1. The Simple Linear Regression model can be written as

Y₁ = B₂+ B₁2, +4

i=1,2,..., n

where 3, and 3, are unknown parameters and e,' s are independent and identically dis-

tributed random variables with E(e) = 0 and Var(e) = 0². Also, B, and 3₁ denotes

the least-squares estimates for the parameters 3, and 3₁.

(a) Derive the formulae for 8, and 3, directly from the sum of squared error (SSE)

formula.

[3 Marks]

(b) Derive the formulae for the variances of the estimators , and 3₁. Express the

results in terms of o2.

[4 Marks]

2. Data on horse racing were collected. The following table gives the attendance (in

thousands of people) at a racetrack and the amount of money (in millions) that was bet

on n = 20 randomly selected days. The data is also available in the file Gambling.xlsx.

(See Test4_supplementary.txt for instructions to import the Excel data into R.)

Attendance Amount Attendance Amount

(thousands (millions) (thousands) (millions)

36.5

149.8

50.6

200.2

234.0

174.3

216.1

250.1

201.8

229.4

257.7

311.3

301.7

308.9

339.4

317.4

51.4

50.9

48.8

58.2

61.5

68.5

66.7

71.2

77.0

307.8

349.2

404.5

Page 2

43.9

57.9

54.5

63.2

64.4

67.9

70.9

83.8

79.7

340.4

455.6

410.3