SECTION 6.2 Confidence Intervals for the Mean (s Unknown) 313 TRY IT YOURSELF 2 Construct 90% and 99% confidence intervals for the population mean temperature of coffee sold in Example 2. Answer: Page 40 Constructing a Confidence Interval You randomly select 36 cars of the same model that were sold at a car dealership and determine the number of days each car was on the dealership’s lot before it was sold. The sample mean is 9.75 days, with a sample standard deviation of 2.39 days. Construct a 99% confidence interval for the population mean number of days the car model is on the dealership’s lot. SOLUTION Because s is unknown, the sample is random, and n = 36 Ú 30, use the t@distribution. Using n = 36, x = 9.75, s = 2.39, c = 0.99, and d.f. = 35, you can use Table 5 to find that tc = 2.724. The margin of error at the 99% confidence level is E = tc s1 n = 2.724# 2.39 2 36 ≈ 1.09. The confidence interval is constructed as shown. Left Endpoint Right Endpoint x - E ≈ 9.75 - 1.09 x + E ≈ 9.75 + 1.09 = 8.66 = 10.84 8.66 6 m 6 10.84 8 8.5 9 9.5 10 10.5 11 x 9.75 8.66 10.84 You can check this answer using technology, as shown below. (When using technology, your answers may differ slightly from those found using Table 5.) STATCRUNCH One sample T summary confidence interval: μ : Mean of population 99% confidence interval results: Mean Sample Mean Std. Err. DF L. Limit U. Limit μ 9.75 0.39833333 35 8.6650174 10.834983 Interpretation With 99% confidence, you can say that the population mean number of days the car model is on the dealership’s lot is between 8.66 and 10.84. TRY IT YOURSELF 3 Construct 90% and 95% confidence intervals for the population mean number of days the car model is on the dealership’s lot in Example 3. Compare the widths of the confidence intervals. Answer: Page A40 See TI-84 Plus steps on page 345. EXAMPLE 3 To explore this topic further, see Activity 6.2 on page 318. 6.2 HISTORICAL REFERENCE William S. Gosset (1876–1937) Developed the t-distribution while employed by the Guinness Brewing Company in Dublin, Ireland. Gosset published his findings using the pseudonym Student.The t-distribution is sometimes referred to as Student’s t-distribution. (See page 35 for others who were important in the history of statistics.)
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