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Notation:
  "_" indicates a subscript, and "^" indicates a superscript.
  For example, "x_i^2" means "x sub i, squared".

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9.4 #9 Here's the book's answer:
  Any value of MSE satisfying 5.099 < MSE < 6.035
I think this should be:
  Any value of MSE satisfying 5.099 < MSE < 6.11

----------------------------------------------------------------------
9.4 #1 Since there's only one observation per treatment
(#replicates K=1), follow the advice on the bottom of the
9.3 (part 2) lecture notes: use SSAB for SSE and MSAB for
MSE. (This advice is also on p. 435 of the textbook.)

----------------------------------------------------------------------
9.3 #5 I suggest a first step of tabulating the data as we did on p. 2
of www.stat.wisc.edu/~jgillett/224/notes/9.3.pdf:
              | B
              | 25                                        | 37
    ----------|-------------------------------------------|-----------
  A NaCl      | 138.40 130.89 94.646 96.653 116.90 88.215 | ...
              |                                           |
    ----------|-------------------------------------------|-----------
    Na_2HPO_4 | ...                                       | ...
              |                                           |
    ----------|-------------------------------------------|-----------

Then find the sample means in this order:
  -- cells
  -- rows and columns
  -- grand
Then do the ANOVA table calculations.

Here are the data in case you want to do arithmetic in Calc or Excel:

Solution  Temperature Yield Stress (MPa)
NaCl      25          138.40 130.89 94.646 96.653 116.90 88.215
NaCl      37          92.312 147.28 116.48 88.802 114.37 90.737
Na_2HPO_4 25          120.18 129.43 139.76 132.75 137.23 121.73
Na_2HPO_4 37          123.50 128.94 102.86 99.941 161.68 136.44

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8.3 #18 Feel free to make a spreadsheet like 8.3.xls, which is posted
in the 8.3 line of the syllabus, to ease the calculations.

Remember that to use "=LINEST(...)", you need to press
"Ctrl-Shift-Enter" (Macintosh: "Command-Enter") (not just
"Enter) after entering the formula so that it will be
treated as an array formula.

[ I want you to use OpenOffice's Calc instead of Excel to
  avoid confusion, but here's a tip for Excel users. In
  Excel (but not in Calc), you need to pre-select space for
  the entire array output. This is 5 rows and (p+1) columns,
  where p is the number of x variables: one column each for
  beta_0 and beta_1 through beta_p.)  Then click in the
  formula bar above the spreadsheet and enter the =LINEST()
  formula. Then type "Ctrl-Shift-Enter" (Macintosh:
  "Command-Enter").
]

Here are the data for #18, all columns at once:
y   x_1 x_2 x_3
730 152 198  91
760 173 201  81
850 166 202  69
840 161 202  72
720 152 198  91
730 153 205  91
840 166 204  70
730 157 204  90
650 136 172  47
850 142 218  59
740 151 207  88
720 145 209  60
710 147 190  63

Here they are again, one column at a time:
y
730
760
850
840
720
730
840
730
650
850
740
720
710

x_1
152
173
166
161
152
153
166
157
136
142
151
145
147

x_2
198
201
202
202
198
205
204
204
172
218
207
209
190

x_3
 91
 81
 69
 72
 91
 91
 70
 90
 47
 59
 88
 60
 63

----------------------------------------------------------------------
8.3 #11
(b) The book's answer, ".3521", is a typo for ".3512".

(e) The "MS" column refers to "mean square", which is the "SS" ("sum
    of squares") column divided by the "DF" ("degrees of freedom")
    column.

----------------------------------------------------------------------
8.2 #3 There's a typo in the answer to (d): "The model ln(y) = ..."
should be "The model y = ..."

Here are the data as two columns:
t   y
0.0 1.000
0.1 0.999
0.2 0.998
0.3 0.997
0.4 0.996
0.5 0.995
0.6 0.993
0.7 0.992
0.8 0.990
0.9 0.988
1.0 0.987
1.1 0.985
1.2 0.983
1.3 0.981
1.4 0.979

Here are the data set two separate columns:
t  
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4

y	 
1.000
0.999
0.998
0.997
0.996
0.995
0.993
0.992
0.990
0.988
0.987
0.985
0.983
0.981
0.979

----------------------------------------------------------------------
8.1 #9 Feel free to make a spreadsheet like 8.1.6.xls, which is posted
in the 8.1 line of the syllabus, to ease the calculations.

