Question

Question 2 (Modified from Sleuth 7.27) Black wheatears, Oenanthe leucura, are small birds of Spain and Morocco. Males of the species demonstrate an exaggerated sexual display by carrying heavy stones to nesting cavities. Different males carry somewhat different sized stones, prompting a study of whether larger stones may signal a higher health status. M. Soler et al. (1999) calculated the average stone mass (grams) carried by each of 21 male black wheatears, along with T-cell response measurements reflecting their immune systems’ strengths. The data are in ex0727.

(a) (1 point) Make a scatter plot of Mass (X) versus Tcell (Y ) including the estimated regression line.

(b) (2 points) Fit the linear model using the lm() function to regress Tcell on Mass (i.e., model the mean of Tcell as a function of Mass). Use the summary() function to view more information about the estimated regression model. Provide an interpretation for the p-values of the regression coefficients.

(c) (1 point) Construct 90% confidence intervals for the regression parameters using the confint() function.

(d) (1 point) Estimate the mean T-cell measurement for a new bird that is observed to carry stones averaging 4 grams in weight by using the predict() function. Construct a 95% confidence interval for mean T-cell measurement for that new bird.

(e) (1 point) Construct a 95% prediction interval for T-cell measurement for the new bird in part (d). How does the prediction interval compare to the confidence interval from part (d)?

Answer 1

(a) The scatter plot of Mass (X) versus Tcell (Y ) including the estimated regression line is:

(b) The linear model is:

y = 0.033 x + 0.087

(c)

Regression output					confidence interval
variables	coefficients	std. error	t (df=19)	p-value	90% lower	90% upper
Intercept	0.0875	0.0787	1.1121	0.2800	-0.0485	0.2235
Mean stone mass (g)	0.0328	0.0106	3.084	.0061	0.0144	0.0512

(d)

Predicted values for: T-cell response (mm)
		95% Confidence Interval
Mean stone mass (g)	Predicted	lower	upper
4	0.218783	0.138391	0.299175

(e)

95% Prediction Interval
lower	upper
0.031116	0.406450

Bird	Mean stone mass (g)	T-cell response (mm)
1	3.33	0.252
2	4.62	0.263
3	5.43	0.251
4	5.73	0.251
5	6.12	0.183
6	6.29	0.213
7	6.45	0.332
8	6.51	0.203
9	6.65	0.252
10	6.75	0.342
11	6.81	0.471
12	7.56	0.431
13	7.83	0.312
14	8.02	0.304
15	8.06	0.37
16	8.18	0.381
17	9.08	0.43
18	9.15	0.43
19	9.35	0.213
20	9.42	0.508
21	9.95	0.411
	r²	0.334
	r	0.578
	Std. Error	0.081
	n	21
	k	1
	Dep. Var.	T-cell response (mm)
ANOVA table
Source	SS	df	MS	F	p-value
Regression	0.0624	1	0.0624	9.51	.0061
Residual	0.1247	19	0.0066
Total	0.1872	20
Regression output					confidence interval
variables	coefficients	std. error	t (df=19)	p-value	95% lower	95% upper
Intercept	0.0875
Mean stone mass (g)	0.0328	0.0106	3.084	.0061	0.0105	0.0551
Predicted values for: T-cell response (mm)
		95% Confidence Interval		95% Prediction Interval
Mean stone mass (g)	Predicted	lower	upper	lower	upper	Leverage
4	0.218783	0.138391	0.299175	0.031116	0.406450	0.225
	r²	0.334
	r	0.578
	Std. Error	0.081
	n	21
	k	1
	Dep. Var.	T-cell response (mm)
ANOVA table
Source	SS	df	MS	F	p-value
Regression	0.0624	1	0.0624	9.51	.0061
Residual	0.1247	19	0.0066
Total	0.1872	20
Regression output					confidence interval
variables	coefficients	std. error	t (df=19)	p-value	90% lower	90% upper
Intercept	0.0875	0.0787	1.1121	0.2800	-0.0485	0.2235
Mean stone mass (g)	0.0328	0.0106	3.084	.0061	0.0144	0.0512