0.12'' + - + - :''(d) In order to calculate the margin of error, we need the p-value of the population.'' + - + - '''5. Hermione Granger is on a mission this year to complain about the astronomical cost of wizarding books to the Hogwart board of administrators. Given that the population mean for book cost is 10 and a standard deviation of 2 galleons, If Hermione were to take a simple random sample of 49 students and make a 68% confidence interval, what would be the range of values for the sample mean or Xbar?''' + - + - '''Choose one answer.''' + - + - :''(a) 8 and 12 galleons'' + - + - :''(b) 9.4 and 10.6 galleons'' + - + - :''(c) 6 and 14 Galleons'' + - + - :''(d) 9.7 and 10.3 galleons'' + - + - '''6. A 95% confidence interval indicates that:''' + - + - '''Choose one answer:''' + - + - :''(a) 95% of the intervals constructed using this process based on samples from this population will include the population mean'' + - + - :''(b) 95% of the time the interval will include the sample mean'' + - + - :''(c) 95% of the possible population means will be included by the interval'' + - + - :''(d) 95% of the possible sample means will be included by the interval'' + - + - '''7. Suppose we want to find out if a coin is not fair. To test this hypothesis we flip the coin 100 times, and in 63 out of 100 flips we get heads. We construct the confidence interval and find it to be (.53,.73). Interpret this confidence interval.''' + - + - '''Choose one answer.''' + - + - :''(a) 95 is the Z score that corresponds to our distribution of sample means'' + - + - :''(b) Confidence is something you learn at fraternity parties'' + - + - :''(c) 95% of the time the true proportion of flips that are heads is between .53 and .73'' + - + - :''(d) If we were to repeat this expirement over and over again, 95 times out of 100 our Confidence interval would cover the true proportion of flips that are heads'' + - + - '''8. A 95% confidence interval is calculated for a sample of weights of 100 randomly selected pigs, and is (42 pounds, 48 pounds). Will the sample mean weight fall within the confidence interval?''' + - + - '''Choose one answer.''' + - + - :''(a) Yes'' + - + - :''(b) We need more information to determine if this is true.'' + - + - :''(c) No'' + - + - + - '''9. The average number of fruit candies in a large bag is estimated. The 95% confidence interval is (40, 48). Based on this information, you know that the best estimate of the population mean is:''' + - + - '''Choose one answer.''' + - + - :''(a) 43'' + - + - :''(b) 40'' + - + - :''(c) 45'' + - + - :''(d) none of the above.'' + - + - :''(e) 44'' + - ==Multiple Regression== + - + - =XI. Analysis of Variance (ANOVA)= + - ==One-Way ANOVA== + - ==Two-Way ANOVA== + - + - =XII. Non-Parametric Inference= + - ==Differences of Medians (Centers) of Two Paired Samples== + - ==Differences of Medians (Centers) of Two Independent Samples== + - ==Differences of Proportions of Two Samples== + - ==Differences of Means of Several Independent Samples== + - ==Differences of Variances of Independent Samples (Variance Homogeneity)== + - + - =XIII. Multinomial Experiments and Contingency Tables= + - ==Multinomial Experiments: Goodness-of-Fit== + - ==Contingency Tables: Independence and Homogeneity== + ==References== ==References==

## Probability and Statistics EBook Practice Problems

The problems provided below may be useful for practicing the concepts, methods and analysis protocols, and for self-evaluation of learning of the materials presented in the EBook.

# I. Introduction to Statistics

## The Nature of Data and Variation

Although natural phenomena in real life are unpredictable, the designs of experiments are bound to generate data that varies because of intrinsic (internal to the system) or extrinsic (due to the ambient environment) effects. How many natural processes or phenomena in real life can we describe that have an exact mathematical closed-form description and are completely deterministic? How do we model the rest of the processes that are unpredictable and have random characteristics?

## Uses and Abuses of Statistics

Statistics is the science of variation, randomness and chance. As such, statistics is different from other sciences, where the processes being studied obey exact deterministic mathematical laws. Statistics provides quantitative inference represented as long-time probability values, confidence or prediction intervals, odds, chances, etc., which may ultimately be subjected to varying interpretations. The phrase Uses and Abuses of Statistics refers to the notion that in some cases statistical results may be used as evidence to seemingly opposite theses. However, most of the time, common principles of logic allow us to disambiguate the obtained statistical inference.

