Question

Based on information from a large insurance company, 67% of all damage liability claims are made by single people under the age of 25. A random sample of 52 claims showed that 43 were made by single people under the age of 25. Does this indicate that the insurance claims of single people under the age of 25 is higher than the national percent reported by the large insurance company? State the null and alternate hypothesis then give the test statistic and your conclusion.

Answer 1

Answer:

We conclude that the insurance claims of single people under the age of 25 is higher than the national percent reported by the large insurance company.

Step-by-step explanation:

We are given that based on information from a large insurance company, 67% of all damage liability claims are made by single people under the age of 25.

A random sample of 52 claims showed that 43 were made by single people under the age of 25.

Let p = population proportion of claims made by single people

So, Null Hypothesis, $H_0$ : p $\leq$ 67%     {means that the insurance claims of single people under the age of 25 is smaller than or equal to the national percent reported by the large insurance company}

Alternate Hypothesis, $H_A$ : p > 67%      {means that the insurance claims of single people under the age of 25 is higher than the national percent reported by the large insurance company}

The test statistics that will be used here is One-sample z-test for proportions;

                          T.S.  =  $\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n} } }$  ~ N(0,1)

where, $\hat p$ = sample proportion of claims made by single people under the age of 25 = $\frac{43}{52}$ = 0.83

           n = sample of claims = 52

So, the test statistics =  $\frac{0.83-0.67}{\sqrt{\frac{0.67(1-0.67)}{52} } }$  

                                    =  2.454

The value of z-test statistics is 2.454.

Since in the question, we are not given the level of significance so we assume it to be 5%. Now, at 0.05 level of significance, the z table gives a critical value of 1.645 for the right-tailed test.

Since the value of our test statistics is more than the critical value of z as 2.454 > 1.645, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.

Therefore, we conclude that the insurance claims of single people under the age of 25 is higher than the national percent reported by the large insurance company.