# EBook Problems Limits CLT

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## Contents |

## EBook Problems Set - The Central Limit Theorem

### Problem 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*

### Problem 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) binomial distribution*

*(c) population distribution*

*(d) uniform distribution*

### Problem 3

All other things being equal, as the sample size increases, the standard error of a statistic

- Choose one answer.

*(a) Approaches the population mean in numerical value.*

*(b) Approaches the standard deviation of the population.*

*(c) Increases.*

*(d) Remains constant if the value of sigma is known.*

*(e) Decreases.*

### Problem 4

Suppose that the distribution of X in the population is strongly skewed to the left. If you took 200 independent and random samples of size 3 from this population, calculated the mean for each of the 200 samples, and drew the distribution of the sample means, what would the sampling distribution of the means look like?

- Choose one answer.

*(a) It will be perfectly normal and the mean will be equal to the median.*

*(b) It will be close to the normal and the mean will be close to the median.*

*(c) On a p-plot, most of the points will be on the line.*

*(d) It will be skewed to the left and the mean will be less than the median.*

### Problem 5

Is the following approach a correct method for teaching the CLT to a class of 40 students?

Ask each of the 40 students to: 1) Ask 50 of their friends, classmates, and relatives for their weight in pounds. 2) Calculate the mean weight or Xbar 3) Draw the histogram of the sample means using the 40 means.

What is the best answer?

- Choose one answer.

*(a) This method is only partially correct because some of the mathematical steps recommended do not meet all of the assumptions and the conditions CLT.*

*(b) This method is not acceptable because the only way one can show how CLT works is to have access to a computer and conduct a lot of simulations.*

*(c) This method is only partially correct because some of the practical steps recommended do not meet all of the assumptions and conditions of CLT.*

*(d) This method is not correct because the student needs to discuss the margin of error as well as the effect of sample size on margin of error and confidence interval.*

### Problem 6

If you take all samples of a particular size from a particular population, find the mean of each sample, and then plot the distribution of the means, what have you created?

- Choose one answer.

*(a) Sample distribution.*

*(b) Sampling distribution.*

*(c) Population distribution.*

### Problem 7

A study was planned to examine the length of a certain species of fish on Gull Lake. The initial plan was to take a random sample of 100 fish from this lake using a special net, and examine our results. We are also interested in investigating fish lengths in Lake Monster. The fish lengths in Lake Monster have a standard deviation that is twice as big as that on Gull Lake. Suppose you still wanted to get an accurate estimate of the mean fish length for Lake Monster (within 0.25 inches). Select the answer that best describes the sample size we need:

- Choose one answer.

*(a) About the same size sample*

*(b) Not enough information to tell*

*(c) A larger sample is needed for Lake Monster than Gull Lake*

*(d) A smaller sample is needed for Lake Monster than Gull Lake*

### Problem 8

The LAPD has been testing a new system of catching speeders on the 405 over the last 10 months. They wanted to see if they really were catching more speeders, so each month they took 20 samples (with replacement) from the tickets issued in this program . Because their sample sizes were always one-fifth of the tickets, they increased in size each month. How did the sampling distribution of the mean change over the 10 months?

- Choose one answer.

*(a) It became just one point - the true population mean*

*(b) It became wider because they had more information*

*(c) It became skewed to the right because very few people get more than 1 or 2 speeding tickets a year*

*(d) It became more like the true distribution of the population of tickets issued*

*(e) It became close to the normal distribution with the mean equal to the population mean*

### Problem 9

What is a practical application of the Central Limit Theorem?

- Choose one answer.

*(a) It gives you the message that you should pick as large of a sample as you can afford.*

*(b) It relates to the normal distribution and the concept of Z and percentile.*

*(c) It shows yo the meaning of standard deviation over repeated samples.*

*(d) It allows you to make inference about the population mean and percentage.*

### Problem 10

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 shape of the raw score distribution*

*(b) It depends on the standard deviation fo the raw scores*

*(c) It equals 100*

*(d) It equals 1*

*(e) It must always be less than the standard deviation of the raw scores*

### Problem 11

In Chesapeake Bay, complex changes in salinity are caused by the mixture of fresh water and sea water during the diurnal tidal cycle. The fresh water from the Chesapeake River floats across the denser brine in the bay, and during low tide it travels farther down the estuary. There is a counterflow, however, along the bottom that carries the dense marine water up the bay during the waning tide. The surface salinity measurements (in parts per thousand) taken at station 11, offshore from Annapolis, Maryland, on July 3-4, 1927 are collected. The normal probability plot and the histogram show that the data are very skewed. Under these circumstances, to conduct a test of hypotheses with this sample of measurements, what extra condition would we need?

- Choose one answer.

*(a) The whole set of measurements in the population of all water is not normal*

*(b) The sample size is 15 or less*

*(c) the sample size is small enough to use the t-test*

*(d) The sample size is large enough to use the Central Limit Theorem result*

### Problem 12

A polling agency has been hired to predict the proportion of voters who favor a certain candidate. The polling agency picks a random sample of 1000 voters of which 400 indicate that they favor the candidate. If they increase the sample size to 2000, how does the standard error change?

- Choose one answer.

*(a) The standard error will decrease by one-fourth*

*(b) The standard error will not change; the margin of error changes*

*(c) Since the sample size is doubled, the standard error will be halved*

*(d) The standard error will decrease not by a factor of 1/2 but by the square of root of 1/2*

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