Question

At a major credit card bank, the percentages of people who historically apply for the Silver, Gold and Platinum cards are 60%, 30% and 10% respectively. In a recent sample of customers responding to a promotion, of 200 customers, 118 applied for Silver, 50 for Gold and 32 for Platinum. Is there evidence to suggest that the percentages for this promotion may be different from the historical proportions? a) What is the expected number of customers applying for each type of card in this sample if the historical proportions are still true? b) Compute the x2 statistic. c) How many degrees of freedom does the x2 statistic have? a) The expected number of silver card customers is (Type an integer or a decimal.) The expected number of gold card customers is (Type an integer or a decimal.) The expected number of platinum card customers is (Type an integer or a decimal.) b) x2 (Round to two decimal places as needed.) c) df = (Type an integer or a decimal.)

Answer 1

We have to test

H0: percentages for this promotion is same as historical proportions

H1: percentages for this promotion are significantly different from historical proportions

Card type	Observed number	Expected percentages	Expected number(total number*expected percentage/100)
Silver	118	60	120
Gold	50	30	60
Platinum	32	10	20
Total	200	100	200

The chi-square test statistic is given by:

(o;-) == x² ei

observed number	expected number	(o-e)2/e
118	120	0.033333
50	60	1.666667
32	20	7.2
200	200	= 8.9 2 x

The test statistic is 8.9

2 x

We have observations for three categories that means we have n=3

The degrees of freedom is n-1 = 3-1 = 2

Based on the above test we can answer the given question:

Answer(a):

The expected number of silver card customers is 120

The expected number of gold card customers is 60

The expected number of platinum card customers is 20

Answer(b):

= 8.9

2 x

Answer(c):

df=2