Question

Please explain how to solve using RStudio, if possible.

In a experiment on relaxation techniques, subject's brain signals were measured before and after the relaxation exercises with the following results:

Person	1	2	3	4	5
Before	41.4	41.5	37.3	41.5	41.5
After	37.3	42.5	35.0	36.4	37.0

Is there sufficient evidence to suggest that the relaxation exercise slowed the brain waves? Assume the population is normally distributed. Select the test statistic, p­-value, Decision to Reject (RH₀) or Failure to Reject (FRH₀)].

Answer 1

We have to perform paired t test.

The following table is obtained:

	Sample 1	Sample 2	Difference = Sample 1 - Sample 2
	41.4	37.3	4.1
	41.5	42.5	-1
	37.3	35	2.3
	41.5	36.4	5.1
	41.5	37	4.5
Average	40.64	37.64	3
St. Dev.	1.868	2.857	2.468
n	5	5	5

For the score differences we have

(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 right-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=4.

Hence, it is found that the critical value for this right-tailed test is tc​=2.132, for α=0.05 and df=4.

The rejection region for this right-tailed test is R=t:t>2.132.

(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=2.718>tc​=2.132, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0265, and since p=0.0265<0.05, 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 0.05 significance level.

Graphically

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