Question

A market researcher working for a bank wants to know if the distribution of applications by card is the same for the past three mailings. She takes a random sample of 200 from each mailing and counts the number applying for​ Silver, Gold, and Platinum. The reported results appear in the accompanying table. Complete parts a through g below.

	silver	gold	platinum	total
mailing 1	119	55	26	200
mailing 2	120	55	25	200
mailing 3	130	54	16	200
total	369	164	67	600

​a) What are the null and alternative​ hypotheses? Choose the correct answer below.

A.

Upper H 0H0​:

The proportions of customers applying for the three types of cards are the same for the three different mailings.

Upper H Subscript Upper AHA​:

The proportions of customers applying for the three types of cards are not the same for the three different mailings.

B.

Upper H 0H0​:

The proportions of customers applying for the three types of cards are not the same for the three different mailings.

Upper H Subscript Upper AHA​:

The proportions of customers applying for the three types of cards are the same for the three different mailings.

C.

Upper H 0H0​:

Card type and mailing are independent.

Upper H Subscript Upper AHA​:

Card type and mailing are not independent.

D.

Upper H 0H0​:

Card type and mailing are not independent.

Upper H Subscript Upper AHA​:

Card type and mailing are independent.

​b) What type of test is​ this?

A.

​Chi-square test of homogeneity

B.

​Chi-square goodness-of-fit test

C.

​Chi-square test of independence

​c) What are the expected numbers for each cell if the null hypothesis is​ true?

	SilverSilver	GoldGold	PlatinumPlatinum
Mailing 1Mailing 1
Mailing 2Mailing 2
Mailing 3Mailing 3	nothing	nothing	nothing

​(Round to three decimal places as​ needed.)

​d) Find the chi squaredχ2 statistic.

chi squaredχ2=__

​(Round to two decimal places as​ needed.)​e) How many degrees of freedom does the chi squaredχ2 statistic​ have?

df=___

​(Type an integer or a​ decimal.)​f)

Find the critical value at alphaαequals=0.05.

The critical value is __

​(Round to two decimal places as​ needed.)

​g) What do you​ conclude?

A.The test statistic is greater than the critical value. Reject Upper H 0H0 and conclude there is sufficient evidence at the​ 5% significance level that the distributions are different for the three mailings.

B.The test statistic is greater than the critical value. Reject Upper H 0H0 and conclude there is sufficient evidence at the​ 5% significance level that card type is not independent of mailing

C.The test statistic is less than the critical value. Fail to reject Upper H 0H0 and conclude there is insufficient evidence at the​ 5% significance level that card type is not independent of mailing.

D.The test statistic is less than the critical value. Fail to reject Upper H 0H0 and conclude there is insufficient evidence at the​ 5% significance level that the distributions are different for the three mailings.

Answer 1

(a) The null and alternative hypotheses are: Option(A)

H0: The proportions of customers applying for the three types of cards are the same for the three different mailings.

HA: The proportions of customers applying for the three types of cards are not the same for the three different mailings.

(b) The type of test is: Option(A)

Chi-square test of homogeneity.

(c) The expected numbers for each cell if the null hypothesis is true are:

	Silver	Gold	Platinum
Mailing 1	123.000	54.667	22.333
Mailing 2	123.000	54.667	22.333
Mailing 3	123.000	54.667	22.333

(d) The chi-squared statistic:

χ^{​​​​​​2} = 3.330

(e) Degrees of freedom:

df = 4

(f) The critical value at α = 0.05:

The critical value is 9.49

(g) The conclusion is: Option(D)

The test statistic is less than the critical value. Fail to reject H0 and conclude there is insufficient evidence at the​ 5% significance level that the distributions are different for the three mailings.