Exercise 5.4.1

Using the trees data frame that comes pre-installed in R, fit the regression model that uses the tree Height to explain the Volume of wood harvested from the tree.

(a) Graph the data

(b) Fit an lm model

## 
## Call:
## lm(formula = Volume ~ Height, data = trees)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -21.274  -9.894  -2.894  12.068  29.852 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -87.1236    29.2731  -2.976 0.005835 ** 
## Height        1.5433     0.3839   4.021 0.000378 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 13.4 on 29 degrees of freedom
## Multiple R-squared:  0.3579, Adjusted R-squared:  0.3358 
## F-statistic: 16.16 on 1 and 29 DF,  p-value: 0.0003784

(c) Print out the table of coefficients with estimate names, estimated value, standard error, and upper and lower 95% confidence intervals.

	Estimate	Std. Error	t value	Pr(>\|t\|)
(Intercept)	-87.12361	29.2731221	-2.976232	0.0058347
Height	1.54335	0.3838693	4.020509	0.0003784

	2.5 %	97.5 %
(Intercept)	-146.993871	-27.253356
Height	0.758249	2.328451

(d) Add the model fitted values to the trees data frame along with the regression model confidence intervals.

Girth	Height	Volume	fit	lwr	upr
8.3	70	10.3	20.91087	14.098550	27.72319
8.6	65	10.3	13.19412	3.254288	23.13395
8.8	63	10.2	10.10742	-1.223363	21.43821
10.5	72	16.4	23.99757	18.159758	29.83538
10.7	81	18.8	37.88772	31.592680	44.18275
10.8	83	19.7	40.97442	33.597379	48.35145
11.0	66	15.6	14.73747	5.471607	24.00333
11.0	75	18.2	28.62762	23.644217	33.61102
11.1	80	22.6	36.34437	30.506556	42.18218
11.2	75	19.9	28.62762	23.644217	33.61102
11.3	79	24.2	34.80102	29.345254	40.25678
11.4	76	21.0	30.17097	25.249799	35.09214
11.4	76	21.4	30.17097	25.249799	35.09214
11.7	69	21.3	19.36752	11.990482	26.74456
12.0	75	19.1	28.62762	23.644217	33.61102
12.9	74	22.2	27.08427	21.918668	32.24987
12.9	85	33.8	44.06112	35.450370	52.67186
13.3	86	27.4	45.60447	36.338602	54.87033
13.7	71	25.7	22.45422	16.159183	28.74926
13.8	64	24.9	11.65077	1.021703	22.27984
14.0	78	34.5	33.25767	28.092067	38.42327
14.2	80	31.7	36.34437	30.506556	42.18218
14.5	74	36.3	27.08427	21.918668	32.24987
16.0	72	38.3	23.99757	18.159758	29.83538
16.3	77	42.6	31.71432	26.730917	36.69772
17.3	81	55.4	37.88772	31.592680	44.18275
17.5	82	55.7	39.43107	32.618747	46.24339
17.9	80	58.3	36.34437	30.506556	42.18218
18.0	80	51.5	36.34437	30.506556	42.18218
18.0	80	51.0	36.34437	30.506556	42.18218
20.6	87	77.0	47.14782	37.207982	57.08765

(e) Graph the data and fitted regression line and uncertainty ribbon.

(f) Add the R-squared value as an annotation to the graph.

Exercise 5.4.2

The data set phbirths from the faraway package contains information on birth weight, gestational length, and smoking status of mother. We’ll fit a quadratic model to predict infant birth weight using the gestational time.

a. Create two scatter plots of gestational length and birth weight, one for each smoking status.

b. Remove all the observations that are premature (less than 36 weeks). For the remainder of the problem, only use these full-term babies.

	black	educ	smoke	gestate	grams
1	FALSE	0	Smoke	40	2898
3	FALSE	2	No Smoke	38	3977
4	FALSE	2	Smoke	37	3040
5	FALSE	2	No Smoke	38	3523
6	FALSE	5	Smoke	40	3100
7	TRUE	6	No Smoke	40	3670

c. Fit the quadratic model

## 
## Call:
## lm(formula = grams ~ poly(gestate, 2) * smoke, data = phbirths2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1433.51  -296.25   -12.25   291.68  1464.49 
## 
## Coefficients:
##                              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                   3364.02      15.45 217.751  < 2e-16 ***
## poly(gestate, 2)1             5770.90     504.11  11.448  < 2e-16 ***
## poly(gestate, 2)2            -2287.74     512.46  -4.464 8.92e-06 ***
## smokeSmoke                    -202.81      32.68  -6.206 7.85e-10 ***
## poly(gestate, 2)1:smokeSmoke  1813.07    1027.39   1.765 0.077904 .  
## poly(gestate, 2)2:smokeSmoke  3654.80     988.42   3.698 0.000229 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 437.7 on 1033 degrees of freedom
## Multiple R-squared:  0.2108, Adjusted R-squared:  0.207 
## F-statistic: 55.19 on 5 and 1033 DF,  p-value: < 2.2e-16

d. Add the model fitted values to the phbirths data frame along with the regression model confidence intervals.

	black	educ	smoke	gestate	grams	fit	lwr	upr
1	FALSE	0	Smoke	40	2898	3243.132	3174.250	3312.013
3	FALSE	2	No Smoke	38	3977	3200.173	3156.090	3244.256
4	FALSE	2	Smoke	37	3040	2804.668	2692.140	2917.196
5	FALSE	2	No Smoke	38	3523	3200.173	3156.090	3244.256
6	FALSE	5	Smoke	40	3100	3243.132	3174.250	3312.013
7	TRUE	6	No Smoke	40	3670	3478.249	3441.992	3514.507

e. On your two scatterplots from part (a), add layers for the model fits and ribbon of uncertainty for the model fits.

f. Create a column for the residuals in the phbirths data set using any of the following:

	black	educ	smoke	gestate	grams	fit	lwr	upr	residuals
1	FALSE	0	Smoke	40	2898	3243.132	3174.250	3312.013	-345.1315
3	FALSE	2	No Smoke	38	3977	3200.173	3156.090	3244.256	776.8272
4	FALSE	2	Smoke	37	3040	2804.668	2692.140	2917.196	235.3318
5	FALSE	2	No Smoke	38	3523	3200.173	3156.090	3244.256	322.8272
6	FALSE	5	Smoke	40	3100	3243.132	3174.250	3312.013	-143.1315
7	TRUE	6	No Smoke	40	3670	3478.249	3441.992	3514.507	191.7509

Chapter 5 - Exercises

Sarah Bosch

29 January 2021