Its key formula is "=LINEST(B2:B11;A2:A11;1;1)", which I
entered in cell E2. Then I pressed "Ctrl-Shift-Enter"
(Macintosh: "Command-Enter") (not just "Enter) so that it
would be treated as an array formula.

Note: the Ctrl-Shift-Enter seems sometimes to fail to give
array output; that is, the output is sometimes only a single
cell, which isn't enough. You can instead use the menu
choice "Insert > Function ..." which brings up a dialog box.
Check the "Array" box in the lower left corner. Then in the
"Formula" text box, type
  =LINEST(
and then use the mouse to select the ranges of cells you
want to use as input. Click "OK".

The function
  LINEST(yvalues; xvalues; allow_const; stats)
is documented at
  http://wiki.services.openoffice.org/wiki/Documentation/How_Tos/Calc:_LINEST_function
You can read more at
  http://wiki.services.openoffice.org/wiki/Documentation/How_Tos/Using_Arrays
and
  http://wiki.services.openoffice.org/wiki/Documentation/How_Tos/Calc:_Statistical_functions

Here is the data set three times, to accomodate your favorite
copy-and-paste method: first, as it appears in the book; second, as
two parallel columns; and third, as two separate columns.

(1) Here's how the data set looks in the book:
Temp  Evap  Temp Evap  Temp Evap  Temp Evap
11.8  2.4   11.8 3.8   18.6 3.5   14.0 1.1
21.5  4.4   24.2 5.0   25.4 5.5   13.6 3.5
16.5  5.0   15.8 2.6   22.1 4.8   25.4 5.1
23.6  4.1   26.8 8.0   25.4 4.8   17.7 2.0
19.1  6.0   24.8 5.4   22.6 3.2   24.7 5.7
21.6  5.9   26.2 4.2   24.4 5.1   24.3 4.7
31.0  4.8   14.2 4.4   15.8 3.3   25.8 5.8
18.9  3.0   14.1 2.2   22.3 4.9   28.3 5.8
24.2  7.1   30.3 5.7   23.2 7.4   29.8 7.8
19.1  1.6   15.2 1.2   19.7 3.3   26.5 5.1

(2) Here's the data set as two parallel columns:
Temp Evap
11.8 2.4
21.5 4.4
16.5 5.0
23.6 4.1
19.1 6.0
21.6 5.9
31.0 4.8
18.9 3.0
24.2 7.1
19.1 1.6
11.8 3.8
24.2 5.0
15.8 2.6
26.8 8.0
24.8 5.4
26.2 4.2
14.2 4.4
14.1 2.2
30.3 5.7
15.2 1.2
18.6 3.5
25.4 5.5
22.1 4.8
25.4 4.8
22.6 3.2
24.4 5.1
15.8 3.3
22.3 4.9
23.2 7.4
19.7 3.3
14.0 1.1
13.6 3.5
25.4 5.1
17.7 2.0
24.7 5.7
24.3 4.7
25.8 5.8
28.3 5.8
29.8 7.8
26.5 5.1

(3) Here's the data set as two separate columns:
Temp
11.8
21.5
16.5
23.6
19.1
21.6
31.0
18.9
24.2
19.1
11.8
24.2
15.8
26.8
24.8
26.2
14.2
14.1
30.3
15.2
18.6
25.4
22.1
25.4
22.6
24.4
15.8
22.3
23.2
19.7
14.0
13.6
25.4
17.7
24.7
24.3
25.8
28.3
29.8
26.5

Evap
2.4
4.4
5.0
4.1
6.0
5.9
4.8
3.0
7.1
1.6
3.8
5.0
2.6
8.0
5.4
4.2
4.4
2.2
5.7
1.2
3.5
5.5
4.8
4.8
3.2
5.1
3.3
4.9
7.4
3.3
1.1
3.5
5.1
2.0
5.7
4.7
5.8
5.8
7.8
5.1

You don't have to use the data directly if you prefer to use these
summary statistics I computed (x=Temperature, y=Evaporation):
  n = 40
  mean(x)=21.507
  mean(y)=4.480
  std(x)=5.244
  std(y)=1.699
  r=.690
But I recommend using the data directly.

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8.1 #1 Find s_x from n and the first sum. Find s_y from n and the
second sum. Find r from n, s_x, s_y, and the third sum. Then use these
numbers, and the formulas in the notes or textbook, to answer the
questions.

