Question

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 1

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 $\pi$, and a confidence interval $1-\alpha$, we have the following confidence interval of proportions.

$\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}$

In which

Z is the zscore that has a pvalue of $1 - \frac{\alpha}{2}$.

For this problem, we have that:

511 homes were surveyed, of which 178 met the minimum recommendations for earthquake preparedness. This means that $n = 511$ and $p = \frac{178}{511} = 0.3483$.

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 $\alpha = 0.10$, z is the value of Z that has a pvalue of $1 - \frac{0.10}{2} = 0.95$, so $z = 1.645$.

The lower limit of this interval is:

$\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.3483 - 1.645\sqrt{\frac{0.3483*0.6517}{511}} = 0.3136$

The upper limit of this interval is:

$\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.3483 + 1.645\sqrt{\frac{0.3483*0.6517}{511}} = 0.3830$

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).