## Statistics The Exploration & Analysis of Data, International Edition, 7th Edition By Roxy Peck – Test Bank

**Chapter 8: Sampling Variability & Sampling Distributions**

**Concept Quiz**

Name _________________________

The following questions are in a True / False format. The answers to these questions will frequently depend on remembering facts, understanding of the concepts, and knowing the statistical vocabulary. Before answering these questions, be sure to read them carefully!

T F

1.

A statistic is a characteristic of a population.

T F

2.

T F

3.

T F

4.

As *n* increases, the mean of the sampling distribution of gets closer to .

T F

5.

The standard deviation of the distribution of decreases as *n* increases.

T F

6.

The sampling distribution oftends to be more spread out for larger sample sizes than for smaller sample sizes.

T F

7.

The distribution of is normal if the population is normal.

T F

8.

The distribution of always has the same shape as the distribution of the population.

T F

9.

For *n* sufficiently large, the distribution of is approximately a standard normal distribution.

T F

10.

The closer is to 0 or 1, the larger *n* must be in order for the distribution of to be approximately normal.

T F

11.

If , a sample size of is large enough for the sampling distribution of to be well approximated by a normal distribution.

T F

12.

The mean of the sampling distribution of isno matter how large *n* is.

**Chapter 8: Sampling Variability & **

**Sampling Distributions**

**Section 8.1-8.2 **

Name ___________________________

- What is the sampling distribution of a statistic?

- Suppose we artificially categorize populations as approximately normal or not approximately normal, and samples as large or small. This categorization results in 4 categories:

- a) Small samples from an approximately normal population
- b) Large samples from an approximately normal population
- c) Small samples from a population that is not approximately normal
- d) Large samples from a population that is not approximately normal

What, if anything, can be said about the __shape__ of the sampling distribution of for each of these 4 situations?

(a)

(b)

(c)

(d)

- The Get-A-Grip tire company claims that the mean lifetime of tires sold on new cars is 23,000 miles and the standard deviation is 2500 miles.

- a) If the claim by Get-A-Grip is true, what is the mean of the sampling distribution of for samples of size ?

- b) If the claim by Get-A-Grip is true, what is the standard deviation of the sampling distribution of for samples of size ?

- c) If the distribution of tire life is approximately normal, what is the probability that the mean of a random sample of
*n*= 4 tire lifetimes will be less than 20,000 miles?**Chapter 8: Sampling Variability &****Sampling Distributions****Section 8.1-8.2**Name ___________________________

- What is the sampling distribution of a statistic?

- Consider sampling from a
__very large____bimodal__population. As the sample size,*n*, increases, some characteristics of the sampling distribution of will change. Does an increasing sample size result in changes in the characteristics of the sampling distribution shown below? If so, specifically how does the sampling distribution change?

- a) The mean of the sampling distribution of

- b) The standard deviation of the sampling distribution of

- c) The shape of the sampling distribution of

- How are the quantities, and , related?

- The HardCore Apple Company claims that the mean firmness of their apples, as measured on the Mohs hardness scale is 7.0 and the standard deviation is 0.3.

- a) If the claim by Hardcore is true, what is the mean of the sampling distribution of for samples of size?

- b) If the claim by HardCore is true, what is the standard deviation of the sampling distribution of for samples of size?

- c) If the distribution of HardCore firmness is approximately normal, and their claim about the mean and standard deviation of firmness is true, what is the probability that the mean firmness of a random sample of size will be less than 6.9?
**Chapter 8: Sampling Variability &****Sampling Distributions****Section 8.3**Name ___________________________

- Consider sampling from a population whose proportion of successes is . As the sample size,
*n*, increases, some characteristics of the sampling distribution of change. Which of the following characteristics will change as*n*increases, and what is the nature of the change?

- a) The mean of the sampling distribution of

- b) The standard deviation of the sampling distribution of

- c) The shape of the sampling distribution of

- One method for estimating abundance of animals is known as line-intercept sampling. The theory of this method, when applied to Alaskan wolverines, predicts that the proportion of attempts to locate wolverine tracks should be successful. Suppose that biologists will make 100 attempts to locate wolverine tracks in random locations in Alaska.

- a) Show that this sample size is large enough for the sampling distribution of to be

approximately normal.

- b) What is the mean of the sampling distribution of if the proportion predicted by line-intercept sampling is correct?

- c) What is the standard deviation of the sampling distribution of if the proportion predicted by intercept sampling is correct?

- d) If the proportion predicted by line-intercept sampling is correct, what is the probability that a sample proportion,, would differ from by as much as 0.05?
**Chapter 8: Sampling Variability &****Sampling Distributions****Section 8.3**Name ___________________________

- Consider sampling from a population whose proportion of successes is . As the sample size,
*n*, increases, some characteristics of the sampling distribution of change. Which of the following characteristics will change as*n*increases, and what is the nature of the change?

- a) The mean of the sampling distribution of

- b) The standard deviation of the sampling distribution of

- c) The shape of the sampling distribution of

- A local owner of apartment buildings is considering raising the rent because of increased costs. He is concerned that apartments may be easy to come by, and will only increase the rent if the current vacancy rate is below 20%. One method for estimating vacancy rate is to take a random sample of similar apartments, and calculate the proportion of apartments that are empty. Suppose the owner has decided to take a sample to estimate the vacancy rate. He is considering a sample size of
*n*= 40. Suppose further that the actual vacancy rate is 0.30.

- a) Show that this sample size is large enough for the sampling distribution of to be

approximately normal.

