36 out of 100 randomly selected taxpayers knew about tax incentives for installing energy-saving furnaces. Find a 90% confidence interval for the population proportion who knew about the incentives. (Round to two places after the decimal point.)

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Question:

36 out of 100 randomly selected taxpayers knew about tax incentives for installing energy-saving furnaces. Find a 90% confidence interval for the population proportion who knew about the incentives. (Round to two places after the decimal point.)

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Answer:The 90% confidence interval for the population proportion who knew about the incentives is (0.28, 0.44).Step-by-step explanation:In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]In whichz is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].For this problem, we have that:[tex]n = 100, \pi = \frac{36}{60} = 0.6[/tex]90% confidence levelSo [tex]\alpha = 0.1[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].The lower limit of this interval is:[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.36 - 1.645\sqrt{\frac{0.36*0.64}{100}} = 0.28[/tex]The upper limit of this interval is:[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.36 + 1.645\sqrt{\frac{0.36*0.64}{100}} = 0.44[/tex]The 90% confidence interval for the population proportion who knew about the incentives is (0.28, 0.44).

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