Question

If you live in California, the decision to purchase earthquake insurance is a critical one. An article in the Annals of the Association of American Geographers investigated many factors that California residents consider when purchasing earthquake insurance. The survey revealed that only 141 of 345 randomly selected residences in Los Angeles County were protected by earthquake insurance. Estimate with 90% confidence the proportion of all L.A. County residences that have earthquake insurance. (Round to 3 decimal places.) Complete the following sentence to provide an interpretation of the confidence interval in the context of the problem_________.

Answer 1

Answer with explanation:

Let p be the  proportion of all L.A. County residences that have earthquake insurance.

Given : The survey revealed that only 141 of 345 randomly selected residences in Los Angeles County were protected by earthquake insurance.

i.e. n= 345

Sample Proportion : $\hat{p}=\dfrac{141}{345}\approx0.409$

Critical value for 90% confidence interval : $z_c=1.645$

Formula to find the confidence interval for population proportion :

$\hat{p}\pm z_c\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

$0.409\pm (1.645)\sqrt{\dfrac{0.409(1-0.409)}{345}}\\\\\approx0.409\pm 0.044\\\\=(0.409-0.044, \ 0.409+0.044)\\\\=(0.369,\ 0.453)$

Hence, the 90% confidence the proportion of all L.A. County residences that have earthquake insurance = (0.369, 0.453)

Interpretation : We are 90% confident that the true proportion of all L.A. County residences that have earthquake insurance lies between 0.369 and 0.453 .