For hints and examples of similar problems, see last year's Assignment #3 and its solutions.
Use R to re-draw Figs. 8-4 (p. 258), 8-8 (p. 262) and 10-4 (p. 357) from the text.
In lecture #23 on 2004-11-08 we saw a computer simulation to demonstrate t confidence intervals for the mean of a normal distribution with unknown variance. How robust are t confidence intervals against non-normality? Repeat the exercise, this time simulating data from an exponential distribution with mean = 10. Show that when n = 5 the coverage is less than the 95% level intended. How large must n be for the actual confidence level to be close enough to 95%?
Generate 1000 samples of size n = 10 from a N(10, 100) distribution and plot the sample mean against the sample standard deviation. Repeat for 1000 samples from an exponential distribution with mean = 10 and compare your results. Relate these result to what you found when you simulated confidence intervals.
(a) If you are estimating a variance, how many degrees of freedom do you need for the upper limit of a 95% confidence interval to be less than 3 times the lower limit?
(b) Cryptosporidium parvum is an opportunistic pathogen that is commonly found in wastewater. If natural occurrences of this pathogen are typically 150 per 500 L, what would be the expected number in a sample of 100 L? Suppose that an effluent source has 150 per 500 L 70% of the time but twice that rate 30% of the time. If 40 pathogens were found in a sample of 100 L, what is the probability that the effluent was at the higher rate of pathogen at the time the sample was collected? State any assumptions you make.
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) Concentration (nanograms/m3) of hexavalent chromium was measured inside and outside 10 different houses in a region of southwestern Ontario.
House: 1 2 3 4 5 6 7 8 9 10 Indoor: 0.22 0.18 0.28 0.34 0.18 0.12 0.29 0.08 0.39 0.28 Outdoor: 0.90 0.66 1.24 0.37 1.55 0.54 0.27 0.68 1.26 0.48(b) Fifteen silicon wafers were randomly assigned to either standard or megasonic cleaning. Only 5 wafers were assigned to megasonic cleaning because it is expensive, the remaining 10 were given standard cleaning. The number of surface defects was noted for each wafer.
Standard: 53 193 113 640 800 140 85 658 140 140 Megasonic: 26 90 546 90 120
The following experimental data show the percentage of water removed from paper (pctwr) as it passes through a dryer, for different exposure times in the dryer (etime in seconds) and at different dryer temperatures (temp in degrees F). Give an interaction plot and a two-factor ANOVA table. Give a 95% confidence interval for the residual variance. State your conclusions.
pctwr 24 26 21 25 39 34 37 40 58 55 56 53 etime 10 10 10 10 20 20 20 20 30 30 30 30 temp 100 100 100 100 100 100 100 100 100 100 100 100pctwr 33 33 36 32 51 50 47 52 75 71 70 73 etime 10 10 10 10 20 20 20 20 30 30 30 30 temp 120 120 120 120 120 120 120 120 120 120 120 120pctwr 45 49 44 45 67 64 68 65 89 87 86 83 etime 10 10 10 10 20 20 20 20 30 30 30 30 temp 140 140 140 140 140 140 140 140 140 140 140 140
Using the data from Question 5, determine if the percentage of water removed can be predicted as a linear function of exposure time when the temperature is held at 100°F. 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 percentage of water removed when the temperature is held at 100°F and the exposure time is, respectively, 0 sec, 25 sec and 60 sec. How reliable do you think your predictions are?
14-4 (p. 519). [Note: you did plots for these data in Assignment #1.]