Question

A company wants to determine whether its consumer product ratings

​(0minus−​10)

have changed from last year to this year. The table below shows the​ company's product ratings from eight consumers for last year and this year. At

alphaαequals=​0.05,

is there enough evidence to conclude that the ratings have​ changed? Assume the samples are random and​ dependent, and the population is normally distributed. Complete parts​ (a) through​ (f).

Consumer	1	2	3	4	5	6	7	8
Rating left parenthesis last year right parenthesisRating (last year)	66	66	44	55	77	99	55	55
Rating​ (this year)	88	88	33	77	66	88	77	88

Answer 1

The following table is obtained:

	Sample 1	Sample 2	Difference = Sample 1 - Sample 2
	66	88	-22
	66	88	-22
	44	33	11
	55	77	-22
	77	66	11
	99	88	11
	55	77	-22
	55	88	-33
Average	64.625	75.625	-11
St. Dev.	17.079	18.996	18.593
n	8	8	8

For the score differences, we have

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

This corresponds to a two-tailed test, for which a t-test for two paired samples be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=7.

Hence, it is found that the critical value for this two-tailed test is tc​=2.365, for α=0.05 and df = 7

The rejection region for this two-tailed test is R={t:∣t∣>2.365}.

(3) Test Statistics

The t-statistic is computed as shown in the following formula:

(4) Decision about the null hypothesis

Since it is observed that |t∣=1.673≤tc​=2.365, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.1382, and since p=0.1382≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is different than μ2​, at the 0.05 significance level.

Graphically

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