- Chapter 6 Q 13: Let be the population mean. Then
so we must evaluate . The definition
so that .
- Chapter 6 Q 15: The moment of order r is which is given
Make the substitution for which
For the special case r=2 we get the formula in the book (using
- According to the fact so that
. Thus is unbiased.
- You square these numbers, add then divide by 20 and get 74.505 as
in the appendix.
- Chapter 6 Q 28:
- The log of the density is
so that the log likelihood is
To maximize this we take the derivative with respect to and
set the result equal to 0 getting
which has root which is the same as
the unbiased estimate above.
- The median, m, of the distribution satisfies
The integral may be done by substituting to get
Solve this to get and then the mle of m
- Chapter 8 Q 26b: We are given so that the
test rejects when t exceeds the t critical value on 7 degrees
of freedom. The nearest curve in Table A.13 is for 6 degrees of
freedom and the quantity d is (4.00-3.5)/1.25= .4
For 6 df this appears to correspond to a of around 0.76
while for 9 df we would get about 0.65. My estimate is about a third
of the way between them (because 7 is a third of the way between
6 and 9) or roughly .72.
- Chapter 8 Q 30b: See the formula on page 319 and plug in
so that n=24 would be needed. Notice that the formula for the sample
size for a two sided level test is just the same as that\
for a one sided level test.
It is also acceptable to do this using the t-test graphs as in the previous
question. In fact, I think this is what the text intended really. If so
you get a sample size of 19.
- Chapter 9 Q 6 b,c: The rejection region of the test is
which can be rewritten as
The power function at is then the area to the
right of the right hand side of this last formula under a standard
normal curve. Plugging in numbers shows that we want the area
to the left of -0.5 which is .31. That is approximately.
To get a probability of type II error equal to 0.1 we need
Putting m=40 and plugging all the other numbers we get
which leads to n of roughly 37 (actually a bit over so rounding up
to 38 would be normal).
- Chapter 9 Q 12: This is just the formula at the foot of
page 351. We get
which we round up to 50 for safety's sake.
- Chapter 9 Q 73: This is a paired comparison problem. You take
the differences, getting 3.2, -3.4, 0.4, 0.5, 0.3, -1.4, -0.3, 0, 3.5,
-3.7, 3.7, 3.9, 1.6, 3.2. The one sample t statistic is 1.22 on 13
degrees of freedom leading to a 1 sided P value of 12% which is only
weak evidence of higher TSI for the treatment than for the placebo.