380 Chapter 8 Confidence Intervals EXAMPLE 8.14 Construct a confidence interval for a proportion In a survey of 517 music teachers, 403 said that the video games Guitar Hero and Rock Band have a positive effect on music education. a. Construct a 95% confidence interval for the proportion of music teachers who believe that these video games have a positive effect. b. A video game manufacturer claims that 80% of music teachers believe that Guitar Hero and Rock Band have a positive effect. Does the confidence interval contradict this claim? Solution a. We begin by checking the assumptions. We have a simple random sample. It is reasonable to believe that the population of music teachers in the United States is at least 20 times as large as the sample. The items in the population can be divided into two categories: those who believe that the games have a positive effect, and those who do not. There are 403 teachers who believe that the games have a positive effect, and 517 − 403 = 114 who do not, so there are 10 or more items in each category. The assumptions are met. Step 1: Compute the point estimate ̂p. The sample size is n = 517 and the number who believe that video games have a positive effect is x = 403. The point estimate is ̂p = 403 517 = 0.779497 Step 2: Find the critical value. The critical value for a 95% confidence interval is z��∕2 = 1.96. Step 3: Compute the margin of error. The margin of error is z��∕2 √ ̂p(1 − ̂p) n We substitute z��∕2 = 1.96, ̂p = 0.779497, and n = 517 to obtain z��∕2 √ ̂p(1 − ̂p) n √ 0.779497(1 − 0.779497) = 1.96 517 = 0.035738 Step 4: Construct the 95% confidence interval. The point estimate is 0.779497 and the margin of error is 0.035738. The 95% confidence interval is 0.779497 − 0.035738 < p < 0.779497 + 0.035738 0.744 < p < 0.815 (rounded to three decimal places) Sept 5: Interpret the result. We are 95% confident that the proportion of music teachers who believe that video games have a positive effect is between 0.744 and 0.815. b. Because the value 0.80 is within the confidence interval, the confidence interval does not contradict the claim. Constructing confidence intervals with technology Example 8.14 presented a 95% confidence interval for the proportion of music teachers who believe that video games have a positive effect on music education. The confidence interval obtained was 0.744 < p < 0.815. Following are the results from a TI-84 Plus calculator.
navidi_monk_elementary_statistics_2e_ch7-9
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