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Chapter 7: Q. 37 (page 440)

Do you drink cereal milk? What sample size would be required to reduce the standard deviation of the sampling distribution to one-half the value you found in Exercise 35(b)? Justify your answer.

Short Answer

Expert verified

Sample size would be required to reduce the standard deviation of the sampling distribution to one-half the value you found in Exercise 35(b) is4048.

Step by step solution

01

Concept Introduction

The sampling distribution's mean difference indicates how much the sample statistic changes from sample to sample.

By a factor of n, it is smaller than the population standard deviation.

Individual observations are more variable than averages.

02

Explanation 

Given:

n=1012

The standard deviation formula is:

蟽p^=p(1-p)n

In order to reduce the standard deviation to half of its existing value, then:

localid="1650086574157" 蟽p^2=12脳p(1鈭�p)n=p(1鈭�p)4n

Let quadrupling the sample size:

localid="1650086582641" 4n=4(1012)n=4048

Hence, the required sample size is 4048.

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Most popular questions from this chapter

The gypsy moth is a serious threat to oak and aspen trees. A state agriculture department places traps throughout the state to detect the moths. When traps are checked periodically, the mean number of moths trapped is only 0.5,but some traps have several moths. The distribution of moth counts is discrete and strongly skewed, with standard deviation0.7.

(a) What are the mean and standard deviation of the average number of moths xin50traps?

(b) Use the central limit theorem to help you find the probability that the average number of moths in 50traps is greater than0.6.

Your mail-order company advertises that it ships 90%of its orders within three working days. You select an SRS of 100of the 5000orders received in the past week for an audit. The audit reveals that 86of these orders were shipped on time.

(a) If the company really ships 90%of its orders on time, what is the probability that the proportion in an SRS of 100orders is as small as the proportion in your sample or smaller? Follow the four-step process.

(b) A critic says, 鈥淎ha! You claim 90%, but in your sample, the on-time percentage is lower than that. So the 90%claim is wrong.鈥� Explain in simple language why your probability calculation in (a) shows that the result of the sample does not refute the 90%claim.

Do you go to church? What sample size would be required to reduce the standard deviation of the sampling distribution to one-third the value you found in Exercise 36 (b)? Justify your answer.

IRS audits The Internal Revenue Service plans to examine an SRS of individual federal income tax returns from each state. One variable of interest is the proportion of returns claiming itemized deductions. The total number of tax returns in each state varies from over 15 million in California to about 240,000in Wyoming.

(a) Will the sampling variability of the sample proportion change from state to state if an SRS of 2000tax returns is selected in each state? Explain your answer.

(b) Will the sampling variability of the sample proportion change from state to state if an SRS of 1%of all tax returns is selected in each state? Explain your answer

Thousands of travelers pass through the airport in Guadalajara, Mexico, each day. Before leaving the airport, each passenger must pass through the Customs inspection area. Customs agents want to be sure that passengers do not bring illegal items into the country. But they do not have time to search every traveler鈥檚 luggage. Instead, they require each person to press a button. Either a red or a green bulb lights up. If the red light shows, the passenger will be searched by Customs agents. A green light means 鈥済o ahead.鈥� Customs agents claim that the proportion of all travelers who will be stopped (red light) is0.30, because the light has probability 0.30of showing red on any push of the button. To test this claim, a concerned citizen watches a random sample of 100travelers push the button. Only 20get a red light.

(a) Assume that the Customs agents鈥� claim is true. Find the probability that the proportion of travelers who get a red light is as small as or smaller than the result in this sample. Show your work.

(b) Based on your results in (a), do you believe the Customs agents鈥� claim? Explain.
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