----------------------------------------------------------------------
7.4 #14 Here are the data (for pasting into Calc):

Automobile Front Rear
1 32.8 41.2 
2 26.6 35.2 
3 35.6 46.1 
4 36.4 46.0 
5 29.2 39.9 
6 40.9 51.7 
7 40.9 51.6 
8 34.8 46.1 
9 36.6 47.3

----------------------------------------------------------------------
7.4 #7,8 In #7, there are 18 cars. Each car goes 1000 miles with one
tread type (A or B), and then another 1000 miles with the other tread
type. (Presumably with the order of treads is chosen randomly for each
car.) In #8, there are 36 cars.

Here are the data for #7 (for pasting into Calc):

Car Tread A Tread B
 1    24.1    20.3
 2    22.3    19.7 
 3    24.5    22.5 
 4    26.1    23.2 
 5    22.6    20.4 
 6    23.3    23.5 
 7    22.4    21.9 
 8    19.9    18.6 
 9    27.1    25.8 
10    23.5    21.4 
11    25.4    20.6 
12    24.9    23.4 
13    23.7    20.3 
14    23.9    22.5 
15    24.6    23.5 
16    26.4    24.5 
17    21.5    22.4 
18    24.6    24.9

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6.7 #8:
(a) The question is, "what's the power, given that the level is
    alpha=.05, n=80, and the true mean is 1.6?"
    One model for the solution is p. 255 Example 6.12.

    But note that here we have
      H_0: mu = (mu_0 = 1.5)
      H_1: mu > 1.5
    For the sake of power calculations, we use mu_1 = 1.6. This means
    that a drawing would have the H_0 solid curve left of the H_1
    dotted curve; and we would reject H_0 for bar{X} > x_{critical},
    so we care about right tail probabilities.

(b) The question is, "what n is required for power=.99 and level=.05,
    given that the true mean is 1.6?"
    Two models for the solution are:
    - the last example in the 6.6and7.pdf lecture notes
    - p. 256 Example 6.13

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6.5 #2 (and others): feel free to make a spreadsheet like 6.5.5.xls,
which is posted in the 6.5 line of the syllabus, to ease the
calculation of X^2.

----------------------------------------------------------------------
6.4 #13a Finding "StDev" requires using "SE Mean" in the next
column. "SE mean" refers to the "standard error of the mean". The
"standard error" of a statistic is its standard deviation, estimated
from a sample. So "SE mean" is the standard deviation of bar{X},
estimated from a sample: s / sqrt(n). It's easy to solve this for s
(which is "StDev").

----------------------------------------------------------------------
6.1 #1 (and others) The book uses null hypotheses of the form
  H_0: mu <= mu_0,
  H_0: mu = mu_0,
and
  H_0: mu >= mu_0
I prefer to use only the second form (H_0: mu = mu_0), because it's
what both the book and I use to do a probability calculation to
evaluate H_0. Therefore I recommend that you understand any of the
book's uses of the first or third form as a use of the second form.

In particular, for 6.1 #1, I understand the book's "H_0: mu >= 5.4" as
"H_0: mu = 5.4".

----------------------------------------------------------------------
5.3 #1
To match the book's answers for parts b and c, use the "plus-four"
interval defined on p. 190 (and in the 5.3 lecture notes).

Parts d and e ask for a sample size. The formulas in the book (in the
last paragraph on p. 191 and in the box on p. 192) have typos: the
author solved for "tilde{n} = ...", but wrote "n = ...".  Since
tilde{n} = n+4, subtract 4 to get correct formulas for n. (The formula
is correct in the lecture notes.)

----------------------------------------------------------------------
4.7 #5, 6 Regarding constructing a probability plot, here's how to do
it with OpenOffice Calc.

Here's how to make a plot for #2, which is an example in the 4.7
lecture notes.

Copy the data (below) and paste in column A. Use the menu Data > Sort
if they're not already sorted. These are {x_i}.

Put "1" (no quotation marks) in cell D1 and "2" in D2. Select D1 and
D2 and drag the square in its bottom right corner to enclose all of
D1-D50. This puts "1 2 3 ... 50" in column D. These are indices {i}.

Put "=(D1-0.5)/50" in C1. Copy this formula to C2:C50 by dragging the
little square from C1 down through C50. These are {c_i}.