## Design of Experiments

Design of experiments is the blueprint for planning a study or experiment, performing the data collection protocol and controlling the study parameters for accuracy and consistency. Data, or information, is typically collected in regard to a specific process or phenomenon being studied to investigate the effects of some controlled variables (independent variables or predictors) on other observed measurements (responses or dependent variables). Both types of variables are associated with specific observational units (living beings, components, objects, materials, etc.)

## Statistics with Tools (Calculators and Computers)

All methods for data analysis, understanding or visualizing are based on models that often have compact analytical representations (e.g., formulas, symbolic equations, etc.) Models are used to study processes theoretically. Empirical validations of the utility of models are achieved by inputting data and executing tests of the models. This validation step may be done manually, by computing the model prediction or model inference from recorded measurements. This process may be possible by hand, but only for small numbers of observations (<10). In practice, we write (or use existent) algorithms and computer programs that automate these calculations for better efficiency, accuracy and consistency in applying models to larger datasets.

# II. Describing, Exploring, and Comparing Data

## Types of Data

There are two important concepts in any data analysis - Population and Sample. Each of these may generate data of two major types - Quantitative or Qualitative measurements.

## Summarizing Data with Frequency Tables

There are two important ways to describe a data set (sample from a population) - Graphs or Tables.

## Pictures of Data

There are many different ways to display and graphically visualize data. These graphical techniques facilitate the understanding of the dataset and enable the selection of an appropriate statistical methodology for the analysis of the data.

## Measures of Central Tendency

There are three main features of populations (or sample data) that are always critical in understanding and interpreting their distributions - Center, Spread and Shape. The main measures of centrality are Mean, Median and Mode(s).

## Measures of Variation

There are many measures of (population or sample) spread, e.g., the range, the variance, the standard deviation, mean absolute deviation, etc. These are used to assess the dispersion or variation in the population.

## Measures of Shape

The shape of a distribution can usually be determined by looking at a histogram of a (representative) sample from that population; Frequency Plots, Dot Plots or Stem and Leaf Displays may be helpful.

## Statistics

Variables can be summarized using statistics - functions of data samples.

## Graphs and Exploratory Data Analysis

Graphical visualization and interrogation of data are critical components of any reliable method for statistical modeling, analysis and interpretation of data.

# III. Probability

Probability is important in many studies and disciplines because measurements, observations and findings are often influenced by variation. In addition, probability theory provides the theoretical groundwork for statistical inference.

## Fundamentals

Some fundamental concepts of probability theory include random events, sampling, types of probabilities, event manipulations and axioms of probability.

## Rules for Computing Probabilities

There are many important rules for computing probabilities of composite events. These include conditional probability, statistical independence, multiplication and addition rules, the law of total probability and the Bayesian rule.

## Probabilities Through Simulations

Many experimental setting require probability computations of complex events. Such calculations may be carried out exactly, using theoretical models, or approximately, using estimation or simulations.

## Counting

There are many useful counting principles (including permutations and combinations) to compute the number of ways that certain arrangements of objects can be formed. This allows counting-based estimation of probabilities of complex events.

# IV. Probability Distributions

There are two basic types of processes that we observe in nature - Discrete and Continuous. We begin by discussing several important discrete random processes, emphasizing the different distributions, expectations, variances and applications. In the next chapter, we will discuss their continuous counterparts. The complete list of all SOCR Distributions is available here.

## Random Variables

To simplify the calculations of probabilities, we will define the concept of a random variable which will allow us to study uniformly various processes with the same mathematical and computational techniques.

## Expectation (Mean) and Variance

The expectation and the variance for any discrete random variable or process are important measures of Centrality and Dispersion. This section also presents the definitions of some common population- or sample-based moments.

## Bernoulli and Binomial Experiments

The Bernoulli and Binomial processes provide the simplest models for discrete random experiments.

## Multinomial Experiments

Multinomial processes extend the Binomial experiments for the situation of multiple possible outcomes.

## Geometric, Hypergeometric and Negative Binomial

The Geometric, Hypergeometric and Negative Binomial distributions provide computational models for calculating probabilities for a large number of experiment and random variables. This section presents the theoretical foundations and the applications of each of these discrete distributions.

## Poisson Distribution

The Poisson distribution models many different discrete processes where the probability of the observed phenomenon is constant in time or space. Poisson distribution may be used as an approximation to the Binomial distribution.

