Question

2. An experiment was conducted by a physiologist to determine whether exercise improves the human immune system. Thirty subjects volunteered to participate in the study. The amount of immunoglobulin known as IgG (an indicator of long-term immunity) and the maximal oxygen uptake (a measure of aerobic fitness level) were recorded for each subject. The data can be found in the file marked AEROBIC. You will need to use the Data Analysis - Regression Function for this problem, as well as some graphing functions.

a. Construct a scattergram for the IgG-maximal oxygen uptake data.

b. Hypothesize a probabilistic model relating IgG to maximal oxygen uptake.

c. Fit the model to the data. Is there sufficient evidence to indicate that the model provides information for the prediction of IgG, y? Test using α = .05.

d. Does a second-order term contribute information for the prediction of y? Test using α = .05.

Subject	IgG	Max Oxy
1	881	34.6
2	1290	45.0
3	2147	62.3
4	1909	58.9
5	1282	42.5
6	1530	44.3
7	2067	67.9
8	1982	58.5
9	1019	35.6
10	1651	49.6
11	752	33.0
12	1687	52.0
13	1782	61.4
14	1529	50.2
15	969	34.1
16	1660	52.5
17	2121	69.9
18	1382	38.8
19	1714	50.6
20	1959	69.4
21	1158	37.4
22	965	35.1
23	1456	43.0
24	1273	44.1
25	1418	49.8
26	1743	54.4
27	1997	68.5
28	2177	69.5
29	1965	63.0
30	1264	43.2

Answer 1

1. Scatterplot is attached below.

2. From the scatter plot it looks like a linear model would fit the data.

Thus we hypothesize the following model

yi = a+ b xi+ei where yi is the ith  amount of immunoglobulin , xi is the the maximal oxygen uptake if the ith subject. i=1,2....30

And ei is a random variable (particularly normally distributed) with mean 0 and fixed variance( sigma^2).

3. We fit the regression model and obtain the output below.

The regression equation is
y = - 100.3 + 32.74 X

thus the estimated coefficients are: a=-100.3 b=32.74

4.To check that whether there is sufficient evidence to indicate that the model provides information for the prediction of IgG,at 5% level of significance we look for overall significance of the model and that by looking to the F-test statistic in ANOVA table of the output.The ANOVA table is given below-

Analysis of Variance

Source	DF	SS	MS	F	P
Regression	1	4472047	4472047	287.21	0.000
Error	28	435982	15571
Total	29	4908029

Thus looking at p value of the F-statistic (p = 0<<0.05) we can reject the Null hypothesis that the model doesn't provide much info for prediction and accept the opposite i.e. it is significantly providing information for prediction at 5% level of significance.