Question

California insurance companies wanted to study factors (e.g., the proximity to a major earthquake fault line) that may influence homeowners’ decisions to purchase earthquake insurance. Surveys were mailed to randomly selected households in three California counties to investigate the possible proximity effect. The data collected are shown below:

Sample size

1000- Los Angeles

1200- San Barnadino

1400-Santa Clara

Numbers with earthquake insurance

377-Los Angekes

469-San Bernadino

390-Santa Clara

1. Los Angeles County is located closest among the three to a major earthquake fault line. Are homeowners in Los Angeles County more likely to purchase earthquake insurance than those in San Bernardino County? Test using α = 0.05.

Answer 1

For LOS ANGELES, we have that the sample size is N1​=1000, the number of favorable cases is X1​=377, so then the sample proportion is p^​1​=​X1/N1​​=377/1000​=0.377

For San Bernadino, we have that the sample size is N2​=1200, the number of favorable cases is X2​=469, so then the sample proportion is p^​2​=X2/N2​​=469/1200​=0.3908

The value of the pooled proportion is computed as pˉ​=(X1​+X2)/N1​+N2​ ​​= (377+469)/1000+1200 ​=0.3845

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

NULL HYPOTHESIS Ho:p1​=p2​

ALTERNATIVE HYPOTHESIS Ha:p1​>p2​

This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is zc​=1.64.

The rejection region for this right-tailed test is R={z:z>1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis: Since it is observed that z=−0.664≤zc​=1.64, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.7467, and since p=0.7467≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion: It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion p1​for LOS ANGELES is greater than p2 FOR San Bernadino​, at the 0.05 significance level.