Question

1. A study investigated the possible link between body mass index (BMI) and plasma testosterone concentrations in a sample of 50 adolescent males between the ages of 14 and 20. Please see columns labeled BMI and testosterone in your HW5a spreadsheet.
2. Report the null and alternate hypothesis
3. Examine the assumptions of normality for the response variable as well as evenness of residuals (show and interpret a residual plot)
4. Show a scatterplot of the data with the best-fitting line
5. Report run the regression and report test statistic, df, and P value
6. Do you accept or reject the null? Interpret your results in one or two sentences.

7. BMI	Testosterone
   21.4	0.78
   19.0	0.70
   18.3	0.63
   19.5	0.60
   20.9	0.60
   23.4	0.69
   25.0	0.76
   24.1	0.58
   24.2	0.50
   22.6	0.48
   20.4	0.49
   16.2	0.43
   17.8	0.42
   21.0	0.38
   18.6	0.35
   20.9	0.35
   22.4	0.32
   23.5	0.31
   18.8	0.28
   19.3	0.25
   19.5	0.23
   20.2	0.24
   21.2	0.24
   21.3	0.26
   22.2	0.27
   28.3	0.30
   27.7	0.24
   28.1	0.19
   29.2	0.17
   33.3	0.18
   33.2	0.23
   34.7	0.24
   35.8	0.06
   37.0	0.15
   37.0	0.17
   39.0	0.18
   41.6	0.17
   42.4	0.15
   47.7	0.12
   45.7	0.25
   41.5	0.25
   38.0	0.25
   38.1	0.32
   37.8	0.35
   34.9	0.37
   34.8	0.39
   34.7	0.46
   32.0	0.49
   31.9	0.42
   30.5	0.36

Answer 1

Hey I have used Excel of solving this as it is better to equipped to handle regression.

Null hypothesis-H_o: Β₁ = 0(slope coefficient is zero i.e there exists no linear relationship between testosterone and BMI)

Alternative hypothesis-H_a: Β₁ ≠ 0(There exists a linear relationship and slope is not zero)

This your data and residuals and predicted values-

BMI	Testosterone	Predicted BMI	Residuals
21.4	0.78	17.67425114	3.725748865
19	0.7	19.66635373	-0.666353727
18.3	0.63	21.40944349	-3.109443495
19.5	0.6	22.15648197	-2.656481966
20.9	0.6	22.15648197	-1.256481966
23.4	0.69	19.91536655	3.484633449
25	0.76	18.17227678	6.827723217
24.1	0.58	22.65450761	1.445492386
24.2	0.5	24.64661021	-0.446610206
22.6	0.48	25.14463585	-2.544635854
20.4	0.49	24.89562303	-4.49562303
16.2	0.43	26.38969997	-10.18969997
17.8	0.42	26.6387128	-8.838712797
21	0.38	27.63476409	-6.634764093
18.6	0.35	28.38180256	-9.781802565
20.9	0.35	28.38180256	-7.481802565
22.4	0.32	29.12884104	-6.728841037
23.5	0.31	29.37785386	-5.877853861
18.8	0.28	30.12489233	-11.32489233
19.3	0.25	30.8719308	-11.5719308
19.5	0.23	31.36995645	-11.86995645
20.2	0.24	31.12094363	-10.92094363
21.2	0.24	31.12094363	-9.920943628
21.3	0.26	30.62291798	-9.32291798
22.2	0.27	30.37390516	-8.173905156
28.3	0.3	29.62686668	-1.326866684
27.7	0.24	31.12094363	-3.420943628
28.1	0.19	32.36600775	-4.266007748
29.2	0.17	32.8640334	-3.664033396
33.3	0.18	32.61502057	0.684979428
33.2	0.23	31.36995645	1.830043548
34.7	0.24	31.12094363	3.579056372
35.8	0.06	35.60317446	0.196825541
37	0.15	33.36205904	3.637940956
37	0.17	32.8640334	4.135966604
39	0.18	32.61502057	6.384979428
41.6	0.17	32.8640334	8.735966604
42.4	0.15	33.36205904	9.037940956
47.7	0.12	34.10909752	13.59090248
45.7	0.25	30.8719308	14.8280692
41.5	0.25	30.8719308	10.6280692
38	0.25	30.8719308	7.128069196
38.1	0.32	29.12884104	8.971158963
37.8	0.35	28.38180256	9.418197435
34.9	0.37	27.88377692	7.016223083
34.8	0.39	27.38575127	7.414248731
34.7	0.46	25.6426615	9.057338499
32	0.49	24.89562303	7.10437697
31.9	0.42	26.6387128	5.261287203
30.5	0.36	28.13278974	2.367210259

This is your residual plot: observations seems evenly distributed above and below the center line i.e they are random not showing any specific pattern.So data adheres to the assumptions of normality.

this is your scatter plot in which we have our line of best fit

green dots are actual values and blue dots are predicted values that's why there form a downward sloping line.

	Coefficients	Standard Error	t Stat	P-value
Intercept	37.0972514	2.38212866	15.5731519	2.14E-20
Testosterone(this is our slope coefficient)	-24.9012824	6.073223821	-4.10017532	0.00015855

T value is -4.100 ignoring the sign we get 4.100

Degrees of freedom are n-2=50-2=48

Test statistic. The test statistic is a t statistic (t) defined by the following equation.

t = b₁ / SE

see p-value (0.000158) is much smaller than 0.05 and even 0.01 so we reject the null hypothesis. and coclude that there exists a linear relation between BMI and testosterone hence slope is not 0.

NOTE- after having your raw data you can go to data analysis tab in excel and then select regression proceed as instructed there and you can get the results in one click.

Regression Statistics
Multiple R	0.50930343
R Square	0.25938998
Adjusted R Square	0.24396061
Standard Error	7.43120421
Observations	50

Please upvote if I am able to help you

Thanks.