Final Exam, 12 December 1996Instructor: Richard Lockhart
Instructions: This is an open book test.
You may use notes, text, other books
and a calculator. Your presentations of statistical analysis will be
marked for clarity of explanation. I expect you to explain what
assumptions you are making and to comment if those assumptions seem
unreasonable. The exam is out of 75.
- The Intelligence Quotient, or IQ, as measured by a certain
psychological test is supposed to have a population distribution with mean
100 and standard deviation 15. Suppose I suspected that the mean IQ of
children in Vancouver city public schools is higher than 100. How large a
sample of children should I draw to give me at least a 90% chance of
concluding that my suspicion is correct if the true mean IQ of such
Vancouver children is 105. Assume that any test will be conducted at
the level 5%. Make whatever assumptions are necessary but be sure to
explain these assumptions. [5 marks]
- Two materials, called A and B for making the soles of shoes
are to be compared. For each of 10 boys a coin is tossed to determine
which sole (left or right shoe) to make out of material A, the other
shoe getting material B. Outputs A, B and C present different analyses of
the data. The response variable is amount of wear, larger values
corresponding to shoes which are more worn out.
- Is there a difference in wear between the two sole materials?
[5 marks]
- Give a 98% confidence interval for the difference in mean wear
between material A and material B.
[3 marks]
- Explain why each boy wore 1 shoe of each type rather than having
5 boys wear type A shoes on both feet and 5 boys wear type B.
[1 mark]
-
What would be wrong with putting A on left shoes and B on the right
on each boy?
[1 mark]
- The attached SAS output, D, shows parts of the input and output for the
analysis of the results of the following experiment. A total of 48 plots
of land were each fertilized with either fertilizer I, II or III - 16
plots of land receiving each fertilizer.
Each plot was then planted with either seed type A or B or C or D. In each
group of 16 plots 4 were chosen at random to plant with each seed type.
Total yield was measured for each plot.
- From the incomplete output produce a complete ANOVA table for the
analysis of this data set. [5 marks]
- State and test relevant hypotheses, describing conclusions in
real world terms in so far as possible. You will have to do fixed
level testing in this problem. [5 marks]
- Which fertilizer produces the highest yield? [2 marks]
- If I ran the analysis again without an interaction term what would
the analysis of variance table be?
You need not give the P-value column. [3 marks]
- One model for regression fits a straight line with no intercept.
The model is
where the 
are independent, have mean 0 and all have the
same variance 
which is unknown. There are n pairs with the
numbers 
for 
being known values of some covariate.
If this model is fitted by least squares, (that is by minimizing
) then the least squares estimate of
is
However, an alternative estimate is
- Show that the estimator
is unbiased. [3 marks] - Compute (give a formula for) the standard error of .
[3 marks]
- Show that the mle of in this model is
, the least
squares estimate, if the have normal distributions.
[4 marks]
- A certain machine is used to measure the breaking strength
of samples of wire. However it is feared that measurements depend on the
operator using the machine. Five batches of wire are made and each is
split into 4 pieces. Each of the four pieces is randomly assigned
to one of 4 machine operators, each operator getting one piece.
The breaking strength of each piece is measured.
Two possible analyses are SAS outputs E and F. Analyse the data in the
most appropriate way, stating real world conclusions clearly and
determining which operators are clearly different from which others.
[10 marks]
- Samples of 7 microwave ovens are obtained from each of
4 companies. The maximum power output in watts is measured for
each oven. Use SAS output G.
- Is there a difference between companies in mean power
output? [5 marks]
- Is the mean power output for company B under 550 watts? [5 marks]
- Give a 90% confidence interval for the ratio of population
standard deviations for companies C and D. [5 marks]
- The yield of a chemical reaction is measured 3 times at each of
four temperatures. A simple linear regression is run; see output H.
- Give a 95% confidence interval for the slope.
[5 marks]
- Give a 90% confidence interval for the average
yield at 200 degrees.
