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Chapter 6: Q73E (page 364)

Bacteria in bottled water. Is the bottled water you drinksafe? The Natural 91影视 Defense Council warns thatthe bottled water you are drinking may contain morebacteria and other potentially carcinogenic chemicals thanallowed by state and federal regulations. Of the more than1,000 bottles studied, nearly one-third exceeded governmentlevels (www.nrdc.org). Suppose that the Natural91影视 Defense Council wants an updated estimate ofthe population proportion of bottled water that violates atleast one government standard. Determine the sample size(number of bottles) needed to estimate this proportion towithin 0.01 with 99% confidence.

Short Answer

Expert verified

The sample size needed to estimate this proportion to within 0.01 with 99% confidence is 14746

Step by step solution

01

Given information

The Natural 91影视 Defense Council warns that the bottled water you are drinking may contain more bacteria and other potentially carcinogenic chemicals than allowed by state and federal regulations. Of the more than 1,000 bottles studied, nearly one-third exceeded government levels

02

Finding the sample size

Here the sample proportion is 1/3.

Therefore,

p^=13=0.333333

q^=1鈭�p^=1鈭�0.333333=0.666667

Here the standard error is 0.01

The critical value for a 99% confidence interval is z伪/2=z0.01/2=z0.005=2.576

SE=z伪/2p^q^nn=z2伪/2p^q^SE2n=2.5762脳0.333333脳0.6666670.012n=14746.17n鈮�14746

The required sample size is 14746

The sample size needed to estimate this proportion to within 0.01 with 99% confidence is 14746

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Most popular questions from this chapter

Suppose you wish to estimate a population mean correct to within .20 with probability equal to .90. You do not know 蟽2, but you know that the observations will range in value between 30 and 34.

a. Find the approximate sample size that will produce the desired accuracy of the estimate. You wish to be conservative to ensure that the sample size will be ample to achieve the desired accuracy of the estimate. [Hint: Using your knowledge of data variation from Section 2.6, assume that the range of the observations will equal 4蟽.]

b. Calculate the approximate sample size, making the less conservative assumption that the range of the observations is equal to 6蟽.

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.

FindZ伪/2for each of the following:

a.= .10

b.= .01

c.= .05

d.= .20

Crash risk of using cell phones while driving. Studies have shown that drivers who use cell phones while operating a motor passenger vehicle increase their risk of an accident. To quantify this risk, the New England Journal of Medicine (January 2, 2014) reported on the risk of a crash (or near crash) for both novice and expert drivers when using a cell phone. In a sample of 371 cases of novices using a cell phone while driving, 24 resulted in a crash (or near crash). In a sample of 1,467 cases of experts using a cell phone while driving, 67 resulted in a crash (or near crash).

a. Give a point estimate of p, the true crash risk (probability) for novice drivers who use a cell phone while driving.

b. Find a 95% confidence interval for p.

c. Give a practical interpretation of the interval, part b.

d. Repeat parts a鈥揷 for expert drivers.

Improving SAT scores. Refer to the Chance (Winter 2001) and National Education Longitudinal Survey (NELS) study of 265 students who paid a private tutor to help them improve their SAT scores, Exercise 2.88 (p. 113). The changes in both the SAT鈥揗athematics and SAT鈥揤erbal scores for these students are reproduced in the table. Suppose the true population meansa change in score on one of the SAT tests for all students who paid a private tutor is 15. Which of the two tests, SAT鈥揗athematics or SAT鈥揤erbal, is most likely to have this mean change? Explain.
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