Question

A random sample of 160 car purchases are selected and categorized by age. The results are listed below. The age distribution of drivers for the given categories is 18% for the under 26 group, 39% for the 26-45 group, 31% for the 45-65 group, and 12% for the group over 65. Calculate the chi-square test statistic to test the claim that all ages have purchase rates proportional to their driving rates. Use α = 0.05.

Answer 1

Answer:

The claim that all ages have purchase rates proportional to their driving rates is false.

Step-by-step explanation:

The complete question is:

A random sample of 160 car accidents are selected and categorized by the age of the driver determined to be at fault. The results are listed below. The age distribution of drivers for the given categories is 18% for the under 26 group, 39% for the 26-45 group, 31% for the 45-65 group, and 12% for the group over 65. Calculate the chi-square test statistic used to test the claim that all ages have crash rates proportional to their driving rates.

Age      >26     26-45       46-65      45<

Drivers 66    39            25          30

  A) 95.431

      B)101.324

      C)85.123

      D)75.101

Solution:

In this case we need to test whether there is sufficient evidence to warrant rejection of the claim that all ages have crash rates proportional to their driving rates.

A Chi-square test for goodness of fit will be used in this case.

The hypothesis can be defined as:

H₀: The observed frequencies are same as the expected frequencies.

Hₐ: The observed frequencies are not same as the expected frequencies.

The test statistic is given as follows:

$\chi^{2}=\sum{\frac{(O-E)^{2}}{E}}$

The values are computed in the table.

The test statistic value is 75.10.

The degrees of freedom of the test is:

n - 1 = 4 - 1 = 3

Compute the p-value of the test as follows:

p-value < 0.00001

*Use a Chi-square table.

p-value < 0.00001 < α = 0.05.

So, the null hypothesis will be rejected at any significance level.

Thus, there is sufficient evidence to warrant rejection of the claim that ages have crash rates proportional to their driving rates.