Question

A manager for an insurance company believes that customers have the following preferences for life insurance products: 30% prefer Whole Life, 30% prefer Universal Life, and 40% prefer Life Annuities. The results of a survey of 330 customers were tabulated. Is it possible to refute the sales manager's claimed proportions of customers who prefer each product using the data?

Product	Number
Whole	109
Universal	96
Annuities	125

 

Step 1 of 10:

State the null and alternative hypothesis.

Answer

 

H0H0: Preferences for life insurance products are not as per the manager's belief.

HaHa: Preferences for life insurance products are as per the manager's belief.

H0H0: Preferences for life insurance products are as per the manager's belief.

HaHa: Preferences for life insurance products are not as per the manager's belief.

Step 2 of 10:

What does the null hypothesis indicate about the proportions of customers who prefer each insurance product?

Answer

 

The proportions of customers who prefer each insurance product are all thought to be equal.
The proportions of customers who prefer each insurance product are different for each category (and equal to the previously accepted values).

State the null and alternative hypothesis in terms of the expected proportions for each category.

 

H0H0:      pWhole=pWhole=,  pUniversal=pUniversal=,  pAnnuities=pAnnuities=HaHa: There is some difference amongst the proportions.

Step 4 of 10:

Find the expected value for the number of customers who prefer Whole Life. Round your answer to two decimal places.

ind the expected value for the number of customers who prefer Universal Life. Round your answer to two decimal places.

Step 6 of 10:

Find the value of the test statistic. Round your answer to three decimal places

Step 7 of 10:

Find the degrees of freedom associated with the test statistic for this problem.

Find the critical value of the test at the 0.05 level of significance. Round your answer to three decimal places.

Make the decision to reject or fail to reject the null hypothesis at the 0.05 level of significance.

 

Fail to Reject Null Hypothesis Reject Null Hypothesis

Step 10 of 10:

State the conclusion of the hypothesis test at the 0.05 level of significance.

 

There is not enough evidence to refute the manager's claim about the proportions of customers who prefer each product.
There is enough evidence to refute the manager's claim about the proportions of customers who prefer each product.

Answer 1

Step 1 of 10:

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H0​: Preferences for life insurance products are as per the manager's belief

Ha​: Preferences for life insurance products are not as per the manager's belief.

Step 2 of 10:

The null hypothesis indicate that "the proportions of customers who prefer each insurance product are different for each category (and equal to the previously accepted values)."

Step 3 of 10:

The null and alternative hypothesis in terms of the expected proportions for each category.

H0: p_Whole​ = 0.30, p_Universal​ = 0.30, p_Annuities​ = 0.40

Ha: There is some difference amongst the proportions.

Step 4 of 10:

From the table, we can see that the expected value for the number of customers who prefer Whole Life is 99

Step 5 of 10:

From the table, we can see that the expected value for the number of customers who prefer Universal Life is 99

Step 6 of 10:

Check table for calculation

Step 7 of 10:

The number of degrees of freedom is df = 3 - 1 = 2

Step 8 of 10:

Critical value corresponding to α = 0.05 and df = 2 can be obtained using online calculator

Critical value is 5.991 (Screenshot of online calculator attached)

Step 9 of 10:

Since it is observed that χ² = 1.472 ≤ χ_c²​ = 5.991, we fail to reject the null hypothesis.

Step 10 of 10:

Therefore, there is not enough evidence to refute the manager's claim about the proportions of customers who prefer each product, at the α=0.05 significance level.