Question

A manager for an insurance company believes that customers have the following preferences for life insurance products: 20% prefer Whole Life, 10% prefer Universal Life, and 70% prefer Life Annuities. The results of a survey of 200 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	94
Universal	68
Annuities	38

Step 1 of 10: State the null and alternative hypothesis.

Step 2 of 10: What does the null hypothesis indicate about the proportions of customers who prefer each insurance product?

Step 3 of 10: State the null and alternative hypothesis in terms of the expected proportions for each category.

Step 4 of 10: Find the expected value for the number of customers who prefer Whole Life. Round your answer to two decimal places.

Step 5 of 10: Find 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.

Step 8 of 10: Find the critical value of the test at the 0.05 level of significance. Round your answer to three decimal places.

Step 9 of 10: Make the decision to reject or fail to reject the null hypothesis at the 0.05 level of significance.

Step 10 of 10: State the conclusion of the hypothesis test at the 0.05 level of significance.

Answer 1

Step 1 of 10: State the null and alternative hypothesis.

The hypothesis being tested is:

H₀: The proportions of customers who prefer each insurance product is the same

H_a: 20% prefer Whole Life, 10% prefer Universal Life, and 70% prefer Life Annuities

Step 2 of 10: What does the null hypothesis indicate about the proportions of customers who prefer each insurance product?

The proportions of customers who prefer each insurance product is the same

Step 3 of 10: State the null and alternative hypothesis in terms of the expected proportions for each category.

The hypothesis being tested is:

H₀: p₁ = p₂ = p₃

H_a: 20% prefer Whole Life, 10% prefer Universal Life, and 70% prefer Life Annuities

Step 4 of 10: Find the expected value for the number of customers who prefer Whole Life. Round your answer to two decimal places.

40

Step 5 of 10: Find the expected value for the number of customers who prefer Universal Life. Round your answer to two decimal places.

20

Step 6 of 10: Find the value of the test statistic. Round your answer to three decimal places.

262.414

Step 7 of 10: Find the degrees of freedom associated with the test statistic for this problem.

2

Step 8 of 10: Find the critical value of the test at the 0.05 level of significance. Round your answer to three decimal places.

5.991

Step 9 of 10: Make the decision to reject or fail to reject the null hypothesis at the 0.05 level of significance.

Since 262.414 > 5.991, we can reject the null hypothesis.

Step 10 of 10: State the conclusion of the hypothesis test at the 0.05 level of significance.

Therefore, we can conclude that 20% prefer Whole Life, 10% prefer Universal Life, and 70% prefer Life Annuities.

	observed	expected	O - E	(O - E)² / E	% of chisq
Whole	94	40	54.000	72.900	27.78
Universal	68	20	48.000	115.200	43.90
Annuities	38	140	-102.000	74.314	28.32
	200	200.000	0.000	262.414	100.00
	262.414	chi-square
	2	df
	1.04E-57	p-value
	5.991	critical value