- b) What is the mean of the sampling distribution of ?

- c) What is the standard deviation of the sampling distribution of ?

- d) What is the approximate probability that a sample proportion,, would be less than 0.20 even though the population proportion is 0.30?
**Chapter 8: Sampling Variability &****Sampling Distributions**Name ___________________________

In question #1 you will be asked to sketch curves representing the distributions of a set of data, as well as the sampling distributions of the mean under different conditions. You need not get these graphs perfectly correct, but should clearly indicate different aspects of the curves, such as location, variability, and shape.

- A very large study of college students’ study habits found that the time (in hours) that freshmen study each week is approximately normal with mean 24 hours and standard deviation 8 hours. Consider random samples of size 16 from the population of freshmen.
- a) On the axes below, sketch curves representing the distribution of the original population and the sampling distribution of for a sample of size. Be sure to indicate which graph is which. What are the mean and standard deviation of the sampling distribution?

- b) What is the probability that a sample of size from this population would result in a sample mean greater than 30 hours?

- c) The study of college freshmen also asked about the students’ time spent playing video games, and a relative frequency histogram of the results appears below. The mean amount of time spent playing video games was 0.8 hours per week.

**Time spent on video games (hours/week)**Suppose we were to take a random sample,

*n*= 4, from this population of freshmen and ask them how much time they spent playing video games per week. Describe the shape, center, and spread of the sampling distribution of , as compared to the distribution of the population.- The principal at John F. Kennedy High School has been asked to provide the average number of classes taken by the students at the school. Since the computer system is down, she takes her alphabetized list of students, randomly selects 50 students, determines the number of classes each of the 50 selected students is taking, and calculates . She then reports “Since I took a large random sample, the
__population__mean number of classes taken by the students at the school is 5.4.” Write a short paragraph to send to her that explains why her statement is not correct.

- Consider the following “population”:. Suppose that a random sample of size is to be selected
__without replacement__from this population. There are 6 possible samples (since the order of selection does not matter). Compute the sample mean for each of these samples and use that information to construct the sampling distribution of . (Display it in table form.)

- Some biologists believe the evolution of handedness is linked to complex behaviors such as tool-use. Under this theory, handedness would be genetically passed on from parents to children. That is, left-handed parents would be more likely to have left-handed children than right-handed parents. An alternate theory asserts that handedness should be random, with left- and right-handedness equally likely. In a recent study using a simple random sample of right-handed parents, 50 of the children born were right-handed. (.) Suppose handedness is a random occurrence with either hand equally likely to be dominant, implying that the probability of a right-handed offspring is .

- a) Show that it is reasonable to approximate the sampling distribution of using a normal distribution.

- b) Assuming left- and right-handed children are equally likely from right-handed parents, what is the probability of observing a sample proportion of at least ?
**Chapter 8: Sampling Variability &****Sampling Distributions**Name ___________________________

In question #1 you will be asked to sketch curves representing the distributions of a set of data, as well as the sampling distributions of the mean under different conditions. You need not get these graphs perfectly correct, but should clearly indicate different aspects of the curves, such as location, variability, and shape.

- The State Fisheries Department wishes to stock the Styx River with fish, and would like the species to not only survive but thrive. The “substrate” (pebble size at bottom) of a river is an important determinant of the quality of spawning habitat. Unknown to the Fisheries officials, the pebble diameters in the Styx River are approximately normally distributed with a mean of 24 mm, and a standard deviation of 8 mm. Fisheries officials will select a random sample of pebbles in an attempt to judge the average pebble size.
- a) On the scale below, sketch two curves, one representing the distribution of the original population and the other the sampling distribution of for a sample of size. Be sure to indicate which graph is which. What are the mean and standard deviation of the sampling distribution?

- b) What is the approximate probability that a sample of from this population would result in a
__sample mean__greater than 30 mm?

- c) The graph below is a relative frequency histogram of the lengths of the gentle Hecate fish in the Styx River, as recorded during a large fish tagging survey.

**Lengths for***Hecate hecate*(feet)Suppose we were to take a random sample,

*n*= 4, from this population of fish and measure their lengths. Describe the shape, center, and spread of the sampling distribution of , as compared to the distribution of the population.- The principal at Thomas Jefferson High School has been asked to estimate the proportion of students at the school who drive to school and use the school parking lot. He takes a random sample of size students and calculates a sample proportion, . “Now,” he exclaims, “since my sample size is greater than 30, the sampling distribution of the sample proportion is approximately normal.” Write a short paragraph that explains why his statement is not correct.

- Consider the following “population”:. Suppose that a random sample of size is to be selected
__without replacement__from this population. There are 6 possible samples (since the order of selection does not matter). Compute the sample mean for each of these samples and use that information to construct the sampling distribution of . (Display it in table form.)

- The first large-scale study of the human sex ratio involved over 6,000 families each having 12 children. (This was done in 19th Century Germany – large families were more common.) 52% of the children they observed were boys. Suppose that 21st Century researchers wish to replicate this observational study to see if the proportion of boys might have changed in the intervening years. Further suppose the researchers track down 50 families with 12 children. From these 600 children, a random sample of 50 children is taken. 30 of the 50 children were boys (.)

- a) If the modern true population proportion of newborn boys is show that it is reasonable to approximate the sampling distribution of using a normal distribution.

- b) If the modern true population proportion of newborn boys is , what is the probability of observing a sample proportion of at least ?

- Consider sampling from a population whose proportion of successes is . As the sample size,

- Consider sampling from a population whose proportion of successes is . As the sample size,

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