MATH253 Week 5 Tutorial
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MATH253 Week 5 Tutorial
This tutorial sheet is related to material covered in chapters 7, 8 and 9. Some of the questions will be discussed in On Campus Workshop in Week 5. Please study the chapters 7, 8 and 9 before attending the On Campus Workshop in Week 5.
Solutions will be available on Canvas on Friday at 5pm.
1. From the definition of the F distribution, show that if U ~ Fm,n , and we define V = 1/U, then V ~ Fn,m . Hence show that Fm,n (0.95) = 
, where Fm,n (α) denotes the value k such that Pr(U > k) = α .
2. For repairs to damaged cars, an insurance company can turn to two garages. To find out whether the two garages offer the same prices for repairs, seven arbitrary cars were sent to both garages to get price quotations and the following quotations were obtained (in hundreds of pounds).
|
Car |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
Garage 1 |
10.6 |
6.0 |
19.0 |
15.2 |
20.4 |
2.6 |
12.6 |
|
Garage 2 |
9.8 |
5.4 |
18.7 |
14.6 |
18.2 |
3.0 |
11.6 |
(a) Is a paired or independent test appropriate here? Explain clearly why.
(b) Compute the difference in the quotations for each car.
(c) Perform an appropriate t-test to test the hypothesis that there is no systematic difference in the quotations from the two garages. Write down your conclusion clearly.
(d) Find a 95% confidence interval for the mean difference in quotations. What is the assumption behind the construction of this interval?
3. In a study, the levels of vitamin B12 (in nanograms) were measured for each of 11 omnivores and 15 vegetarians. The recorded data are as follows.
Omnivores 0.78, 0.80, 0.76, 0.78, 0.58, 0.64, 0.92, 0.86, 0.62, 0.54, 0.46
Vegetarians 0.50, 0.32, 0.78, 0.68, 0.46, 0.32, 0.50, 0.68, 0.32, 0.86, 0.40, 0.36, 0.44, 0.62, 0.26
The investigators want to test the hypothesis that vitamin B12 level in omnivores is higher than the level in vegetarians, on average.
(a) Is a paired or independent test appropriate here? Explain clearly why.
(b) Obtain the sample mean and sample standard deviation for each of the two samples.
(c) Perform an appropriate t-test and write your conclusion clearly.
(d) State the assumptions for the t-test you performed in part (c).
(e) Compute 95% confidence intervals for each of the population variances σ 1(2) (for omnivores) and σ2(2) (for vegetarians), and also the corresponding 95% confidence intervals for the standard deviations σ 1 , σ2 .
(f) Test H0 : σ 1 = σ2 against the alternative σ 1 
σ2 . (Note: if the F distributions with the correct degrees of freedom does no appear in your tables, use the F distribution with the nearest DF values which does appear in the tables.)
(g) Discuss the relevance of the result in (f) to the test performed in (c).
2022-11-09