An endocrinologist was interested in exploring the relationship between the level of a steroid (Y ) and age (X) in healthy female subjects whose ages ranged from 8 to 25 years. She collected a sample of 27 healthy females in this age range. The data are given below, where the first column represents X = age and the second column represents Y = steroid level. For all R programming, print input and output codes and values, data to be used is below.

"age" "steroid"

15 14.1
10 8.5
13 10.8
16 18.4
10 4.7
18 23.3
16 16.4
10 9.4
16 17.7
23 35.8
19 25.4
18 24.9
24 42.1
19 26.5
24 40
12 10.7
13 11.6
10 3.6
23 37.9
17 16.8
19 24
23 37.7
20 29.6
14 13.7
19 23.1
11 8.3
17 19.6
9 7.8
11 7.1
13 13.3
18 20.8
25 44.4
9 9.7
12 12.5
22 34.9
8 4.3
9 5.9
8 6
22 36.2
15 11.7
10 5.3
15 15.6
9 6.6
14 15.7
13 10.5
17 20.7
23 36.8
23 37.2
8 5
16 19.6
16 18.9
15 16.1
10 7.7
14 11.9
12 9
8 4.4
8 2.7
8 5.2
16 19.3
20 27.5
20 27.8
13 12.9
12 12.8
13 9.3
15 16.1
19 25
13 10.5
13 9
18 22.3
22 33.6
9 4.9
19 28.4
15 14
21 30.6
19 24.8

(a) Create a quadratic regression in R. Write down the fitted equation, multiple R2, and the p-value for β1 from the summary() output.
(b) Calculate the correlation between X and X2 using the cor() function. If the value is close to 1, what might this number indicate? Create a new centered variable xi = Xi? X(should be X with a bar over it) and calculate the correlation between x and x2.
(c) Fit a quadratic regression using the centered variable xi. Compare the fitted equation, multiple R2 and p-value for β1 to part (a).