Question

3. UTRGV researchers would like to determine if there is a significant difference between the grades achieved in the fall versus spring semester. Grades were recorded for a group of 7 students in both semester and are listed below (α = .05). Participant Fall Spring 1 85 89 2 68 84 3 76 79 4 77 82 5 89 90 6 93 91 7 90 90 a. Will we need a one- or two-tailed hypothesis test? b. State your null hypothesis c. State your alternative hypothesis d. Provide the SPSS output for your test and identify (circle or highlight) the t-obtained and the p-value e. Did you reject or fail to reject the null hypothesis? f. What can you conclude? g. Calculate the 95% confidence interval for the mean differences h. Calculate Cohen’s d

Answer 1

Solution:

Here, we have to use paired t test for checking the given claim. The null and alternative hypothesis for this test is given as below:

a. Will we need a one- or two-tailed hypothesis test?

We will need a two-tailed hypothesis test.

b. State your null hypothesis

Null hypothesis: H₀: There is no significant difference between the grades achieved in the fall and spring semester.

c. State your alternative hypothesis

Alternative hypothesis: H_a: There is a significant difference between the grades achieved in the fall and spring semester.

H₀: µ_d = 0 vs. H_a: µ_d ≠ 0

This is a two-tailed test.

We are given

Level of significance = α = 0.05

d. Provide the SPSS output for your test and identify (circle or highlight) the t-obtained and the p-value

The SPSS output for this test is given as below:

Paired Samples Statistics
		Mean	N	Std. Deviation	Std. Error Mean
Pair 1	Fall	82.5714	7	9.10782	3.44243
	Spring	86.4286	7	4.72077	1.78429

Paired Samples Correlations
		N	Correlation	Sig.
Pair 1	Fall & Spring	7	.823	.023

Paired Samples Test
		Paired Differences					t	df	Sig. (2-tailed)
		Mean	Std. Deviation	Std. Error Mean	95% Confidence Interval of the Difference
					Lower	Upper
Pair 1	Fall - Spring	-3.85714	5.87164	2.21927	-9.28751	1.57322	-1.738	6	.133

From above SPSS output, we have

t-obtained = -1.738

P-value = 0.133

e. Did you reject or fail to reject the null hypothesis?

Here,

P-value = 0.133

α = 0.05

P-value > α = 0.05

So, we do not reject the null hypothesis that there is no significant difference between the grades achieved in the fall and spring semester.

f. What can you conclude?

There is sufficient evidence to conclude that there is no significant difference between the grades achieved in the fall and spring semester.

There is insufficient evidence to conclude that there is a significant difference between the grades achieved in the fall and spring semester.

g. Calculate the 95% confidence interval for the mean differences

The 95% confidence interval for mean difference by using SPSS output is given as below:

Confidence interval = (-9.28751, 1.57322)

h. Calculate Cohen’s d

Formula is given as below:

d = ( X1bar – X2bar )/ sqrt(S1^2 + S2^2 – 2*r*S1*S2)

We have

X1bar = 82.57143

X2bar = 86.42857

S1 = 9.10782

S2 = 4.720775

r = 0.82289

d = (82.57143 - 86.42857)/sqrt(9.10782^2 + 4.720775^2 - 2*0.82289*9.10782*4.720775)

d = -0.65691

Cohen’s d = -0.65691