Question

Provide an appropriate response.

Test the claim that the paired sample data is from a population with a mean difference of 0. Assume the samples are random and dependent, and the populations are normally distributed. Use .

Select one:

A. Reject H₀: There is sufficient evidence to reject the claim.

B. Fail to reject H₀: There is sufficient evidence to reject the claim.

C. Fail to reject H₀: There is not sufficient evidence to reject the claim.

D. Reject H₀: There is not sufficient evidence to reject the claim.

Answer 1

The following table is obtained:

	Sample 1	Sample 2	Difference = Sample 1 - Sample 2
	30	28	2
	28	24	4
	47	25	22
	43	35	8
	31	22	9
Average	35.8	26.8	9
St. Dev.	8.585	5.07	7.81
n	5	5	5

From the sample data, it is found that the corresponding sample means are:

Xˉ1​=35.8

Xˉ2​=26.8

Also, the provided sample standard deviations are:

s_1 = 8.585

s2​=5.07

and the sample size is n = 5. For the score differences we have

Dˉ=9

sD​=7.81

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μD​ = 0

Ha: μD​ ≠ 0

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

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df = 4.

Hence, it is found that the critical value for this two-tailed test is

t_c = 2.776

The rejection region for this two-tailed test is

R={t:∣t∣>2.776}.

(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∣ = 2.577 ≤ tc​ = 2.776,

it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is p = 0.0615, and

since p=0.0615≥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.

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