Question

Problem 8.4: Refer to Muscle Mass Problem 1.27. Second-order regression model (8.2) with independent normal error terms is expected to be appropriate.  

A.    Fit regression model (8.2). Plot the fitted regression function and the data. Does the quadratic regression function appear to be a good fit here? Find R^2.

B.    Test whether or not there is regression relation; use α= .05. State the alternatives, decision rule and conclusion.
C.    Estimate the mean muscle mass for women aged 48 years, use a 95 percent confidence interval. Interpret your interval.
D.    Predict the muscle mass for women whose age is 48 years; use a 95 percent prediction interval. Interpret your interval.
E. Test whether the quadratic term can be dropped from the regression model; use α= .05. State the alternatives, decision rule, and conclusion.
F.    Express the fitted regression function obtained in part (a) in terms of the original variable X.
G.    Calculate the coefficient of simple correlation between X and X^2 and between x and x^2. Is the use of a centered variable helpful here?

Data is:

106    43
106    41
97    47
113    46
96    45
119    41
92    47
112    41
92    48
102    48
107    42
107    47
102    43
115    44
101    42
87    55
91    57
97    56
82    59
78    57
95    54
98    53
94    52
96    53
100    54
84    60
70    59
104    51
76    59
93    57
73    68
73    63
76    60
80    63
84    63
71    64
64    66
88    65
79    60
88    65
73    65
74    69
76    61
87    70
70    68
69    78
54    78
62    78
78    72
65    70
64    73
74    76
87    78
63    78
82    71
80    75
52    77
56    76
70    72
74    76

Answer 1

(a) The plot is:

Yes, it is a good fit.

R² = 0.763

(b) The regression output is:

	R²	0.763
	Adjusted R²	0.754
	R	0.873
	Std. Error	8.043
	n	60
	k	2
	Dep. Var.	mass
ANOVA table
Source	SS	df	MS	F	p-value
Regression	11,843.7322	2	5,921.8661	91.55	1.59E-18
Residual	3,687.1178	57	64.6863
Total	15,530.8500	59
Regression output					confidence interval
variables	coefficients	std. error	t (df=57)	p-value	95% lower	95% upper
Intercept	207.8060
age	-2.9802	1.0052	-2.965	.0044	-4.9931	-0.9674
age2	0.0150	0.0084	1.788	.0792	-0.0018	0.0317

The hypothesis being tested is:

H₀: β₂ = 0

H₁: β₂ ≠ 0

Since the p-value (0.0792) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Therefore, we cannot conclude that there is a regression relation.

(c) Mean muscle mass = 99.244

The 95% confidence interval is between 96.267 and 102.221.

Predicted values for: mass
			95% Confidence Interval		95% Prediction Interval
age	age2	Predicted	lower	upper	lower	upper	Leverage
48	2,304	99.244	96.267	102.221	82.866	115.622	0.034

(d) Mean muscle mass = 99.244

The 95% prediction interval is between 82.866 and 115.622.

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