Question

A large car insurance company selected samples of single and married male policyholders and recorded the number who made an insurance claim over the preceding three-year period. Single Policyholders Married Policyholders n1 = 450 n2 = 925 # making claim = 67 # making claim = 93 Using alpha = 0.05, determine whether the claim rates are higher for single male policyholders verses married male policyholders. Solve using the p-value approach only.

Answer 1

Answer:

The null hypothesis is rejected.

There is enough evidence to support the claim that rates are higher for single male policyholders verses married male policyholders (P-value = 0.004).

Step-by-step explanation:

This is a hypothesis test for the difference between proportions.

The claim is that rates are higher for single male policyholders verses married male policyholders.

Then, the null and alternative hypothesis are:

$H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2> 0$

The significance level is 0.05.

The sample 1 (single group), of size n1=450 has a proportion of p1=0.1489.

$p_1=X_1/n_1=67/450=0.1489$

The sample 2 (married group), of size n2=925 has a proportion of p2=0.1005.

$p_2=X_2/n_2=93/925=0.1005$

The difference between proportions is (p1-p2)=0.0483.

$p_d=p_1-p_2=0.1489-0.1005=0.0483$

The pooled proportion, needed to calculate the standard error, is:

$p=\dfrac{X_1+X_2}{n_1+n_2}=\dfrac{67.005+93}{450+925}=\dfrac{160}{1375}=0.1164$

The estimated standard error of the difference between means is computed using the formula:

$s_{p1-p2}=\sqrt{\dfrac{p(1-p)}{n_1}+\dfrac{p(1-p)}{n_2}}=\sqrt{\dfrac{0.1164*0.8836}{450}+\dfrac{0.1164*0.8836}{925}}\\\\\\s_{p1-p2}=\sqrt{0.0002+0.0001}=\sqrt{0.0003}=0.0184$

Then, we can calculate the z-statistic as:

$z=\dfrac{p_d-(\pi_1-\pi_2)}{s_{p1-p2}}=\dfrac{0.0483-0}{0.0184}=\dfrac{0.0483}{0.0184}=2.62$

This test is a right-tailed test, so the P-value for this test is calculated as (using a z-table):

$P-value=P(z>2.62)=0.004$

As the P-value (0.004) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that rates are higher for single male policyholders verses married male policyholders.