Question

Height	Weight	Age	Shoe Size	Waist Size	Pocket Change
64	180	39	7	36	18
66	140	31	9	30	125
69	130	31	9	25	151
63	125	36	7	25	11
68	155	24	8	31	151
62	129	42	6	32	214
63	173	30	8	34	138
60	102	26	6	25	67
66	180	33	8	30	285
66	130	31	9	30	50
63	125	32	8	26	32
68	145	33	10	28	118
75	235	44	12	40	60
68	138	43	8	27	50
65	165	55	9	30	22
64	140	24	7	31	95
78	240	40	9	38	109
71	163	28	7	32	14
68	195	24	10	36	5
66	122	33	9	26	170
53	115	25	7	25	36
71	210	30	10	36	50
78	108	23	7	22	75
69	126	23	8	24	175
77	215	24	12	36	41
68	125	23	8	30	36
62	105	50	6	24	235
69	126	42	9	27	130
55	140	42	8	29	14
67	145	30	8	30	50

Explain the correlation coefficient of determination.

a. Height Vs. Weight with an alpha of 0.05/2

b. Weight vs age alfpha 0.01/2

c. Height vs shoe sz an alpha of =0.02/2

Answer 1

a ) Let's find correlation coefficient between Height ad Weight using excel.

First copy and paste the given data in excel:

Using "=CORREL(A:A,B:B)" this excel command, we get the correlation between Height and Weight as 0.534

Note that: I put "Height " and "Weight" in "A " and "B" columns of excel.

Similarly do for "Weight and age"; Height and shoe size.

See the following image.

Let's test the hypothesis of correlation coefficient.

THe null hypothesis ( H₀ ) and the alternative hypothesis ( H_a ) are as follows:

Where is the notation of population correlation coefficient.

a) We find correlation coefficient between Height ad Weight ( r ) = 0.534

Degrees of freedom = n - 2 = 30 - 2 = 28

For   = 0.05, the critical r ( r_c ) value from pearson correlation table is 0.361

Decision rule :

1) If r > r_c then we reject null hypothesis .

2) If r < r_c then we fail to reject null hypothesis .

Here r = 0.534 > r_c = 0.361  

so we reject the null hypothesis.

Conclusion: At 5% level of significance the data provide sufficient evidence in the favor of alternative hypothesis.

That is there is correlation between Height and Weight .

The correlation coefficient of determination between Height and Weight = 0.534² = 0.2852

b) From the above excel output, the correlation between Weight and age ( r ) is 0.110

Degrees of freedom = 28

For = 0.01 and df = 28, the critical value of correlation coefficient ( r_c ) is 0.463

Here r < r_c, so we fail to reject null hypothesis at 1% level of significance.

At 5% level of significance the data does not provide sufficient evidence in the favor of alternative hypothesis.

That is there is no correlation between Weight and age.

The correlation coefficient of determination between Weight and age is 0.110² = 0.0121

c)  

From the above excel output, the correlation between Weight and Shoe Size( r ) is 0.551

Degrees of freedom = 28

For = 0.02 and df = 28, the critical value of correlation coefficient ( r_c ) is 0.423

Here r > r_c, so we reject null hypothesis at 2% level of significance.

Conclusion:

At 5% level of significance the data provide sufficient evidence in the favor of alternative hypothesis.

That is there is correlation between Height and Shoe Size..

The correlation coefficient of determination between Weight and age is 0.551² = 0.3035