Five hundred eleven (511) homes in a certain southern California community are randomly surveyed to determine if they meet minimal earthquake preparedness recommendations. One hundred seventy-eight (178) of the homes surveyed met the minimum recommendations for earthquake preparedness and 333 did not. Find the confidence interval at the 90% confidence level for the true population proportion of southern California community homes meeting at least the minimum recommendations for earthquake preparedness. (Round your answers to three decimal places.)

Answer:The confidence interval at the 90% confidence level for the true population proportion of southern California community homes meeting at least the minimum recommendations for earthquake preparedness is (0.3136, 0.3830).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 interval [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:511 homes were surveyed, of which 178 met the minimum recommendations for earthquake preparedness. This means that [tex]n = 511[/tex] and [tex]p = \frac{178}{511} = 0.3483[/tex].Find the confidence interval at the 90% confidence level for the true population proportion of southern California community homes meeting at least the minimum recommendations for earthquake preparedness.So [tex]\alpha = 0.10[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.10}{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}} = 0.3483 - 1.645\sqrt{\frac{0.3483*0.6517}{511}} = 0.3136[/tex]The upper limit of this interval is:[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.3483 + 1.645\sqrt{\frac{0.3483*0.6517}{511}} = 0.3830[/tex]The confidence interval at the 90% confidence level for the true population proportion of southern California community homes meeting at least the minimum recommendations for earthquake preparedness is (0.3136, 0.3830).