# EBook Problems

### From Socr

(→XI. Analysis of Variance (ANOVA)) |
(→XII. Non-Parametric Inference) |
||

Line 922: | Line 922: | ||

=XII. Non-Parametric Inference= | =XII. Non-Parametric Inference= | ||

- | ==Differences of Medians (Centers) of Two Paired Samples== | + | 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]]. |

- | ==Differences of Medians (Centers) of Two Independent Samples== | + | ==[[AP_Statistics_Curriculum_2007_NonParam_2MedianPair | Differences of Medians (Centers) of Two Paired Samples]]== |

- | ==Differences of Proportions of Two 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. |

- | ==Differences of Means of Several Independent Samples== | + | ===[[EBook_Problems_NonParam_2MedianPair | Problems]]=== |

- | ==Differences of Variances of Independent Samples (Variance Homogeneity)== | + | |

+ | ==[[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]]=== | ||

+ | |||

+ | ==[[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]]=== | ||

+ | |||

+ | ==[[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]]=== | ||

+ | |||

+ | ==[[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]]=== | ||

=XIII. Multinomial Experiments and Contingency Tables= | =XIII. Multinomial Experiments and Contingency Tables= |

## Revision as of 01:03, 27 November 2008

## 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?

### Problems

## 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.

### Problems

## 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.)

### Problems

## 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.

### Problems

# 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.

### Problems

## Summarizing Data with Frequency Tables

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

### Problems

## 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.

### Problems

## 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).

### Problems

## 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.

### Problems

## 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.

### Problems

## Statistics

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

### Problems

## 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.

### Problems

# 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.

### Problems

## 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.

### Problems

## 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.

### Problems

## 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.

### Problems

# 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.

### Problems

## 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.

### Problems

**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.**

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.**

*(a) normal distribution*

*(b) uniform distribution*

*(c) population distribution*

*(d) binomial distribution*

## Bernoulli and Binomial Experiments

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

### Problems

## Multinomial Experiments

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

### Problems

## 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.

### Problems

## 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.

### Problems

# 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.

**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.**

*(a) 131 and 139 pounds.*

*(b) 133 and 137 pounds.*

*(c) 119 and 151 pounds*

*(d) 125 and 145 pounds.*

*(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*

**7. A set of test scores are normally distributed. The mean is 100 and the standard deviation is 20. These scores are converted to z-scores. What are the z-scores of the mean and median?**

**Choose one answer.**

*(a) 1*

*(b) 100*

*(c) 0*

*(d) 50*

**8. In Japan there is an annual turkey dog eating contest. The number of turkey dogs that contestants eat are normally distributed with a mean of 36 turkey dogs and a standard deviation of 6 turkey dogs. A contestant eats 27 turkey dogs. What is his z-score?**

**Choose one answer.**

*(a) 6*

*(b) -1.5*

*(c) 9*

*(d) 1.5*

*(e) -9*

### Problems

## 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.

### Problems

## 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.

### Problems

# 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.

**1. Which of the following would make the sampling distribution of the sample mean narrower? Check all answers that apply.**

**Choose at least one answer.**

*(a) A smaller population standard deviation*

*(b) A smaller sample size*

*(c) A larger standard error*

*(d) A larger sample size*

*(e) A larger population standard deviation*

### Problems

## 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?

### Problems

## 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.

**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*

### Problems

## 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.

### Problems

## 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.

### Problems

## Normal Approximation to Poisson

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

### Problems

# 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.

### Problems

## Estimating a Population Mean: Large Samples

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

### Problems

## 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.

### Problems

## 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.

### Problems

### 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.

**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*

**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%**

**Choose one answer.**

*(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%.*

*(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%.*

*(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%.*

*(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%.*

### Problems

## 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.

### Problems

# 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.)

**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*

*(b) Chi-Square*

*(c) Paired Difference t*

*(d) Two-Sample t*

**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?**

**Choose One Answer.**

*(a) No way to tell*

*(b) The new p-value would be the same as before*

*(c) The new p-value would be smaller than before*

*(d) The new p-value would be larger than before*

**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:**

**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.*

### Problems

## 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.

**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*

**2. A utility company serves 50,000 households. As part of a survey of customer attitudes, they take a simple random sample of 750 of these households. The average number of television sets in the sample households turns out to be 1.86, and the standard deviation in the sample is 0.80. What sample size would be necessary for the standard error of the sample mean to be 0.02?**

**Choose one answer**

*(a) 5,000*

*(b) 1,600*

*(c) 10,000*

*(d) 1,000*

### Problems

## Testing a Claim about a Mean: Small Samples

We continue with the discussion on inference for the population mean for 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.*

### Problems

## Testing a Claim about a Proportion

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

**1. A random sample of 1000 Americans aged 65 and older was collected in 1980 and found that 15% had "hazardous" levels of drinking, which is defined as regularly drinking an amount of alcohol that could cause health problems given the subject's medical conditions. Researchers wanted to know if this proportion has changed since 1980 and so collected a random sample of 1500 Americans aged 65 and older in 2004. They found that 12% drank at hazardous levels. Which of the following is closest to the value of a test statistic that could be used to test the hypothesis that the proportion of hazardous drinkers over the age of 65 has declined since 1980?**

