#### Example 1, Section 11.3 notes:

# times until exhaustion when running on treadmill:
x <- c(8.4, 9.0, 9.6, 10.0, 11.0)  

# 20 km ski race times:
y <- c(71.4, 68.7, 69.4, 63.0, 62.6)

# Least-squares estimates:

lm(y ~ x)

## Section 11.4 calculations:

summary(lm(y ~ x))

# Note:  10.112^2 = estimate of V(beta_0-hat)

# Note:  1.049^2 = estimate of V(beta_1-hat)

# Note:  2.077^2 = MSE, the estimate of sigma^2

## Section 11.6 calculations:

lsfit <- lm(y ~ x)

95% CI for beta_1:

LCL <- summary(lsfit)[[4]]["x","Estimate"] - qt(.025, df=lsfit$df, lower=F)*summary(lsfit)[[4]]["x","Std. Error"]
UCL <- summary(lsfit)[[4]]["x","Estimate"] + qt(.025, df=lsfit$df, lower=F)*summary(lsfit)[[4]]["x","Std. Error"]
print(round(c(LCL,UCL),3))

# 90% CI for E(Y) at x* = 10.5:
predict(lsfit, newdata=data.frame(x=10.5), interval="confidence", level=0.90)

# 90% PI for Y* at x* = 10.5:
predict(lsfit, newdata=data.frame(x=10.5), interval="prediction", level=0.90)