Page 497 - Probability and Statistical Inference

474 9. Confidence Interval Estimation

(feet) each car travelled from the moment the brakes were applied to the
moment the car came to a complete stop was recorded. The summary statis-
tics are shown below:
Car Sample Size s
Make A n = 12 37.1 3.1
A
Make B n = 10 39.6 4.3
B
Construct a 95% two–sided confidence interval for µ – µ based on sufficient
A
B
2
statistics. Assume that the elapsed times are distributed as N(µ , σ ) and N(µ ,
A
B
σ ) respectively for the Make A and B cars with all parameters unknown. Is the
2
interval shortest on the average among all 95% two–sided confidence intervals
for µ – µ depending on sufficient statistics for (µ , µ , σ)?
B
A
A
B
9.3.7 In an experiment in nutritional sciences, a principal investigator con-
sidered 8 overweight men with comparable backgrounds which included eat-
ing habits, family traits, health condition and job related stress. A study was
conducted to estimate the average weight reduction for overweight men fol-
lowing a regimen involving nutritional diet and exercise. The technician weighed
in each individual before they entered this program. At the conclusion of the
two–month long study, each individual was weighed. The data follows:
ID# of Weight (x , pounds) Weight (x , pounds)
1 2
Individual Before Study After Study
1 235 220
2 189 175
3 156 150
4 172 160
5 165 169
6 180 170
7 170 173
8 195 180

Here, it is not reasonable to assume that in the relevant target population, X 1
and X are independent. The nutritional scientist believed that the assumption
2
of a bivariate normal distribution would be more realistic. Obtain a 95% two–
sided confidence interval for µ – µ where E(X ) = µ , i = 1, 2.
i
i
2
1
9.3.8 (Exercise 9.3.7 Continued) Suppose that the scientist believed be-
fore running this experiment that on the average the weight would go down
by at least 10 pounds when such overweight men went through their regimen
of diet and exercise for a period of two months. Can the 95% confidence