Question

Automobile collision insurance is used to pay for any claims made against the driver in the event of an accident. This type of insurance will typically pay to repair any assets that your vehicle damages. A random sample of 40 collision claims of 20- to 24-year-old drivers results in a mean claim of $4590 with standard deviation of $2302. An independent random sample of 40 collision claims of 30-to 59-year-old drivers results in a mean claim of $3669 with a standard deviation of $2029. Using the concept of hypothesis testing, determine if a higher insurance premium should be paid by 20- to 24-year-old drivers. Use a a=0.10 level of significance, and let population 1 be 20- to 24-year old drivers and population 2 be 30-to 59-year old drivers. Complete parts (a) through (e) below. (a) Collision claims tend to be skewed right. Why do you think this is the case? O A. There are no very large collision claims. B. There are many large collision claims relative to the majority of claims. C. There are a few very large collision claims relative to the majority of claims. (b) What type of test should be used? O A. A hypothesis test regarding the difference of two means using a matched-pairs design O B. A hypothesis test regarding the difference between two population proportions from independent samples O C. A hypothesis test regarding the difference of two means using Welch's approximatet O D. A hypothesis test regarding two population standard deviations

Answer 1

Collision claims tend to be skewed right due to a few very large claims. A hypothesis test regarding the difference of two means using Welch's approximate t-test should be used.

Step-by-step explanation:

(a) The correct answer is C. There are a few very large collision claims relative to the majority of claims. Skewness refers to the symmetry or lack of symmetry in a distribution. In this case, the distribution of collision claims is skewed right because there are a few very large claims that pull the mean higher than the majority of claims.

(b) The correct answer is C. A hypothesis test regarding the difference of two means using Welch's approximate t-test should be used. This is because we are comparing the means of two independent samples with different variances. Welch's t-test accounts for unequal variances in the two populations.

Answer 2

(a) C. There are a few very large collision claims relative to the majority of claims.

(b) C. A hypothesis test regarding the difference of two means using Welch's approximation.

Step-by-step explanation:

(a) The correct choice is C. In the context of collision claims being skewed right, it means there are a few very large collision claims relative to the majority of claims. This indicates that while most claims might be relatively moderate, there exist a handful of claims that are substantially larger, creating the skewness towards the higher end.

(b) The appropriate test to use in this scenario is a hypothesis test regarding the difference of two means using Welch's approximation. We're comparing the means of two independent samples (20- to 24-year-old drivers vs. 30-to 59-year-old drivers) with unknown and potentially unequal variances. Welch's test accommodates this situation and is robust even when assumptions regarding equal variances might not hold. It allows for an accurate comparison of means between these two groups.

To conduct this test, we first compute the test statistic, which is given by:

`\[ t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{(s_1^2)/(n_1) + (s_2^2)/(n_2)}} \]`

Where
`\( \bar{x}_1 \)` and
`\( \bar{x}_2 \)` are the sample means ,
`\( s_1 \)` and
`\( s_2 \)` are the sample standard deviations, and
`\( n_1 \)` and
`\( n_2 \)` are the sample sizes for the respective groups. Then, we compare this test statistic to the critical value from the t-distribution with degrees of freedom calculated using Welch's approximation. If the calculated t-value falls beyond the critical value at a significance level of 0.10, we reject the null hypothesis, suggesting that a higher insurance premium should be paid by 20- to 24-year-old drivers.