Question

1. A direct-mail retailer experimented with three different ways of incorporating order forms into its catalog. In type 1 catalogs, the form was at the end of the catalog; in type 2, it was in the middle; and in type 3, there were forms both in the middle and at the end. Each form was sent to a sample of 1,000 potential customers, none of whom had previously bought from the retailer. The observed contingency table is given below:

Received	Type 1	Type 2	Type 3
No	944	961	915
Yes	56	39	85

a)   Is there significant evidence that the proportion of order forms received differs for the three types of forms, specify null and alternative hypothesis?

b) Compare the actual cell counts with the expected cell counts to explain where the dependency from part a) comes from

c) Calculate a 95% confidence interval for the probability of receiving the form back, for each of the three methods.

Answer 1

(a) The hypothesis being tested is:

H0: The proportion of order forms received is the same for the three types of forms

Ha: The proportion of order forms received differs for the three types of forms

(b)

observed	expected	O - E	(O - E)² / E
56	60.000	-4.000	0.267
39	60.000	-21.000	7.350
85	60.000	25.000	10.417
180	180.000	0.000	18.033
18.03	chi-square
2	df
.0001	p-value

The p-value is 0.0001.

Since the p-value (0.0001) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the proportion of order forms received differs for the three types of forms.

(c) The 95% confidence interval for the probability of receiving the form back, for type 1 is between 0.0417 and 0.0703.

Observed
0.056	p (as decimal)
56/1000	p (as fraction)
56.	X
1000	n
0.0158	std. error
0.0417	confidence interval 95.% lower
0.0703	confidence interval 95.% upper
0.0143	margin of error

The 95% confidence interval for the probability of receiving the form back, for type 2 is between 0.027 and 0.051.

Observed
0.039	p (as decimal)
39/1000	p (as fraction)
39.	X
1000	n
0.0158	std. error
0.027	confidence interval 95.% lower
0.051	confidence interval 95.% upper
0.012	margin of error

The 95% confidence interval for the probability of receiving the form back, for the type 3 is between 0.0677 and 0.1023.

Observed
0.085	p (as decimal)
85/1000	p (as fraction)
85.	X
1000	n
0.0158	std. error
0.0677	confidence interval 95.% lower
0.1023	confidence interval 95.% upper
0.0173	margin of error

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