Section 8.1 Confidence Intervals for a Population Mean, Standard Deviation Known 359 CAUTION Always round the sample size up. For example, if the value of n given by the formula is 84.01, round it up to 85. SUMMARY Let m be the desired margin of error. Let �� be the population standard deviation, and let z��∕2 be the critical value for a confidence interval. The sample size n needed so that the confidence interval will have margin of error m is given by n = (z��∕2 ⋅ �� m )2 If the value of n given by the formula is not a whole number, round it up to the nearest whole number. By rounding up, we can be sure that the margin of error is no greater than the desired value m. EXAMPLE 8.7 Finding the necessary sample size Scientists want to estimate the mean weight of mice after they have been fed a special diet. From previous studies, it is known that the weight is normally distributed with standard deviation 3 grams. How many mice must be weighed so that a 95% confidence interval will have a margin of error of 0.5 gram? Solution Since we want a 95% confidence interval, we use z��∕2 = 1.96. We are also given �� = 3 and m = 0.5. We therefore use the formula as follows: n = (z��∕2 ⋅ �� m )2 = (1.96 ⋅ 3 0.5 )2 = 138.30; round up to 139 We must weigh 139 mice in order to obtain a 95% confidence interval with a margin of error of 0.5 gram. Factors that limit sample size Since larger sample sizes result in narrower confidence intervals, it is natural to wonder why we don’t always collect a large sample when we want to construct a confidence interval. In practice, the size of the sample that is feasible to obtain is often limited. In some cases, an expensive experimental procedure must be repeated each time an observation is made. For example, studies of automobile safety that require the crashing of new cars are not likely to have large sample sizes. Sometimes ethical considerations restrict the sample size. For example, when a new drug is being tested, there is a risk of adverse health effects to the subjects who take the drug. It is important that the sample size not be larger than necessary, to limit the health risk to as few people as possible. Check Your Understanding 13. A machine used to fill beverage cans is supposed to put exactly 12 ounces of beverage in each can, but the actual amount varies randomly from can to can. The population standard deviation is �� = 0.05 ounce. A simple random sample of filled cans will have their volumes measured, and a 95% confidence interval for the mean fill volume will be constructed. How many cans must be sampled for the margin of error to be equal to 0.01 ounce? 97 14. An IQ test is designed to have scores that have a standard deviation of �� = 15. A simple random sample of students at a large university will be given the test in order to construct a 98% confidence interval for the mean IQ of all students at the university. How many students must be tested so that the margin of error will be equal to 3 points? 136 Answers are on page 366.
navidi_monk_elementary_statistics_2e_ch7-9
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