# Tukey Stem-and-Leaf, ANOVA, Regression

Question A1: The smoke concentration (in mgm-3) is monitored over a northern industrial region every day at 16.00 hours for 75 consecutive weekdays. The results are tabulated below.

Smoke Concentration (mgm-3)

71 85 95 84 65 89 87 73 67 72 96 87 69 42 56

100 68 56 92 65 71 86 60 80 62 94 68 64 58 69

78 79 83 68 68 60 99 81 78 76 79 71 89 56 45

83 58 63 77 78 63 70 94 75 74 76 66 54 54 58

102 72 71 86 98 73 67 81 95 69 65 68 46 63 51

(a) Construct a Tukey stem-and-leaf diagram, using values 20, 30, 40,..., 100 as the stem, and hence summarise these results in the form of a frequency distribution table.

(b) Using this frequency distribution table alone (not the original tabulated results), and applying a suitable coding technique, calculate the mean and variance of this sample of smoke concentrations.

(c) Assuming that the smoke concentration is Normally distributed, construct a 94% confidence interval for m, the true (population) mean smoke concentration.

(d) Assuming now that the smoke concentration is Normally distributed with m = 72 and standard deviation s = 12 (mgm-3), evaluate the probability that the smoke concentration is

(i) at most 90 mgm-3

(ii) between 70 and 80 mgm-3.

Question B2

Identical specimens of metal are treated against corrosion by one of four methods. They are then exposed to equal volumes of acid and the amounts of corrosion assessed. The results are tabulated below.

Method Corrosion of Metal Specimens

1 96 101 93 102 94 109

2 101 104 89 95 88

3 105 105 111 100 106 103 121

4 102 104 99 114 109

Carry out an Analysis of Variance (ANOVA) to determine whether the mean level of corrosion differs significantly across these four treatment methods.

State your conclusions clearly and indicate which method is preferred (if any).

Question B3: A recent investigation into traffic flow through a single-lane system monitored both the speed and density of traffic over a number of five-minute intervals. The results are tabulated below.

Average Density Average Speed

(Vehicles/Mile) (Miles/Hour)

69 16.8

88 14.8

65 19.1

84 14.9

63 21.6

53 25.7

82 16.8

53 23.1

70 19.7

34 31.8

42 29.1

(a) Plot these observations on a scattergram.

(b) Fit the least-squares regression line of speed on density of traffic, and plot the regression line on the scattergram.

Hence predict (to the same accuracy as the original data) the speed of traffic when the density is 45 vehciles/mile.

(c) Use Analysis of Variance (ANOVA) to test the validity of this regression line.

Please see attached for full question.

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#### Solution Summary

3 questions have been answered dealing with. 1) Tukey stem-and-leaf diagram 2) Analysis of Variance (ANOVA) 3) Least-squares regression line