548 Inferences About Normal Linear Models
EXERCISES
9.6.1.Obtain the least squares estimates for the modelyi=α∗+βxi+eiby min-
imizing the sum of squares given in expression (9.6.11). Determine the distribution
of ˆα∗.
9.6.2.Students’ scores on the mathematics portion of the ACT examination,x,
and on the final examination in the first-semester calculus (200 points possible),y,
are:
x 25 20 26 26 28 28 29 32 20 25
y 138 84 104 112 88 132 90 183 100 143
x 26 28 25 31 30
y 141 161 124 118 168The data are also in the rda fileregr1.rda. Use R or another statistical package
for computation and plotting.
(a)Calculate the least squares regression line for these data.(b)Plot the points and the least squares regression line on the same graph.(c)Obtain the residual plot and comment on the appropriateness of the model.(d)Find 95% confidence interval forβunder the usual assumptions. Comment
in terms of the problem.9.6.3(Telephone Data).Consider the data presented below. The responses (y)for
this data set are the numbers of telephone calls (tens of millions) made in Belgium
for the years 1950 through 1973. Time, the years, serves as the predictor variable
(x). The data are discussed on page 172 of Hettmansperger and McKean (2011)
and are in the filetelephone.rda.
Year 50 51 52 53 54 55
No. Calls 0.44 0.47 0.47 0.59 0.66 0.73
Year 56 57 58 59 60 61
No. Calls 0.81 0.88 1.06 1.20 1.35 1.49
Year 62 63 64 65 66 67
No. Calls 1.61 2.12 11.90 12.40 14.20 15.90
Year 68 69 70 71 72 73
No. Calls 18.20 21.20 4.30 2.40 2.70 2.90(a)Calculate the least squares regression line for these data.(b)Plot the points and the least squares regression line on the same graph.(c)What is the reason for the poor least squares fit?9.6.4.Show that the covariance between ˆαandβˆis zero.
9.6.5.Find (1−α)100% confidence intervals for the parametersαandβin Model
(9.6.1).