MATH253 Week 9 Tutorial
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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 + β1 xi + ei for i = 1, 2, . . . , 12,
where ei ’s are independent with ei N(0, σ2).
The following summary statistics were computed:
Σ xi = 136 Σ yi = 692
Σ xi(2) = 1958 Σ yi(2) = 46656 Σ xi yi = 9432
(a) Find the equation of the itted 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 ind the 90% prediction interval for this value.
(d) Is the slope parameter signiicant? Give an interpretation of the estimated slope parameter value.
(e) Calculate a 95% conidence interval for the intercept parameter β0. Hence test the null hypothesis H0 : β0 = 0 against H1 : β0 0 at the 5% signiicance 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 proit (y, in thousands of pounds) and research expenditure (x, in thousands of pounds) often irms:
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 + β1 xi + ei for i = 1, 2, . . . , 10,
where ei ’s are independent with ei N(0, σ2). The following summary statistics were computed:
Σ xi = 434 Σ yi = 559
Σ xi(2) = 19794 Σ yi(2) = 32279 Σ xi yi = 25231
(a) Compute the least squares estimates 0 and 1 .
(b) Write down the itted regression line.
(c) Compute an estimate of the error variance σ 2 .
(d) Calculate a 95% conidence interval for the slope parameter β1 . Hence test the null hypothesis H0 : β1 = 0 against H1 : β1 0 at the 5% signiicance level, and write down your conclusion clearly.
(e) Calculate a 95% conidence interval for the mean proit of airm with research expenditure $55000. Interpret this interval.
2023-12-19