[5 marks]
SAS output A
DATA
A B A shoe
13.2 14.0 L
8.2 8.8 L
10.9 11.2 R
14.3 14.2 L
10.7 11.8 R
6.6 6.4 L
9.5 9.8 L
10.8 11.3 L
8.8 9.3 R
13.3 13.6 L
CODE
options pagesize=60 linesize=80;
data shoes;
infile 'shoes.dat';
input A B Ashoe $ ;
proc glm data=shoes;
model B = A;
run;
OUTPUT
General Linear Models Procedure
Dependent Variable: B
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 55.74764638 55.74764638 333.73 0.0001
Error 8 1.33635362 0.16704420
Corrected Total 9 57.08400000
R-Square C.V. Root MSE B Mean
0.976590 3.702087 0.4087104 11.040000
Source DF Type III SS Mean Square F Value Pr > F
A 1 55.74764638 55.74764638 333.73 0.0001
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 0.247447347 0.41 0.6931 0.60475347
A 1.015291877 18.27 0.0001 0.05557678
SAS output B
DATA
13.2 A
14.0 B
8.2 A
8.8 B
10.9 A
11.2 B
14.3 A
14.2 B
10.7 A
11.8 B
6.6 A
6.4 B
9.5 A
9.8 B
10.8 A
11.3 B
8.8 A
9.3 B
13.3 A
13.6 B
CODE
data shoes;
infile 'shoes1.dat';
input wear material ;
proc sort data=shoes;
by material;
proc ttest cochran;
class material;
run;
OUTPUT
TTEST PROCEDURE
Variable: WEAR
MATERIAL N Mean Std Dev Std Error
A 10 10.63000000 2.45132617 0.77517740
B 10 11.04000000 2.51846514 0.79640861
Variances T Method DF Prob>|T|
Unequal -0.3689 Satterthwaite 18.0 0.7165
Cochran 9.0 0.7207
Equal -0.3689 18.0 0.7165
For H0: Variances are equal, F' = 1.06 DF = (9,9) Prob>F' = 0.9372
SAS output C
DATA
As in A
CODE
data shoes;
infile 'shoes.dat';
input A B Ashoe $;
diff=A-B;
proc means mean std stderr t prt maxdec=2;
run;
OUTPUT
TTEST PROCEDURE
Variable: WEAR
MATERIAL N Mean Std Dev Std Error
A 10 10.63000000 2.45132617 0.77517740
B 10 11.04000000 2.51846514 0.79640861
Variances T Method DF Prob>|T|
Unequal -0.3689 Satterthwaite 18.0 0.7165
Cochran 9.0 0.7207
Equal -0.3689 18.0 0.7165
For H0: Variances are equal, F' = 1.06 DF = (9,9) Prob>F' = 0.9372
Variable Mean Std Dev Std Error T Prob>|T|
A 10.63 2.45 0.78 13.71 0.0001
B 11.04 2.52 0.80 13.86 0.0001
DIFF -0.41 0.39 0.12 -3.35 0.0085
SAS output D
DATA
8.83 I A
9.20 I A
9.22 I A
9.16 I A
9.80 I B
10.10 I B
9.87 I B
9.67 I B
9.16 I C
9.20 I C
9.54 I C
9.73 I C
9.20 I D
9.66 I D
9.58 I D
9.52 I D
8.98 II A
8.76 II A
9.08 II A
8.53 II A
9.92 II B
9.51 II B
9.29 II B
10.22 II B
9.18 II C
8.95 II C
8.83 II C
9.08 II C
9.42 II D
10.02 II D
9.66 II D
9.03 II D
8.49 III A
8.44 III A
8.29 III A
8.53 III A
8.80 III B
9.01 III B
9.03 III B
8.76 III B
8.53 III C
8.61 III C
8.57 III C
8.49 III C
8.80 III D
8.98 III D
8.83 III D
8.89 III D
CODE
data yield;
infile 'seedfert.dat';
input yield fert $ seed $ ;
proc glm data=yield;
class fert seed;
model yield = fert|seed;
means fert / tukey cldiff alpha=0.05;
means seed / tukey ;
run;
OUTPUT
General Linear Models Procedure
Dependent Variable: YIELD
Sum of Mean
Source DF Squares Square
FERT 5.22792917
SEED 3.56582292
0.40427083
Error 1.95137500
Corrected Total 11.14939792
Tukey's Studentized Range (HSD) Test for variable: YIELD
Alpha= 0.05 Confidence= 0.95 df= 36 MSE= 0.054205
Critical Value of Studentized Range= 3.457
Minimum Significant Difference= 0.2012
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
FERT Confidence Between Confidence
Comparison Limit Means Limit
I - II -0.01495 0.18625 0.38745
I - III 0.57317 0.77438 0.97558 ***
II - I -0.38745 -0.18625 0.01495
II - III 0.38692 0.58812 0.78933 ***
III - I -0.97558 -0.77438 -0.57317 ***
III - II -0.78933 -0.58812 -0.38692 ***
Tukey's Studentized Range (HSD) Test for variable: YIELD
Alpha= 0.05 df= 36 MSE= 0.054205
Critical Value of Studentized Range= 3.809
Minimum Significant Difference= 0.256
Means with the same letter are not significantly different.