Put "=NORMINV(C1; AVERAGE(A$1:A$50); STDEV(A$1:A$50))" in B1. Copy
this formula to B2:B50.  These are {y_i}. (If X ~ N(mu, sigma^2), then
x=NORMINV(p, mu, sigma) sets x so that P(X < x) = p. Note that this
step combines the two steps at the top of p. 2 of the 4.7 lecture
notes, which calculated z_i and then y_i. Here we calculate y_i
directly, because the NORMINV() function gives probabilities for any
N(mu, sigma^2), not just for N(0, 1).)

Select columns A and B by clicking on A and dragging to B.
Select the menu Insert > Chart ...
1. Chart Type: choose "XY (Scatter)" and click Next
2. Data Range: (it should already be "$Sheet1.$A$1:$B$50") click Next
3. Data Series: click Next
4. Chart Elements: Choose appropriate titles and labels, click Finish

Right-click on one of the data points on the chart and choose "Insert
trend line ..." and select "Linear" regression. Check "Show
equation". Click OK.

Here are the data for #2 (from lecture 4.7):
11.6
12.6
12.7
12.8
13.1
13.3
13.6
13.7
13.8
14.1
14.3
14.3
14.6
14.8
15.1
15.2
15.6
15.6
15.7
15.8
15.8
15.9
15.9
16.1
16.2
16.2
16.3
16.4
16.5
16.5
16.5
16.6
17.0
17.1
17.3
17.3
17.4
17.4
17.4
17.6
17.7
18.1
18.3
18.3
18.3
18.5
18.5
18.8
19.2
20.3

Here are data for #5 and #6 (from Table 1.2, p. 21):
7.59
6.28
6.07
5.23
5.54
3.46
2.44
3.01
13.63
13.02
23.38
9.24
3.22
2.06
4.04
17.11
12.26
19.91
8.50
7.81
7.18
6.95
18.64
7.10
6.04
5.66
8.86
4.40
3.57
4.35
3.84
2.37
3.81
5.32
5.84
2.89
4.68
1.85
9.14
8.67
9.52
2.68
10.14
9.20
7.31
2.09
6.32
6.53
6.32
2.01
5.91
5.60
5.61
1.50
6.46
5.29
5.64
2.07
1.11
3.32
1.83
7.56

For #6, if you have data {x_i} in column A, you fill column B with
{ln(x_i}} by putting "=LN(A1)" (no quotes) in cell B1. Copy this
formula to all of column B. Then make a plot as if your data are in
column B.

I've not looked up how to put two plots in the same spreadsheet. For
now, you can save your first spreadsheet with the name "4.7.5". Then
use File > Save As ... to save it a second time as "4.7.6", and then
put the second plot (only) in this second file.

----------------------------------------------------------------------
3.3 #9 Integration by parts works for (a). Start with
  mu_X = E(X) = INTEGRAL_0^infinity t * .1e^{-.1t} dt

Let u = t and dv = .1 e^{-.1t}dt
  => du = dt and v = - e^{-.1t}

So mu_X = INTEGRAL u dv
  = uv - INTEGRAL v du
  = [-t e^{-.1t} |_0^infinity ] - INTEGRAL -e^{-.1t} dt
  = ...

For (b), it's probably easiest to start with the alternate form for
the variance, the book's equation 3.21 (also on the top of page 3 in
the handout). So we need

  sigma_X = sqrt(sigma_X^2)
    = sqrt( [INTEGRAL t^2 .1 e^{-.1t} dt] - mu_X^2 )

Integration by parts works again, this time letting u = t^2 and
dv = .1e^{-.1t}.

----------------------------------------------------------------------
1.2 #14 (c) This problem is easier if you consider the second version
of the standard deviation formula (p. 13 equation 1.5):
  s = sqrt{ [1/(n-1)] [ sum X_i^2 - n Xbar^2 ] }

Let C be the sum of the first nine terms in the sum:
  C = sum_{i=1}^{9} X_i^2

Then, writing the sum as the first nine terms plus the 10th, we have
  s = sqrt{ [1/(n-1)] [ C + X_{10}^2 - n Xbar^2 ] }
Use the original values of s, X_{10} = 100,000, and Xbar to solve for C.

Then use the new values of X_{10} = 1,000,000 and Xbar, along with
your value for C, to get the new s.