265
APPENDIX C
SOLUTIONS TO PROBLEMS
C.1 (i) This is just a special case of what we covered in the text, with n = 4: E(
Y
) = µ and Var(
Y
) =
2/4.
C.2 (i) E(Wa) = a1E(Y1) + a2E(Y2) + + anE(Yn) = (a1 + a2 + + an)µ. Therefore, we must
have a1 + a2 + + an = 1 for unbiasedness.
C.3 (i) E(W1) = [(n – 1)/n]E(
Y
) = [(n – 1)/n]µ, and so Bias(W1) = [(n – 1)/n]µ – µ = –µ/n.
Similarly, E(W2) = E(
Y
)/2 = µ/2, and so Bias(W2) = µ/2 – µ = –µ/2. The bias in W1 tends to
zero as n → , while the bias in W2 is –µ/2 for all n. This is an important difference.
C.4 (i) Using the hint, E(Z|X) = E(Y/X|X) = E(Y|X)/X =
X/X =
. It follows by Property CE.4,
the law of iterated expectations, that E(Z) = E(
) =
.
Y
Y
266
(ii) This follows from part (i) and the fact that the sample average is unbiased for the
population average: write
(iii) In general, the average of the ratios, Yi/Xi, is not the ratio of averages,
2/.W Y X=
(This
(iv) For the n = 17 observations given in the table – which are, incidentally, the first 17
C.5 (i) While the expected value of the numerator of G is E(
Y
) =
, and the expected value of
the denominator is E(1 –
Y
) = 1 –
, the expected value of the ratio is not the ratio of the
C.6 (i) H0: µ = 0.
267
(iv) The estimated reduction, about 33 ounces, does not seem large for an entire year’s
(v) The implicit assumption is that other factors that affect liquor consumption – such as
income, or changes in price due to transportation costs, are constant over the two years.
C.7 (i) The average increase in wage is
d
d
= .24, or 24 cents. The sample standard deviation is
(iii) We have the mean and standard error from part (i): t = .24/.1164
2.062. The 5%
critical value for a one-tailed test with df = 14 is 1.761, while the 1% critical value is 2.624.
C.8 (i) For Mark Price,
y
= 188/429
.438.
C.9 (i) X is distributed as Binomial(200,.65), and so E(X) = 200(.65) = 130.
268
(iv) The evidence is pretty strong against the dictator’s claim. If 65% of the voting
C.10 Since
y
y
)
.024. We can use the standard normal approximation for the