Question

Below are 5 participants’ ratings of how likable two comedians were. Based on these data, conduct a related-samples t test to analyze whether likable ratings differ between groups.

Participants	Ratings for humorous comedian	Ratings for non-humorous comedian
1	8	2
2	6	3
3	7	4
4	4	5
5	5	1
	=6	=3

A. Write down the null and alternative hypotheses using proper notation

B. Calculate , given estimated standard error = 1.14 (You will earn maximally 3 bonus points if you also show the steps of getting this value 1.14)

C. What is the value of for α = 0.05 with a two-tailed test?

D. Please complete the test by stating your decision as well as your interpretation.

Decision:

Interpretation:

Answer 1

Step 1: null hypothesis (H0): μ1 = μ2 or µd = 0 (means are equal)

Alternative hypothesis (H1): μ1 ≠ μ2 or µd ≠ 0 (means are unequal)...this is our claim

Step 2: Calculate the t-score

Where,

ΣD = Sum of the differences = 6+3+3-1+4 = 15

Σ (D2) =Sum of the squared differences = 36+9+9+1+16=71

(ΣD) 2=Sum of the differences squared = 15x15=225

So, test statistic = (15/5) / sqrt ((71-225/5)/ (5x4)) = 2.63

METHOD-2(USING STANDARD ERROR)

Standard error of mean difference (SE) = SD / sqrt (n) = 2.55 /sqrt (5) = 1.14

Where SD is the standard deviation of the differences = sqrt ((D - mean difference) 2 / (n -1)) = 2.55

, mean of differences = D bar = ΣD / n = 15/5 =3

And sample size = n= 5

Then, test statistics = (sample mean difference - mean differences) / SE = (3 - 0)/1.14 = 2.63 (WHICH IS SAME AS COMPUTED ABOVE)

Step 3: degrees of freedom = sample size – 1 = 5-1 = 4

Step 4: p-value from the t-table, using the degrees of freedom and alpha level, use 0.05 (5%).

For this problem, with DF=4, the critical t-value is 2.776

Step 5: Comparing critical value (2.776) to the calculated t-value (2.63).

Since, the calculated t-value is lesser than the critical value at an alpha level of .05 and DF = 4

Or comparing the p-value with alpha level we get p value = 0.058186. > 0.05

Both of which tells that the test statistic lies outside the rejection regions.

Step 6 :( Conclusion)

So, we can fail to reject the null hypothesis that there is no difference between means.

Or there is not sufficient evidence to support the claim that the two means are different.

Interpretation: both ratings are equal