322 CHAPTER 6 Confidence Intervals Constructing a Confidence Interval for p Use the data in Example 1 to construct a 95% confidence interval for the population proportion of U.S. adults who plan on traveling this summer. SOLUTION From Example 1, np = 0.70. So, the point estimate for the population proportion of failures is nq = 1 - 0.70 = 0.30. Using n = 540, you can verify that the sampling distribution of np can be approximated by a normal distribution. nnp = 1540210.702 = 378 7 5 and nnq = 1540210.302 = 162 7 5 Using zc = 1.96, the margin of error is E = zcBnpnq n = 1.96B10.70210.302 540 ≈ 0.039. Next, find the left and right endpoints and form the 95% confidence interval. Left Endpoint Right Endpoint np - E ≈ 0.70 - 0.039 = 0.661 np + E ≈ 0.70 + 0.039 = 0.739 0.661 6 p 6 0.739 x 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.70 0.661 0.739 You can check this answer using technology, as shown below. (When using technology, your answers may differ slightly from those found using Table 4.) STATCRUNCH 95% confidence interval results: Proportion Count Total Sample Prop. Std. Err. L. Limit U. Limit p 378 540 0.7 0.019720 0.66135 0.73865 Interpretation With 95% confidence, you can say that the population proportion of U.S. adults who plan on traveling this summer is between 66.1% and 73.9%. TRY IT YOURSELF 2 Use the data in Try It Yourself 1 to construct a 90% confidence interval for the population proportion of U.S. adults who are very concerned about how much personal data their smart speakers collect. Answer: Page A40 The confidence level of 95% used in Example 2 is typical of opinion polls. The result, however, is usually not stated as a confidence interval. Instead, the result of Example 2 would be stated as shown. A survey found that 70% of U.S. adults plan on traveling this summer. The margin of error for the survey is {3.9%. Minitab and TI-84 Plus steps are shown on pages 344 and 345. EXAMPLE 2 Study Tip Notice in Example 2 that the confidence interval for the population proportion p is rounded to three decimal places. This round-off rule will be used throughout the text.
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