Quiz 7 Scripts

1. Study the lecture notes and screen cast for Regression, Part 1: plot and cor (4:09).

Consider the bivariate sample of (x, y) pairs, (0, 0),(1, 2.5),(2, 2),(3, 3.5),(4, 6),(5, 7.5),(6, 5),(7, 10.5),(8, 8),(9, 6.5),(10, 5),(11, 10.5),(12, 8),(13, 6.5),(14, 9),(15, 13.5),(16, 14),(17, 13.5),(18, 14),(19, 9.5),(20, 15),(21, 15.5),(22, 16),(23, 11.5),(24, 18),(25, 13.5),(26, 18),(27, 14.5),(28, 16) (x’s alone: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28; y’s alone: 0,2.5,2,3.5,6,7.5,5,10.5,8,6.5,5,10.5,8,6.5,9,13.5,14,13.5,14,9.5,15,15.5,16,11.5,18,13.5,18,14.5,16) . Find the correlation between x and y.

x = c(0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)
y = c(0,2.5,2,3.5,6,7.5,5,10.5,8,6.5,5,10.5,8,6.5,9,13.5,14,13.5,14,9.5,15,15.5,16,11.5,18,13.5,18,14.5,16)
cor(x,y)
# The answer is 0.9103867

2. Study Part 2: lm and abline (4:12).

Find the slope of the least-squares regression line.

lm(y ~ x)
# The slope is from the coefficients, where it gives us x = 0.533

lm(y~x) prints out: Call: lm(formula = y ~ x)

Coefficients: (Intercept) x
2.641 0.533

3. Find the y-intercept of the line.

lm(y ~ x)
# The slope is from the coefficients, where it gives us Intercept = 2.641

4. Study Part 3: predict and qqnorm (10:41).

Predict the values of y for x=13, 14, and 15. For x=13, y^ is

model = lm(x~y)
d = data.frame(y = seq(13, 15, 1))
y.hat = predict(model, d)

yhat13 = 2.641+0.533*13
yhat13

[1] 9.57

5. For x=14, y^ is

yhat14 = 2.641+0.533*14
yhat14

[1] 10.103

6. For x=15, y^ is

yhat15 = 2.641+0.533*15
yhat15

[1] 10.636

7. Study Part 4: multiple regression (8:42).

Make a mulitple regression model of qsec (the 1/4-mile time) vs. disp and hp (the engine displacement and horsepower) from the mtcars data frame. What is the slope in the disp direction?

m3 = lm(qsec~ disp + hp, data = mtcars)
m3
# summary(m3)

Call: lm(formula = qsec ~ disp + hp, data = mtcars)

Residuals: Min 1Q Median 3Q Max -2.0266 -0.9156 0.1675 0.6109 4.1752

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept) 20.454114 0.531156 38.509 < 2e-16 disp 0.004870 0.002954 1.649 0.11
hp -0.025422 0.005339 -4.761 4.93e-05 — Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.247 on 29 degrees of freedom Multiple R-squared: 0.5443, Adjusted R-squared: 0.5129 F-statistic: 17.32 on 2 and 29 DF, p-value: 1.124e-05

8. What is the slope in the hp direction?

m3 = lm(qsec~ disp + hp, data = mtcars)
# we check the estimate of the slope for hp, which in this case is -0.025422