Question

4. A manager for an insurance company believes that customers have the following preferences for life insurance products: 40% prefer Whole Life, 30% prefer Universal Life, and 30% prefer Life Annuities. The results of a survey of 320 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         24
Universal   96
Annuities   200

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

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.

or

H0: Preferences for life insurance products are not as per the manager's belief.

Ha: Preferences for life insurance products are 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?

a. The proportions of customers who prefer each insurance product are all thought to be equal.
b. 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: State the null and alternative hypothesis in terms of the expected proportions for each category.

P whole=

P universal=

P 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.

E( whole) =

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

E (annuities)

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 0.05level of significance.

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

Answer 1

1)
A)
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.

2)
b. The proportions of customers who prefer each insurance product are different for each category

3)
P whole= 0.4

P universal=0.3

P annuities =0.3
4)
   Oi   Ei   (Oi-Ei)^2/Ei
0.4   24   128   84.5
0.3   96   96   0
0.3   200   96   112.6666667
          
1   320   TS   197.1666667
       critical   5.991464547
       p-value   1.53392E-43
E(whole) = N * p_whole = 320*0.4 = 128

5)
E(annuities) = 96

6)
TS = 197.1666

7)
df = r-1 = 2
8)
critical value = 5.9914
9)
TS > critical value
hence reject the null hypothesis

10)
we conclude that
b)
. There is enough evidence to refute the manager's claim about the proportions of customers who prefer each product.