Question

1. Now assume that Data Set A depicts the scores of five subjects who received both Treatment 1 and Treatment 2. Calculate a t-test for dependent means to determine whether the means for the two treatments were significantly different. The correlation between the two treatments is +1.00. In your complete answer, remember to include your t-statistic, critical value, and your decision about whether to reject the null hypothesis. For this question, you should assume that the same participants received each of the two treatments. (15 points)

Treatment 1	Treatment 2
45	60
50	70
55	80
60	90
65	100

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u_d = 0

Alternative hypothesis: u_d ≠ 0

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a matched-pairs t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard deviation of the differences (s), the standard error (SE) of the mean difference, the degrees of freedom (DF), and the t statistic test statistic (t).

s = 7.9057

SE = s / sqrt(n)

S.E = 3.5355

DF = n - 1 = 5 -1

D.F = 4

t = [ (x₁ - x₂) - D ] / SE

t = - 7.07

t_critical = + 2.777

Rejection region is - 2.777 > t > + 2.777

where d_i is the observed difference for pair i, d is mean difference between sample pairs, D is the hypothesized mean difference between population pairs, and n is the number of pairs.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 4 degrees of freedom is more extreme than 7.07; that is, less than - 7.07 or greater than 7.07.

Thus, the P-value = less than 0.0001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

Reject H0.

From the above test we have sufficient evidence in the favor of the claim that the means for the two treatments were significantly different.