# V. Normal Probability Distribution

The Normal Distribution is perhaps the most important model for studying quantitative phenomena in the natural and behavioral sciences - this is due to the Central Limit Theorem. Many numerical measurements (e.g., weight, time, etc.) can be well approximated by the normal distribution.

## The Standard Normal Distribution

The Standard Normal Distribution is the simplest version (zero-mean, unit-standard-deviation) of the (General) Normal Distribution. Yet, it is perhaps the most frequently used version because many tables and computational resources are explicitly available for calculating probabilities.

## Nonstandard Normal Distribution: Finding Probabilities

In practice, the mechanisms underlying natural phenomena may be unknown, yet the use of the normal model can be theoretically justified in many situations to compute critical and probability values for various processes.

## Nonstandard Normal Distribution: Finding Scores (Critical Values)

In addition to being able to compute probability (p) values, we often need to estimate the critical values of the Normal Distribution for a given p-value.

## Multivariate Normal Distribution

The multivariate normal distribution (also known as multivariate Gaussian distribution) is a generalization of the univariate (one-dimensional) normal distribution to higher dimensions (2D, 3D, etc.) The multivariate normal distribution is useful in studies of correlated real-valued random variables.

# VI. Relations Between Distributions

In this chapter, we will explore the relations between different distributions. This knowledge will help us to compute difficult probabilities using reasonable approximations and identify appropriate probability models, graphical and statistical analysis tools for data interpretation. The complete list of all SOCR Distributions is available here and the SOCR Distributome applet provides an interactive graphical interface for exploring the relations between different distributions.

## The Central Limit Theorem

The exploration of the relation between different distributions begins with the study of the sampling distribution of the sample average. This will demonstrate the universally important role of normal distribution.

## Law of Large Numbers

Suppose the relative frequency of occurrence of one event whose probability to be observed at each experiment is p. If we repeat the same experiment over and over, the ratio of the observed frequency of that event to the total number of repetitions converges towards p as the number of experiments increases. Why is that and why is this important?

## Normal Distribution as Approximation to Binomial Distribution

Normal Distribution provides a valuable approximation to Binomial when the sample sizes are large and the probability of successes and failures are not close to zero.

## Poisson Approximation to Binomial Distribution

Poisson provides an approximation to Binomial Distribution when the sample sizes are large and the probability of successes or failures is close to zero.

## Binomial Approximation to Hypergeometric

Binomial Distribution is much simpler to compute, compared to Hypergeometric, and can be used as an approximation when the population sizes are large (relative to the sample size) and the probability of successes is not close to zero.

## Normal Approximation to Poisson

The Poisson can be approximated fairly well by Normal Distribution when λ is large.

# VII. Point and Interval Estimates

Estimation of population parameters is critical in many applications. Estimation is most frequently carried in terms of point-estimates or interval (range) estimates for population parameters that are of interest.

## Method of Moments and Maximum Likelihood Estimation

There are many ways to obtain point (value) estimates of various population parameters of interest, using observed data from the specific process we study. The method of moments and the maximum likelihood estimation are among the most popular ones frequently used in practice.

## Estimating a Population Mean: Large Samples

This section discusses how to find point and interval estimates when the sample-sizes are large.

## Estimating a Population Mean: Small Samples

Next, we discuss point and interval estimates when the sample-sizes are small. Naturally, the point estimates are less precise and the interval estimates produce wider intervals, compared to the case of large-samples.

## Student's T distribution

The Student's T-Distribution arises in the problem of estimating the mean of a normally distributed population when the sample size is small and the population variance is unknown.

## Estimating a Population Proportion

Normal Distribution is appropriate model for proportions, when the sample size is large enough. In this section, we demonstrate how to obtain point and interval estimates for population proportion.

## Estimating a Population Variance

In many processes and experiments, controlling the amount of variance is of critical importance. Thus the ability to assess variation, using point and interval estimates, facilitates our ability to make inference, revise manufacturing protocols, improve clinical trials, etc.

# VIII. Hypothesis Testing

Hypothesis Testing is a statistical technique for decision making regarding populations or processes based on experimental data. It quantitatively answers the possibility that chance alone might be responsible for the observed discrepancy between a theoretical model and the empirical observations.

## Fundamentals of Hypothesis Testing

In this section, we define the core terminology necessary to discuss Hypothesis Testing (Null and Alternative Hypotheses, Type I and II errors, Sensitivity, Specificity, Statistical Power, etc.)

