Section 8.3 Confidence Intervals for a Population Proportion 383 and obtain n = 0.25 (1.96 0.03 )2 = 1067.1 (round up to 1068) We estimate that we need to sample 1068 teachers to obtain a 95% confidence interval with a margin of error of 0.03. This estimate is larger than the one in Example 8.15 because we used a value of 0.5 for ̂p, which provides a sample size large enough to guarantee that the margin of error will be no greater than 0.03 no matter what the true value of p is. Check Your Understanding 2. In a preliminary study, a simple random sample of 100 computer chips was tested, and 17 of them were found to be defective. Now another sample will be drawn in order to construct a 95% confidence interval for the proportion of chips that are defective. Use the results of the preliminary study to estimate the sample size needed so that the confidence interval will have a margin of error of 0.06. 151 3. A pollster is going to sample a number of voters in a large city and construct a 98% confidence interval for the proportion who support the incumbent candidate for mayor. Find a sample size so that the margin of error will be no larger than 0.05. 542 Answers are on page 390. The margin of error does not depend on the population size In 2013, there were about 18 million registered voters in the state of California, and about 0.27 million registered voters in the state of Wyoming. A simple random sample of 1000 Wyoming voters is selected to estimate the proportion of voters who favor the Democratic candidate for president. Another simple random sample of 1000 California voters is selected to determine the proportion of Democratic voters in that state. Which estimate has the smaller standard error? Because California has a much larger population of registered voters than does Wyoming, it might seem that a larger sample would be needed in California to produce the same standard error. Surprisingly enough, this is not the case. In fact, the standard errors for the two estimates will be about the same. This is clear from the formula for the standard error: The population size does not enter into the calculation. Since the standard errors are about the same, the margins of error will be about the same if confidence intervals of the same level are constructed for both population proportions. Intuitively, we can see that population size doesn’t matter by considering an analogy with testing the water in a swimming pool. To determine whether the chemical balance is correct, one withdraws a few drops of water to test. As long as the contents of the pool are well mixed, so that the water removed constitutes a simple random sample of molecules from the pool, it doesn’t matter how large the pool is. One doesn’t need to sample more water from a bigger pool. EXAMPLE 8.17 The margin of error does not depend on the population size A pollster has conducted a poll using a sample of 500 drawn from a town with population 25,000. He now wants to conduct the poll in a larger town with population 250,000, and to obtain approximately the same margin of error as in the smaller town. How large a sample must he draw? Solution He should draw a sample of 500, just as in the small town. The population size does not affect the margin of error.
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