Here are data for height, measured in centimeters, among a group of 20 male coll
ID: 3220721 • Letter: H
Question
Here are data for height, measured in centimeters, among a group of 20 male college students and each one's father. The column labeled N gives the number of father/son pairs with each combination of heights. Answer the following questions, being sure to show your work clearly and concisely. (a) Calculate the correlation coefficient for height between fathers and sons for this group of students. (b) Infer the correlation coefficient for height between fathers and sons for the population of male college students. (c) Is the population correlation coefficient statistically significantly different from zero? Explain.Explanation / Answer
(a) Pearson’s Correlation Co-efficient is calculated using the below formula:
Let,
X denote Height of Parent
Y denote Height of Child
In our case,
n=20
Thus, we have the below table for calculating Correlation Coefficient:
N
Parent (Xi)
Offspring (Yi)
Xi2
Yi2
XiYi
1
172
174
29584
30276
29928
2
175
175
30625
30625
30625
3
175
178
30625
31684
31150
4
176
170
30976
28900
29920
5
176
174
30976
30276
30624
6
176
177
30976
31329
31152
7
178
178
31684
31684
31684
8
179
176
32041
30976
31504
9
180
178
32400
31684
32040
10
180
179
32400
32041
32220
11
180
179
32400
32041
32220
12
180
181
32400
32761
32580
13
181
178
32761
31684
32218
14
181
178
32761
31684
32218
15
181
180
32761
32400
32580
16
182
172
33124
29584
31304
17
182
175
33124
30625
31850
18
183
175
33489
30625
32025
19
184
179
33856
32041
32936
20
184
185
33856
34225
34040
Total
3585
3541
642819
627145
634818
Therefore,
n = 20
xi = 3585
yi = 3541
xi2 = 642819
yi2 = 627145
xiyi = 634818
Substituting the above values in the formula we get,
rxy = 0.4478
(b) Since the correlation between heights of fathers and their sons is 0.4478, it indicates that there's a moderate correlation between the two and the positive sign infers that height of son will tend to be more if height of father is more.
(c) We need to test if, rxy = 0 (two-sided test. We assume a significance level of 0.05, i.e. = 0.05)
Therefore, our hypothesis becomes
Ho: rxy = 0
H1: rxy 0
We use t-test to test the significance of correlation co-efficient. The t-statistic is given as follows:
where,
rxy = 0.4478
n = 20
Substituting the values in the above formula we get,
t = 2.125
To test the significance of our t-statistics, we refer to the critical value of t-distribution at 5% significance level and n-2, i.e. 18, degrees of freedom, i.e. t(0.025,18) = 2.1. (0.025 is taken as we have a two-sided test at 0.05 significance level, i.e. 0.05/2 = 0.025)
Since, value of value of t-statistic (2.125) > critical value of t (2.1), we reject our null hypothesis (Ho) and conclude that our correlation coe-fficient is significantly different from 0.
N
Parent (Xi)
Offspring (Yi)
Xi2
Yi2
XiYi
1
172
174
29584
30276
29928
2
175
175
30625
30625
30625
3
175
178
30625
31684
31150
4
176
170
30976
28900
29920
5
176
174
30976
30276
30624
6
176
177
30976
31329
31152
7
178
178
31684
31684
31684
8
179
176
32041
30976
31504
9
180
178
32400
31684
32040
10
180
179
32400
32041
32220
11
180
179
32400
32041
32220
12
180
181
32400
32761
32580
13
181
178
32761
31684
32218
14
181
178
32761
31684
32218
15
181
180
32761
32400
32580
16
182
172
33124
29584
31304
17
182
175
33124
30625
31850
18
183
175
33489
30625
32025
19
184
179
33856
32041
32936
20
184
185
33856
34225
34040
Total
3585
3541
642819
627145
634818
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