Solutions to STA 4032 Assignment 5
7-38.a)
b)
c)The value of alpha that minimizes the standard error is:
d)With a = 4 and n1=2n2, the value of alpha to choose is 8/9. The arbitrary value of =0.5 is too small and will result in a larger standard error. With =8/9 the standard error is
If =0.5 the standard error is
7-44.
7-46.
7-52a) cannot be unbiased since it will always be less than a.
b) bias = .
c)
d) P(Yy)=P(X1, …,Xn y)=(P(X1y))n=. Thus, f(y) is as given. Thus,
bias=E(Y)-a = .
e) For any n >1, n(n+2) > 3n so the variance of is less than that of . It is in this sense that the second estimator is better than the first.
8-4a) 95% CI for
b) .95% CI for
c) 99% CI for
d) 99% CI for
e) When n is larger, the CI will get narrower. The higher the confidence level, the wider the CI.
8-14.a) 95% Two-sided CI on the true mean life of a 75-watt light bulb
For = 0.05, z/2 = z0.025 = 1.96 , and x = 1014, =25 , n=20
b) 95% One-sided CI on the true mean life
For = 0.05, z = z0.05 =1.65 and x = 1014, =25 , n=20
The lower bound of the one sided confidence interval is lower than the lower bound of the two-sided confidence interval even though the level of significance is the same. This is because all of the Type I probability (or is in the left tail (or in the lower bound).
8-3295% confidence interval on mean peak power
8-42.95% lower bound confidence for the mean wall thickness
given = 4.05 s = 0.08 n = 25
t,n-1 = t0.05,24 = 1.711
4.023
There is high confidence that the true mean wall thickness is greater than 4.023 mm.
8-56a) 99% two-sided confidence interval on 2
and
b) 99% lower confidence bound for 2
For = 0.01 and n = 10, 21.67
c) 90% lower confidence bound for 2
For = 0.1 and n = 10, 14.68
d) The lower confidence bound of the 99% two-sided interval is less than the one-sided interval. The lower confidence bound for 2 is in part (c) is greater because the confidence is lower.
8-64a) 95% Confidence Interval on the true proportion of helmets showing damage
b)
c)