**Choose one answer.**

*(a) -2.13*

*(b) 0.014*

*(c) 0.418*

*(d) 4.54*

### Problems

## 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.

### Problems

# 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**.

### Problems

## 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.

### Problems

## Comparing Two Variances

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

### Problems

## Inferences about Two Proportions

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

### Problems

# 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.

**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.*

*(b) This is sometimes true.*

*(c) This is never true.*

**2. The correlation between high school algebra and geometry scores was found to be + 0.8. Which of the following statements is not true?**

*(a) Most of the students who have above average scores in algebra also have above average scores in geometry.*

*(b) Most people who have above average scores in algebra will have below average scores in geometry*

*(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.*

*(d) Most students who have below average scores in algebra also have below average scores in geometry.*

**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?**

**Choose one answer.**

*(a) Most people who spend a lot of hours watching television have low cardiovascular endurance.*

*(b) Most people who have good cardiovascular endurance spend a lot of time running and little time watching television.*

*(c) Based on the correlation, if you increase your running hours per week, your cardiovascular endurance will decrease.*

*(d) Based on the correlation, if you increases your television watching time, your cardiovascular endurance will decrease.*

*(e) Most people with a lot of miles ran per week have high cardiovascular endurance.*

**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?**

**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.*

**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.*

**8. If the correlation coefficient is 0.80, then:**

**Choose one answer.**

*(a) The explanatory variable is usually less than the response variable.*

*(b) The explanatory variable is usually more than the response variable.*

*(c) None of the statements are correct.*

*(d) Below-average values of the explanatory variable are more often associated with below-average values of the response variable.*

*(e) Below-average values of the explanatory variable are more often associated with above-average values of the response variable.*

**9. Given the following data, what is the best estimate for the coefficient of correlation between the ages of the husbands and wives?**

**There are 50 couples (husband and wife). The age range for men is from 50 to 70 years old. The age range for women is from 48 to 68 years old. For all of the couples, the husband is two years older than the wife. For instance, in one couple the husband is 50 years old and the wife is 48 years old.**

**Choose one answer.**

*(a) The coefficient of correlation between the age of husband and wife is equal to +1.*

*(b) We need the actual data to compute the coefficient of correlatin between the age of the husband and wife.*

*(c) The coefficient of correlation between the age of husband and wife is equal to zero.*

*(d) The coefficient of correlation between the age of husband and wife is equal to +0.50.*

*(e) The coefficient of correlation between the age of husband and wife is equal to -1.*

### Problems

## 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.

**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*

**5. Suppose that wildlife researchers monitor the local alligator population by taking aerial photograhs on a regular schedule. They determine that the best fitting linear model to predict weight in pounds from the length of the gators inches is:**

**Weight = -393 + 5.9*Length with r2 = 0.836.**

**Which of the following statements is true?**

**Choose one answer.**

*(a) A gator that is about 10 inches above average in length is about 59 pounds above the average weight of these gators.*

*(b) The correlation between a gator's length and weight is 0.836.*

*(c) The correlation between a gator's height and weight cannot be determined without the actual data.*

*(d) The correlation between a gator's height and weigth is about -0.914.*

### Problems

## Variation and Prediction Intervals

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

**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*

**10. Suppose we plan to take a random sample of adults in the U.S. and determine the percent of them who have attended church in the last 30 days. We calculate a 90% confidence interval for the proportion of all adults in the U.S. who attended church in the last 30 days. Which of the following changes in our plans would result in a wider confidence interval? Check all that apply.**

**Choose one answer.**

*(a) Using an 85% confidence level.*

*(b) Using a 95% confidence level.*

*(c) Using a larger sample.*

*(d) Using a smaller sample.*

**11. Kevin has always, ever since he was a wee lad, wondered what proportion of the candies in M&M chocolate candies bags are yellow. However, his persistent calls to the M&M headquarter were of no avail. Now that he wields the awesome power of being a TA for Stat 10, he makes each of his 200 students go buy a M&M bag, count the colors, and compute a 99% confidence intervals for the yellow candy proportion. Assume that each M&M bag is a random sample, approximately how many of the 200 confidence intervals will not capture the true population proportion for yellow M&M's?**

**Choose one answer.**

*(a) Not enough information for an answer*

*(b) 0 to 4*

*(c) 4 to 8*

*(d) 12 to 14*

*(e) 8 to 12*

**12. A 95% confidence interval for the proportion of U.S. adults who favor the death penalty is given by (0.03, 0.09). Is the following statement true or false?**

**"There is a 95% probability that an adult in the US is in favor of the death penalty."**

*(a) True*

*(b) False*

### Problems

## 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}.

### Problems

# 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.

### Problems

## 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**.

### Problems

# 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.

### Problems

## 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.

### Problems

## Differences of Proportions of Two Samples

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

### Problems

## 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.

### Problems

## 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**.

### Problems

# XIII. Multinomial Experiments and Contingency Tables

## Multinomial Experiments: Goodness-of-Fit

## Contingency Tables: Independence and Homogeneity

## References

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