312 CHAPTER 6 Confidence Intervals 156 158 160 162 164 166 168 x 162.0 156.7 167.3 Confidence Intervals and t-Distributions Constructing a confidence interval for m when s is not known using the t@distribution is similar to constructing a confidence interval for m when s is known using the standard normal distribution—both use a point estimate x and a margin of error E. When s is not known, the margin of error E is calculated using the sample standard deviation s and the critical value tc. So, the formula for E is E = tc s1 n . Margin of error for m (s unknown) Before using this formula, verify that the sample is random, and either the population is normally distributed or n Ú 30. Constructing a Confidence Interval for a Population Mean (S Unknown) In Words In Symbols 1. Verify that s is not known, the sample is random, and either the population is normally distributed or n Ú 30. 2. Find the sample statistics n, x, and s. x = Σx n , s = CΣ1x - x22 n - 1 3. Identify the degrees of freedom, d.f. = n - 1 the level of confidence c, and the Use Table 5 in Appendix B. critical value tc. 4. Find the margin of error E. E = tc 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 You randomly select 16 coffee shops and measure the temperature of the coffee sold at each. The sample mean temperature is 162.0°F with a sample standard deviation of 10.0°F. Construct a 95% confidence interval for the population mean temperature of coffee sold. Assume the temperatures are approximately normally distributed. SOLUTION Because s is unknown, the sample is random, and the temperatures are approximately normally distributed, use the t@distribution. Using n = 16, x = 162.0, s = 10.0, c = 0.95, and d.f. = 15, you can use Table 5 to find that tc = 2.131. The margin of error at the 95% confidence level is E = tc s1 n = 2.131# 10.0 2 16 ≈ 5.3. The confidence interval is shown below and in the figure at the left. Left Endpoint Right Endpoint x - E ≈ 162 - 5.3 = 156.7 x + E ≈ 162 + 5.3 = 167.3 156.7 6 m 6 167.3 Interpretation With 95% confidence, you can say that the population mean temperature of coffee sold is between 156.7°F and 167.3°F. See Minitab steps on page 344. EXAMPLE 2 Study Tip Remember that you can calculate the sample standard deviation s using the formula s = CΣ1x - x22 n - 1 or the alternative formula s = CΣx2 - 1Σx22 n n - 1 . However, the most convenient way to find the sample standard deviation is to use technology.
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