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Chapter 7: Q. T7.8 (page 438)

Which of the following statements about the sampling distribution of the sample mean is incorrect?

a. The standard deviation of the sampling distribution will decrease as the sample size increases.

b. The standard deviation of the sampling distribution measures how far the sample mean typically varies from the population mean.

c. The sample mean is an unbiased estimator of the population mean.

d. The sampling distribution shows how the sample mean is distributed around the population mean.

e. The sampling distribution shows how the sample is distributed around the sample mean.

Short Answer

Expert verified

(e). The sampling distribution is incorrect because it is the distribution of all possible sample means, which cannot be centred on the sample mean.

Step by step solution

01

Given Information

The choices are,

a. As the sample size grows, the sampling distribution's standard deviation decreases.

b. The sampling distribution's standard deviation indicates how far the sample mean normally differs from the population mean.

c. The sample mean is a reliable predictor of the population average.

d. The sampling distribution depicts the distribution of the sample mean around the population mean.

e. The sampling distribution depicts the distribution of the sample around the sample mean.

02

Explanation for correct option

a. Yes, because a bigger sample size provides more information about the population and so allows for more precise forecasts.

b. Correct, the variability of the sample mean over all potential samples is the standard deviation of the sampling distribution of the sample mean.

c. Yes, the sample mean is a reliable predictor of the population mean.

d. Correct, because the variation is determined by the sample distribution's standard deviation.

Because the sampling distribution represents the distribution of all potential sample means, it cannot be centred on the sample mean. e. Incorrect.

As a result, the best solution is (e)

03

Explanation for incorrect option

a. The standard deviation of the sampling distribution will decrease as the sample size increases is not the answer.

b. The standard deviation of the sampling distribution measures how far the sample mean typically varies from the population mean is not the answer.

c. The sample mean is an unbiased estimator of the population meanis not the answer.

d. The sampling distribution shows how the sample mean is distributed around the population mean is not the answer.

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Most popular questions from this chapter

According to government data, 22% of American children under the age of 6 live in households with incomes less than the official poverty level. A study of learning in early childhood chooses an SRS of 300 children from one state and finds that p鈭�p^=0.29.

a. Find the probability that at least 29% of the sample are from poverty-level households, assuming that 22% of all children under the age of 6 in this state live in poverty-level households.

b. Based on your answer to part (a), is there convincing evidence that the percentage of children under the age of 6 living in households with incomes less than the official poverty level in this state is greater than the national value of 22%? Explain your reasoning.

Lightning strikes The number of lightning strikes on a square kilometer of open ground in a year has mean 6 and standard deviation 2.4. The National Lightning Detection Network (NLDN) uses automatic sensors to watch for lightning in 1-square-kilometer plots of land. Find the probability that the total number of lightning strikes in a random sample of 50 square-kilometer plots of land is less than 250.

The number of hours a lightbulb burns before failing varies from bulb to bulb. The population distribution of burnout times is strongly skewed to the right. The central limit theorem says that

a. as we look at more and more bulbs, their average burnout time gets ever closer to the mean 渭 for all bulbs of this type.

b. the average burnout time of a large number of bulbs has a sampling distribution with the same shape (strongly skewed) as the population distribution.

c. the average burnout time of a large number of bulbs has a sampling distribution with a similar shape but not as extreme (skewed, but not as strongly) as the population distribution.

d. the average burnout time of a large number of bulbs has a sampling distribution that is close to Normal.

e. the average burnout time of a large number of bulbs has a sampling distribution that is exactly Normal.

You work for an advertising agency that is preparing a new television commercial to appeal to women. You have been asked to design an experiment to compare the effectiveness of three versions of the commercial. Each subject will be shown one of the three versions and then asked to reveal her attitude toward the product. You think there may be large differences in the responses of women who are employed and those who are not. Because of these differences, you should use

a. a block design, but not a matched pairs design.

b. a completely randomized design.

c. a matched pairs design.

d. a simple random sample.

e. a stratified random sample.

Tall girls? To see if the claim made in Exercise 12is true at their high school, an Ap Statistics class chooses an SRS of twenty 16-year-old females at the school and measures their heights. In their sample, the mean height is 64.7inches. Does this provide convincing evidence that 16-year-old females at this school are taller than 64inches, on average?

a. What is the evidence that the average height of all 16-year-old females at this school is greater than 64inches, on average?

b. Provide two explanations for the evidence described in part (a).

We used technology to simulate choosing 250SRSs of size n=20from a population of three hundred 16-year-old females whose heights follow a Normal distribution with mean localid="1654113150676" 渭=64inches and standard deviation 渭=2.5inches. The dotplot shows x=the sample mean height for each of the 250simulated samples.

c. There is one dot on the graph at 62.5. Explain what this value represents.

d. Would it be surprising to get a sample mean of x=64.7or larger in an SRS of size 20when 渭=64inches and 蟽=2.5inches? Justify your answer.

e. Based on your previous answers, is there convincing evidence that the average height of all 16-year-old females at this school is greater than 64inches? Explain your reasoning.
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