Question

A market research firm used a sample of individuals to rate the purchase potential of a particular product before and after the individuals saw a new television commercial about the product. The purchase potential ratings were based on a 0 to 10 scale, with higher values indicating a higher purchase potential. The null hypothesis stated that the mean rating "after" would be less than or equal to the mean rating "before." Rejection of this hypothesis would show that the commercial improved the mean purchase potential rating. Use

α = 0.05

and the following data to test the hypothesis and comment on the value of the commercial.

Individual	Purchase Rating
	After	Before
1	6	5
2	6	5
3	7	7
4	4	3
5	3	7
6	9	8
7	7	5
8	6	6

Answer 1

Let be the  true mean rating "before" and    be the true mean rating "after" the individuals saw a new television commercial about the product.

The null hypothesis stated that the mean rating "after" would be less than or equal to the mean rating "before." Rejection of this hypothesis would show that the commercial improved the mean purchase potential rating.

That is we want to test the following hypotheses

Since the same set of individuals are used to collect the before and after ratings, this is a paired sample and hence we will use the test for difference between the means for dependent (paired) samples.

Let be the mean improvement in the rating measured as the difference between rating after and rating before viewing the commercial.

We can rewrite the hypotheses as

From the sample calculate the difference between the rating as below

individual	Purchase Rating
	After	Before	Difference (after - before) (d)
1	6	5	1
2	6	5	1
3	7	7	0
4	4	3	1
5	3	7	-4
6	9	8	1
7	7	5	2
8	6	6	0

Next we calculate the following

n=8 is the sample size

The sample mean difference is

The sample standard deviation of difference is

We estimate the unknown population standard deviation as

The standard error of mean difference is

is the hypothesized value of mean difference (from the null hypothesis)

We will use t test as the sample size is less than 30 and we do not know the population standard deviation.

the test statistics is

The degrees of freedom for t statistics is n-1=8-1=7

This is a right tailed test (from the alternative hypothesis having ">"). The critical value of t is such that the area under the right tail is 0.05. The critical value of t for alpha=0.05 is

Using table df=7 and area under the right tail=0.05 we get the critical value of t=1.895

We will reject the null hypothesis if the test statistics is greater than the critical value.

Here the test statistics is 0.386 and it is less than the critical value 1.895. Hence we do not reject the null hypothesis.

We conclude that at 5% level of significance, there is no sufficient evidence to support the claim that he commercial improved the mean purchase potential rating.

The new commercial does not seem to improve the  purchase potential of this particular product.