SECTION 6.1 Confidence Intervals for the Mean (s Known) 301 Confidence Intervals for a Population Mean Using a point estimate and a margin of error, you can construct an interval estimate of a population parameter such as m. This interval estimate is called a confidence interval. A c@confidence interval for a population mean M is x - E 6 m 6 x + E. The probability that the confidence interval contains m is c, assuming that the estimation process is repeated a large number of times. DEFINITION Constructing a Confidence Interval for a Population Mean (S Known) In Words In Symbols 1. Verify that s is known, the sample is random, and either the population is normally distributed or n Ú 30. 2. Find the sample statistics n and x. x = Σx n 3. Find the critical value zc that corresponds Use Table 4 in Appendix B. to the given level of confidence. 4. Find the margin of error E. E = zc s1 n 5. Find the left and right endpoints Left endpoint: x - E and form the confidence interval. Right endpoint: x + E Interval: x - E6 m 6 x + E GUIDELINES Constructing a Confidence Interval Use the data in Example 1 and the result of Example 2 to construct a 95% confidence interval for the mean number of hours spent per week on required athletic activities by all student-athletes in the conference. SOLUTION In Examples 1 and 2, you found that x ≈ 19.9 and E ≈ 0.7. The confidence interval is constructed as shown. Left Endpoint Right Endpoint x - E ≈ 19.9 - 0.7 x + E ≈ 19.9 + 0.7 = 19.2 = 20.6 19.2 6 m 6 20.6 x 19 19.5 20 20.5 21 19.9 19.2 20.6 Interpretation With 95% confidence, you can say that the population mean number of hours spent on required athletic activities is between 19.2 and 20.6 hours. See Minitab steps on page 344. EXAMPLE 3 Study Tip When you construct a confidence interval for a population mean, the general round-off rule is to round off to the same number of decimal places as the sample mean. Study Tip Other ways to represent a confidence interval are 1x - E, x + E2 and x { E. For instance, in Example 3, you could write the confidence interval as 119.2, 20.62 or 19.9 { 0.7.
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