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Chapter 6: Q62E (page 363)

In each case, find the approximate sample size required to construct a 95% confidence interval for p that has sampling error of SE = .08.

a. Assume p is near .2.

b. Assume you have no prior knowledge about p, but you wish to be certain that your sample is large enough to achieve the specified accuracy for the estimate.

Short Answer

Expert verified

a. The approximate sample size is 97.

b. The approximate sample size is 151.

Step by step solution

01

Given information

There have 95% confidence interval for p that has a sampling error is 0.08

02

Finding the approximate sample size

a.

p=0.2, α=0.05,  and  SE=0.08

The approximate sample size obtained as,

localid="1658293941430" n=(zα2)2(p(1−p))(SE)2=(1.960)2(0.2×0.8)(0.08)2=0.6146560.0064=96.04

n≈97

Therefore, the approximate sample size is 96.

03

Finding the approximate sample size

b.

Assume that p is 0.5

p=0.5, α=0.05,  and  SE=0.08

The approximate sample size obtained as,

localid="1658293955933" n=(zα2)2(p(1−p))(SE)2=(1.960)2(0.5×0.5)(0.08)2=0.96040.0064=150.0625

n≈151

Therefore, the approximate sample size is 150.

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Most popular questions from this chapter

A random sample of 70 observations from a normally distributed population possesses a sample mean equal to 26.2 and a sample standard deviation equal to 4.1.

a. Find an approximate 95% confidence interval for

b. What do you mean when you say that a confidence coefficient is .95?

c. Find an approximate 99% confidence interval for

d. What happens to the width of a confidence interval as the value of the confidence coefficient is increased while the sample size is held fixed?

e. Would your confidence intervals of parts a and c be valid if the distribution of the original population was not normal? Explain

U.S. Postal Service’s performance. The U.S. Postal Service (USPS) reports that 95% of first-class mail within the same city is delivered on time (i.e., within 2 days of the time of mailing). To gauge the USPS performance, Price Waterhouse monitored the delivery of first-class mail items between Dec. 10 and Mar. 3—the most difficult delivery season due to bad weather conditions and holidays. In a sample of 332,000 items, Price Waterhouse determined that 282,200 were delivered on time. Comment on the performance of USPS first-class mail service over this time period.

The following sample of 16 measurements was selected from a population that is approximately normally distributed:

  1. Construct an 80% confidence interval for the population mean.
  2. Construct a 95% confidence interval for the population mean and compare the width of this interval with that of part a.
  3. Carefully interpret each of the confidence intervals and explain why the 80% confidence interval is narrower.

Do social robots walk or roll? Refer to the InternationalConference on Social Robotics(Vol. 6414, 2010) studyof the trend in the design of social robots, Exercise 6.48(p. 357). Recall that you used a 99% confidence interval toestimate the proportion of all social robots designed withlegs, but no wheels. How many social robots would need tobe sampled in order to estimate the proportion to within.075 of its true value?

Are you really being served red snapper? Refer to the Nature (July 15, 2004) study of fish specimens labeled “red snapper,” Exercise 3.75 (p. 196). Recall that federal law prohibits restaurants from serving a cheaper, look-alike variety of fish (e.g., vermillion snapper or lane snapper) to customers who order red snapper. A team of University of North Carolina (UNC) researchers analyzed the meat from each in a sample of 22 “red snapper” fish fillets purchased from vendors across the United States in an effort to estimate the true proportion of fillets that are really red snapper. DNA tests revealed that 17 of the 22 fillets (or 77%) were not red snapper but the cheaper, look-alike variety of fish.

a. Identify the parameter of interest to the UNC researchers.

b. Explain why a large-sample confidence interval is inappropriate to apply in this study.

c. Construct a 95% confidence interval for the parameter of interest using Wilson’s adjustment.

d. Give a practical interpretation of the confidence interval.
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