SECTION 6.1 Confidence Intervals for the Mean (s Known) 299 In Example 1, the probability that the population mean is exactly 19.9 is virtually zero. So, instead of estimating m to be exactly 19.9 using a point estimate, you can estimate that m lies in an interval. This is called making an interval estimate. An interval estimate is an interval, or range of values, used to estimate a population parameter. DEFINITION Although you can assume that the point estimate in Example 1 is not equal to the actual population mean, it is probably close to it. To form an interval estimate, use the point estimate as the center of the interval, and then add and subtract a margin of error. For instance, if the margin of error is 0.6, then an interval estimate would be given by 19.9 { 0.6 or 19.3 6 m 6 20.5. The point estimate and interval estimate are shown in the figure. x 21 20.5 20 Interval Estimate 19.5 19 Point estimate x = 19.9 Left endpoint 19.3 Right endpoint 20.5 Before finding a margin of error for an interval estimate, you should first determine how confident you need to be that your interval estimate contains the population mean m. The level of confidence c is the probability that the interval estimate contains the population parameter, assuming that the estimation process is repeated a large number of times. DEFINITION You know from the Central Limit Theorem that when n Ú 30, the sampling distribution of sample means approximates a normal distribution. The level of confidence c is the area under the standard normal curve between the critical values, -zc and zc. Critical values are values that separate sample statistics that are probable from sample statistics that are improbable, or unusual. You can see from the figure shown below that c is the percent of the area under the normal curve between -zc and zc. The area remaining is 1 - c, so the area in one tail is 1 211 - c2. Area in one tail For instance, if c = 90%, then 5% of the area lies to the left of -zc = -1.645 and 5% lies to the right of zc = 1.645, as shown in the table. 1 2 c −zc z = 0 zc (1 − c) 1 2 (1 − c) z If c = 90%: c = 0.90 Area in blue region 1 - c = 0.10 Area in yellow regions 1 211 - c2 = 0.05 Area in one tail -zc = -1.645 Critical value separating left tail zc = 1.645 Critical value separating right tail Study Tip In this text, you will usually use 90%, 95%, and 99% levels of confidence. Here are the z@scores that correspond to these levels of confidence. Level of Confidence zc 90% 1.645 95% 1.96 99% 2.575 For help with intervals on the number line, see Integrated Review at MyLab® Statistics
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