## Testing a Claim about a Mean: Large Samples

As we already saw how to construct point and interval estimates for the population mean in the large sample case, we now show how to do hypothesis testing in the same situation.

## Testing a Claim about a Mean: Small Samples

We continue with the discussion on inference for the population mean for small samples.

## Testing a Claim about a Proportion

When the sample size is large, the sampling distribution of the sample proportion $\hat{p}$ is approximately Normal, by CLT. This helps us formulate hypothesis testing protocols and compute the appropriate statistics and p-values to assess significance.

## Testing a Claim about a Standard Deviation or Variance

The significance testing for the variation or the standard deviation of a process, a natural phenomenon or an experiment is of paramount importance in many fields. This chapter provides the details for formulating testable hypotheses, computation, and inference on assessing variation.

# IX. Inferences from Two Samples

In this chapter, we continue our pursuit and study of significance testing in the case of having two populations. This expands the possible applications of one-sample hypothesis testing we saw in the previous chapter.

## Inferences About Two Means: Dependent Samples

We need to clearly identify whether samples we compare are Dependent or Independent in all study designs. In this section, we discuss one specific dependent-samples case - Paired Samples.

## Inferences About Two Means: Independent Samples

Independent Samples designs refer to experiments or observations where all measurements are individually independent from each other within their groups and the groups are independent. In this section, we discuss inference based on independent samples.

## Comparing Two Variances

In this section, we compare variances (or standard deviations) of two populations using randomly sampled data.

### Problems

This section presents the significance testing and inference on equality of proportions from two independent populations.

# X. Correlation and regression

Many scientific applications involve the analysis of relationships between two or more variables involved in a process of interest. We begin with the simplest of all situations where Bivariate Data (X and Y) are measured for a process and we are interested on determining the association, relation or an appropriate model for these observations (e.g., fitting a straight line to the pairs of (X,Y) data).

## Correlation

The Correlation between X and Y represents the first bivariate model of association which may be used to make predictions.

## Regression

We are now ready to discuss the modeling of linear relations between two variables using Regression Analysis. This section demonstrates this methodology for the SOCR California Earthquake dataset.

## Variation and Prediction Intervals

In this section, we discuss point and interval estimates about the slope of linear models.

## Multiple Regression

Now, we are interested in determining linear regressions and multilinear models of the relationships between one dependent variable Y and many independent variables Xi.

# XI. Analysis of Variance (ANOVA)

## One-Way ANOVA

We now expand our inference methods to study and compare k independent samples. In this case, we will be decomposing the entire variation in the data into independent components.

## Two-Way ANOVA

Now we focus on decomposing the variance of a dataset into (independent/orthogonal) components when we have two (grouping) factors. This procedure called Two-Way Analysis of Variance.

# XII. Non-Parametric Inference

To be valid, many statistical methods impose (parametric) requirements about the format, parameters and distributions of the data to be analyzed. For instance, the Independent T-Test requires the distributions of the two samples to be Normal, whereas Non-Parametric (distribution-free) statistical methods are often useful in practice, and are less-powerful.

## Differences of Medians (Centers) of Two Paired Samples

The Sign Test and the Wilcoxon Signed Rank Test are the simplest non-parametric tests which are also alternatives to the One-Sample and Paired T-Test. These tests are applicable for paired designs where the data is not required to be normally distributed.

## Differences of Medians (Centers) of Two Independent Samples

The Wilcoxon-Mann-Whitney (WMW) Test (also known as Mann-Whitney U Test, Mann-Whitney-Wilcoxon Test, or Wilcoxon rank-sum Test) is a non-parametric test for assessing whether two samples come from the same distribution.

## Differences of Proportions of Two Samples

Depending upon whether the samples are dependent or independent, we use different statistical tests.

## Differences of Means of Several Independent Samples

We now extend the multi-sample inference which we discussed in the ANOVA section, to the situation where the ANOVA assumptions are invalid.

## Differences of Variances of Independent Samples (Variance Homogeneity)

There are several tests for variance equality in k samples. These tests are commonly known as tests for Homogeneity of Variances.

# XIII. Multinomial Experiments and Contingency Tables

## Multinomial Experiments: Goodness-of-Fit

The Chi-Square Test is used to test if a data sample comes from a population with specific characteristics.

## Contingency Tables: Independence and Homogeneity

The Chi-Square Test may also be used to test for independence (or association) between two variables.