358 Chapter 8 Confidence Intervals The confidence level measures the success rate of the method used to construct the confidence interval The center diagram in Figure 8.8 presents 100 different 95% confidence intervals. When we construct a confidence interval with level 95%, we are essentially getting a look at one of these confidence intervals. However, we don’t get to see any of the other confidence intervals, nor do we get to see the vertical line that indicates where the true value �� is. Therefore, we cannot be sure whether we got one of the confidence intervals that covers ��, or whether we were unlucky enough to get one of the unsuccessful ones. What we do know is that our confidence interval was constructed by a method that succeeds 95% of the time. The confidence level describes the success rate of the method used to construct a confidence interval, not the success of any particular interval. Check Your Understanding 11. To determine how well a new method of teaching vocabulary is working in a certain elementary school, education researchers plan to give a vocabulary test to a sample of 100 sixth-graders. It is known that scores on this test have a standard deviation of 8. The researchers plan to compute the sample mean ̄x, then construct a 95% confidence interval for the population mean test score. a. What is the critical value z��∕2 for this confidence interval? 1.96 b. Find the margin of error for this confidence interval. 1.568 c. Let m represent the margin of error for this confidence interval. For what percentage of all samples will the confidence interval ̄x ± m cover the true population mean? 95% 12. The researchers now plan to construct a 99% confidence interval for the test scores described in Exercise 11. a. What is the critical value z��∕2 for this confidence interval? 2.576 b. Find the margin of error for this confidence interval. 2.061 c. Let m represent the margin of error for this confidence interval. For what percentage of all samples will the confidence interval ̄x ± m cover the true population mean? 99% Answers are on page 366. Objective 4 Find the sample size necessary to obtain a confidence interval of a given width Finding the Necessary Sample Size We have seen that we can make the margin of error smaller if we are willing to reduce our level of confidence. We can also reduce the margin of error by increasing the sample size. We can see this by looking at the formula for margin of error: m = z��∕2 �� √ n Since the sample size n appears in the denominator, making it larger will make the value of m smaller. We will show how we can manipulate this formula using algebra to express the sample size n in terms of the margin of error m. m = z��∕2 �� √ n √ n = m z��∕2 ⋅ �� √ n √ n (Multiply both sides by √ n ) √ n m m = z��∕2 ⋅ �� m (Divide both sides by m) n = (z��∕2 ⋅ �� m )2 (Square both sides) With this formula, if we know how small we want the margin of error to be, we can compute the sample size needed to achieve the desired margin of error.
navidi_monk_elementary_statistics_2e_ch7-9
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