Question

Claim: There is a significant correlation between test 1 scores and the hours a student spent studying.

Use the data-analysis add-in to complete the hypothesis test for the correlation coefficient using level of significance of 0.05.

Hours Spent Studying	Test 1 Scores
8	91
6	83
6	75
12	85
4	90
6	80
7	67
7	82
5	88
9	87
5	95
3	91
2	73
13	80
4	83
10	92
12	94
0	68
3	75
8	91
3	79
11	95
0	38
8	76
9	91
7	85
0	59
2	70
1	69
5	78

2. Hypothesis Test			identify the claim
Hypothesis Statements	Ho
	Ha
Standardized Test Statistic
P-value

Insert Data Analysis test here

Answer 1

First we calculate the correlation coefficient between test 1 scores and the hours a student spent studying:

X	Y	X*Y	X2	Y2
	8	8	64	64	64
	6	6	36	36	36
	6	6	36	36	36
	12	12	144	144	144
	4	4	16	16	16
	6	6	36	36	36
	7	7	49	49	49
	7	7	49	49	49
	5	5	25	25	25
	9	9	81	81	81
	5	5	25	25	25
	3	3	9	9	9
	2	2	4	4	4
	13	13	169	169	169
	4	4	16	16	16
	10	10	100	100	100
	12	12	144	144	144
	0	0	0	0	0
	3	3	9	9	9
	8	8	64	64	64
	3	3	9	9	9
	11	11	121	121	121
	0	0	0	0	0
	8	8	64	64	64
	9	9	81	81	81
	7	7	49	49	49
	0	0	0	0	0
	2	2	4	4	4
	1	1	1	1	1
	5	5	25	25	25
Sum =	176	176	1430	1430	1430

The correlation coefficient r is computed using the following expression:

where

In this case, based on the data provided, we get that

Therefore, based on this information, the sample correlation coefficient is computed as follows

The following needs to be tested:

H0​:ρ=0

HA​:ρ̸0

where \rhoρ corresponds to the population correlation.

The sample size is n=30, so then the number of degrees of freedom is df=n−2=30−2=28

The corresponding t-statistic to test for the significance of the correlation is:

The p-value is computed as follows:

p=Pr(∣t28​∣>)<0.0001

Since we have that p<0.0001, it is concluded that the null hypothesis H0​ is rejected.

Therefore, based on the sample correlation provided, it is concluded that there is enough evidence to claim that the population correlation ρ is different than 0, at the 0.05 significance level.

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