1. Consider again the data from Question 5 from the Week 8 lab. In a manufacturing plant, filtersare used to remove pollutants. We are interested in comparing the lifespan of 5 different types of filters. Six filters of each type are tested, and the time to failure in hours is given in the dataset filters (on the website, in csv format).
(a) Calculate s2.
(b) Calculate a 95% confidence interval for the difference in lifespan between filter types 3 and 4.
(c) Show that the hypothesis that the filters all have the same lifespan is testable.
(d) Test this hypothesis, using matrix theory.
(e) Test the same hypothesis using the linearHypothesis function from the car package.
2. Derive a formula for a prediction interval for tTβ in the less than full rank model.
3. Consider a one-way classification model with three levels. To test τ1 = τ2 = τ3 we could use either of the following matrices:
.
Show that the test statistics formed using these two matrices are the same.
Hint: find A such that AC1 = C2.
4. An industrial psychologist is investigating absenteeism among production-line workers, based ondifferent types of work hours: (1) 4-day week with a 10-hour day, (2) 5-day week with a flexible 8hour day, and (3) 5-day week with a structured 8-hour day. A study is conducted and the following data obtained of the average number of days missed:
Work plan
Number100 85 90
They also find s2 = 110.15.
(a) Test the hypothesis that the work plan has no effect on the absenteeism.
Test the hypothesis that work plans 1 and 3 have the same rate of absenteeis
