7-2 Estimating a Population Mean 339 3. Use Prior Results Estimate the value of s by using the results of some other earlier study. In addition, we can sometimes be creative in our use of other known results. For example, Wechsler IQ tests are designed so that the standard deviation is 15. Statistics students have IQ scores with a standard deviation less than 15, because they are a more homogeneous group than people randomly selected from the general population. We do not know the specific value of s for statistics students, but we can be safe by using s = 15. Using a value for s that is larger than the true value will make the sample size larger than necessary, but using a value for s that is too small would result in a sample size that is inadequate. When determining the sample size n, any errors should always be conservative in the sense that they make the sample size too large instead of too small. EXAMPLE 5 IQ Scores of Statistics Students Assume that we want to estimate the mean IQ score for the population of statistics students. How many statistics students must be randomly selected for IQ tests if we want 95% confidence that the sample mean is within 3 IQ points of the population mean? SOLUTION For a 95% confidence interval, we have a = 0.05, so za>2 = 1.96. Because we want the sample mean to be within 3 IQ points of m, the margin of error is E = 3. Also, we can assume that s = 15 (see the discussion that immediately precedes this example). Using Formula 7-4, we get n = c za>2s E d 2 = c 1.96# 15 3 d 2 = 96.04 = 97 (rounded up) INTERPRETATION Among the thousands of statistics students, we need to obtain a simple random sample of at least 97 of their IQ scores. With a simple random sample of only 97 statistics students, we will be 95% confident that the sample mean x is within 3 IQ points of the true population mean m. YOUR TURN. Do Exercise 23 “Ages of Moviegoers.” Means: Confidence Intervals and Sample Size Determination Access tech supplements, videos, and data sets at www.TriolaStats.com TECH CENTER Confidence Interval 1. Click Analysis in the top menu. 2. Select Confidence Intervals from the dropdown menu and select Mean One Sample from the submenu. 3. Using Summary Statistics: Select the Use Summary Statistics tab and enter the desired confidence level, sample size, sample mean, and sample standard deviation. Using Sample Data: Select the Use Data tab and select the desired data column. 4. Click Evaluate. TIP: Statdisk will automatically choose between the normal and t distributions, depending on whether a value for the population standard deviation is entered. Sample Size Determination 1. Click Analysis in the top menu. 2. Select Sample Size Determination from the dropdown menu and select Estimate Mean from the submenu. 3. Enter the confidence level, margin of error E, and population standard deviation. Also enter the known population size if you are sampling without replacement from a finite population. 4. Click Evaluate. Statdisk rGo Figure Robocalls: In a recent year, there were 1517 robocalls made every second. There were 47,839,232,200 robocalls made in the year. 39.36% of the robocalls were scams. continued
RkJQdWJsaXNoZXIy NjM5ODQ=