SECTION 6.1 Confidence Intervals for the Mean (s Known) 303 μ The horizontal segments represent 90% confidence intervals for different samples of the same size. In the long run, 9 of every 10 such intervals will contain m. In Examples 3 and 4, and Try It Yourself 4, the same sample data were used to construct confidence intervals with different levels of confidence. Notice that as the level of confidence increases, the width of the confidence interval also increases. In other words, when the same sample data are used, the greater the level of confidence, the wider the interval. For a normally distributed population with s known, you may use the normal sampling distribution for any sample size (even when n 6 302, as shown in Example 5. Constructing a Confidence Interval A college admissions director wishes to estimate the mean age of all students currently enrolled. In a random sample of 20 students, the mean age is found to be 22.9 years. From past studies, the standard deviation is known to be 1.5 years, and the population is normally distributed. Construct a 90% confidence interval for the population mean age. SOLUTION Because s is known, the sample is random, and the population is normally distributed, use the formula for E given in this section. Using n = 20, x = 22.9, s = 1.5, and zc = 1.645, the margin of error at the 90% confidence level is E = zc s1 n = 1.645# 1.52 20 ≈ 0.6. The 90% confidence interval can be written as x { E ≈ 22.9 { 0.6 or as shown below. Left Endpoint Right Endpoint x - E ≈ 22.9 - 0.6 x + E ≈ 22.9 + 0.6 = 22.3 = 23.5 22.3 6 m 6 23.5 x 23.5 24 23 22.5 22 22.9 22.3 23.5 Interpretation With 90% confidence, you can say that the mean age of all the students is between 22.3 and 23.5 years. TRY IT YOURSELF 5 Construct a 90% confidence interval for the population mean age for the college students in Example 5 with the sample size increased to 30 students. Compare your answer with that of Example 5. Answer: Page A40 After constructing a confidence interval, it is important that you interpret the results correctly. Consider the 90% confidence interval constructed in Example 5. Because m is a fixed value predetermined by the population, it is either in the interval or not. It is not correct to say, “There is a 90% probability that the actual mean will be in the interval (22.3, 23.5).” This statement is wrong because it suggests that the value of m can vary, which is not true. The correct way to interpret this confidence interval is to say, “With 90% confidence, the mean is in the interval (22.3, 23.5).” This means that when many samples are collected and a confidence interval is created for each sample, approximately 90% of these intervals will contain m, as shown in the figure at the left. This correct interpretation refers to the success rate of the process being used. See TI-84 Plus steps on page 345. EXAMPLE 5 Tech Tip Here are instructions for constructing a confidence interval in Excel. First, click Formulas at the top of the screen and click Insert Function in the Function Library group. Select the category Statistical and select the CONFIDENCE.NORM function. In the dialog box, enter the values of alpha, the standard deviation, and the sample size (see below). Then click OK. The value returned is the margin of error, which is used to construct the confidence interval. 0.551700678 A 1 =CONFIDENCE.NORM(0.1,1.5,20) Alpha is the level of significance, which will be explained in Chapter 7. When using Excel in Chapter 6, you can think of alpha as the complement of the level of confidence. So, for a 90% confidence interval, alpha is equal to 1 - 0.90 = 0.10.
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