Question

The manufacturer of an MP3 player wanted to know whether a 10% reduction in price is enough to increase the sales of its product. To investigate, the owner randomly selected eight outlets and sold the MP3 player at the reduced price. At seven randomly selected outlets, the MP3 player was sold at the regular price. Reported below is the number of units sold last month at the regular and reduced prices at the randomly selected outlets.

Regular price	132	127	86	116	117	123	99
Reduced price	122	138	155	136	115	121	134	127

At the 0.100 significance level, can the manufacturer conclude that the price reduction resulted in an increase in sales? Hint: For the calculations, assume reduced price as the first sample.

1. Compute the pooled estimate of the variance. (Round your answer to 3 decimal places.)

2. Compute the test statistic. (Round your answer to 2 decimal places.)

3. State your decision about the null hypothesis.

Answer 1

The sample size is 8. The provided sample data along with the data required to compute the sample mean and sample variance are shown in the table below:

	Reduced Price	Reduced Price2
	122	14884
	138	19044
	155	24025
	136	18496
	115	13225
	121	14641
	134	17956
	127	16129
Sum =	1048	138400

The sample mean is computed as follows:

Also, the sample variance is

Therefore, the sample standard deviation s is

The sample size is 7. The provided sample data along with the data required to compute the sample mean and sample variance are shown in the table below:

	Regular Price	Regular Price2
	132	17424
	127	16129
	86	7396
	116	13456
	117	13689
	123	15129
	99	9801
Sum =	800	93024

The sample mean is computed as follows:

Also, the sample variance is

Therefore, the sample standard deviation s is

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​

Ha: μ1​ > μ2​

This corresponds to a right-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.1, and the degrees of freedom are df = 13. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:

Hence, it is found that the critical value for this right-tailed test is t_c = 1.35 , for α=0.1 and df = 13 .

(3) Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

t = 2.238

(4) Decision about the null hypothesis

Since it is observed that t = 2.238 > t_c = 1.35 , it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0217 , and since p = 0.0217 < 0.1 , it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μ1​ is greater than μ2​, at the 0.1 significance level.

Or, the sale has increased