Use R to re-draw Figs. 8-4 (p. 258), 8-8 (p. 262) and 10-4 (p. 357) from the text.
How robust are t confidence intervals against non-normality? Generate 1000 samples of n = 4 observations from a Weibull distribution with shape parameter = 30 and scale parameter = 2 and, for each, compute the 95% t confidence interval for the mean. Is the coverage close to the 95% level intended? How large must n be for the actual confidence level to be close enough to 95%?
(a) If you are estimating a variance, how many degrees of freedom do you need for the upper limit of a 99% confidence interval to be less than 4 times the lower limit?
(b) The mean power output of a diesel engine you manufacture is supposed to be 40 kw. From past experience, you know that the output varies between engines, with a standard deviation of 1.3 kw. If you want to test the hypothesis that the mean power output is 40 kw at the 1% level of significance and be 90% certain to detect when the mean is above 40.5 kw or below 39.5 kw, how many engines would you have to test? If you could only afford to test 10 engines, what would be your probability of Type II error? Would the test still be worth doing?
(c) A company produces 10% of a certain product domestically and 90% offshore. They are packed in lots of 100 in the factory. Domestic production has a 2% defective rate while for offshore production the rate is only 1%. If a given lot has 3 defective items, what is the probability that it was produced domestically?
Carry out appropriate analyses for the following two data sets. Give graphs. State any assumptions you make. As far as possible, test each assumption. State your conclusions.
(a) A new coal liquefaction process is being studied. It is claimed that the new process results in a higher yield of distillate synthetic fuel than the current process. The following data give the number of kg of synthetic fuel produced per kg of hydrogen consumed in the process.
New process:
16.4
17.7
15.9
11.3
12.8
12.2
14.7
14.1
15.4
18.7
Old process:
11.1
12.8
12.1
14.2
10.5
13.2
14.5
15.3
10.9
12.6
(b) It is thought that the heat loss in glass pipes is smaller than that in steel pipes of the same size. To verify this contention, nine pairs of 50-m pipe segments of assorted diameters were obtained. Each pair consisted of a glass segment and a steel segment of the same diameter. The heat loss was measured in each case.
Pair:
1
2
3
4
5
6
7
8
9
Steel:
4.6
3.7
4.2
1.9
4.8
6.1
4.7
5.5
5.4
Glass:
2.5
1.3
2.0
1.8
2.7
3.2
3.0
3.5
3.4
Ozonization as a secondary treatment for effluent, following absorption by ferrous chloride, was studied for three reaction times (time, in min) and three pH levels. The study yielded the following results for effluent decline (effdecl, in %). Give an interaction plot and a two-factor ANOVA table. Give a 95% confidence interval for the residual variance. State your assumptions and your conclusions.
effdecl
23
21
16
18
14
13
20
22
14
13
12
11
21
20
13
12
10
13
time
20
20
20
20
20
20
40
40
40
40
40
40
60
60
60
60
60
60
ph
7
7
9
9
10.5
10.5
7
7
9
9
10.5
10.5
7
7
9
9
10.5
10.5
Using the data from Question 5 above, determine if the effluent decline can be predicted as a linear function of pH when the reaction time is held at 40 min. Present your analysis in an ANOVA table with F-tests for non-linearity and for the slope of the regression line. Give a 95% confidence interval for the residual variance. State your assumptions and your conclusions.
Predict the effluent decline when the reaction time is 40 min and the pH is 8. How reliable do you think your prediction is?
14-8 (p 520). [Note: you did plots for these data in Assignment #1.]