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MATH253 Week 9 Tutorial
This tutorial sheet is related to material covered in chapter 12. Some of the questions will be discussed in
On Campus Workshop in Week 9. Please study the chapter 12 before attending the On Campus Workshop
in Week 9.
Solutions will be available on Canvas on Friday 5pm.
1. Consider the following data on the numbers of hours that twelve persons studied for a test and their
scores on the test.
Hours studied x 6 9 12 14 3 9 12 22 1 17 18 13
Test score y 36 50 64 64 18 34 68 103 32 71 89 63
It has been suggested that y and x are linearly related and the relationship can be modelled as
yi = β0 + β1xi + i for i = 1, 2, . . . , 12,
where i
’s are independent with i ～ N

0, σ2

.
The following summary statistics were computed:
Pxi = 136 Pyi = 692
Px
2
i = 1958 Py
2
i = 46656 Pxiyi = 9432
(a) Find the equation of the fitted regression line.
(b) Compute an estimate of the error variance σ
2
.
(c) Predict the test score of a person who studied 10 hours for the test and find the 90% prediction
interval for this value.
(d) Is the slope parameter significant? Give an interpretation of the estimated slope parameter value.
(e) Calculate a 95% confidence interval for the intercept parameter β0. Hence test the null hypothesis
H0 : β0 = 0 against H1 : β0 6= 0 at the 5% significance level, and write down your conclusion
clearly.
(f) Perform an appropriate test to test whether there is evidence that the intercept β0 is larger than
10. Give a practical interpretation of the result.
2. The following data are observations on the profit (y, in thousands of pounds) and research expenditure
(x, in thousands of pounds) of ten firms:
Firm: 1 2 3 4 5 6 7 8 9 10
x : 40 45 30 48 60 41 36 60 42 32
y : 50 60 40 65 70 55 48 72 54 45
It has been suggested that y and x are linearly related and the relationship can be modelled as
yi = β0 + β1xi + i for i = 1, 2, . . . , 10,
where i
’s are independent with i ～ N

0, σ2

. The following summary statistics were computed:
Pxi = 434 Pyi = 559
Px
2
i = 19794 Py
2
i = 32279 Pxiyi = 25231
(a) Compute the least squares estimates βb0 and βb1.
(b) Write down the fitted regression line.
(c) Compute an estimate of the error variance σ
2
.
(d) Calculate a 95% confidence interval for the slope parameter β1. Hence test the null hypothesis
H0 : β1 = 0 against H1 : β1 6= 0 at the 5% significance level, and write down your conclusion
clearly.
(e) Calculate a 95% confidence interval for the mean profit of a firm with research expenditure ￡55000.
Interpret this interval.
2