Tukey Grouping Mean N SEED
A 9.49833 12 B
A
A 9.29917 12 D
B 8.98917 12 C
B
B 8.79250 12 A
SAS output E
DATA
89 1 A
88 1 B
97 1 C
94 1 D
84 2 A
77 2 B
92 2 C
79 2 D
81 3 A
87 3 B
87 3 C
85 3 D
87 4 A
92 4 B
89 4 C
84 4 D
79 5 A
81 5 B
80 5 C
88 5 D
CODE
data strength;
infile 'bhhp281q1.dat';
input strength batch $ operator $ ;
proc glm data=strength;
class operator;
model yield = operator;
means operator / tukey cldiff alpha=0.05;
run;
OUTPUT
General Linear Models Procedure
Dependent Variable: STRENGTH
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 70.00000000 23.33333333 0.76 0.5318
Error 16 490.00000000 30.62500000
Corrected Total 19 560.00000000
R-Square C.V. Root MSE STRENGTH Mean
0.125000 6.434867 5.5339859 86.000000
Source DF Type III SS Mean Square F Value Pr > F
OPERATOR 3 70.00000000 23.33333333 0.76 0.5318
Tukey's Studentized Range (HSD) Test for variable: STRENGTH
Alpha= 0.05 Confidence= 0.95 df= 16 MSE= 30.625
Critical Value of Studentized Range= 4.046
Minimum Significant Difference= 10.014
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
OPERATOR Confidence Between Confidence
Comparison Limit Means Limit
C - D -7.014 3.000 13.014
C - B -6.014 4.000 14.014
C - A -5.014 5.000 15.014
D - C -13.014 -3.000 7.014
D - B -9.014 1.000 11.014
D - A -8.014 2.000 12.014
B - C -14.014 -4.000 6.014
B - D -11.014 -1.000 9.014
B - A -9.014 1.000 11.014
A - C -15.014 -5.000 5.014
A - D -12.014 -2.000 8.014
A - B -11.014 -1.000 9.014
SAS output F
CODE
data strength;
infile 'bhhp281q1.dat';
input strength batch $ operator $ ;
proc glm data=strength;
class batch operator;
model yield = batch operator;
means batch / tukey cldiff alpha=0.05;
means operator / tukey cldiff alpha=0.05;
run;
OUTPUT
General Linear Models Procedure
Dependent Variable: STRENGTH
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 7 334.00000000 47.71428571 2.53 0.0754
Error 12 226.00000000 18.83333333
Corrected Total 19 560.00000000
R-Square C.V. Root MSE STRENGTH Mean
0.596429 5.046208 4.3397389 86.000000
Source DF Type III SS Mean Square F Value Pr > F
BATCH 4 264.00000000 66.00000000 3.50 0.0407
OPERATOR 3 70.00000000 23.33333333 1.24 0.3387
Tukey's Studentized Range (HSD) Test for variable: STRENGTH
Alpha= 0.05 Confidence= 0.95 df= 12 MSE= 18.83333
Critical Value of Studentized Range= 4.508
Minimum Significant Difference= 9.781
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
BATCH Confidence Between Confidence
Comparison Limit Means Limit
1 - 4 -5.781 4.000 13.781
1 - 3 -2.781 7.000 16.781
1 - 2 -0.781 9.000 18.781
1 - 5 0.219 10.000 19.781 ***
4 - 1 -13.781 -4.000 5.781
4 - 3 -6.781 3.000 12.781
4 - 2 -4.781 5.000 14.781
4 - 5 -3.781 6.000 15.781
3 - 1 -16.781 -7.000 2.781
3 - 4 -12.781 -3.000 6.781
3 - 2 -7.781 2.000 11.781
3 - 5 -6.781 3.000 12.781
2 - 1 -18.781 -9.000 0.781
2 - 4 -14.781 -5.000 4.781
2 - 3 -11.781 -2.000 7.781
2 - 5 -8.781 1.000 10.781
5 - 1 -19.781 -10.000 -0.219 ***
5 - 4 -15.781 -6.000 3.781
5 - 3 -12.781 -3.000 6.781
5 - 2 -10.781 -1.000 8.781
Tukey's Studentized Range (HSD) Test for variable: STRENGTH
Alpha= 0.05 Confidence= 0.95 df= 12 MSE= 18.83333
Critical Value of Studentized Range= 4.199
Minimum Significant Difference= 8.1485
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
OPERATOR Confidence Between Confidence
Comparison Limit Means Limit
C - D -5.149 3.000 11.149
C - B -4.149 4.000 12.149
C - A -3.149 5.000 13.149
D - C -11.149 -3.000 5.149
D - B -7.149 1.000 9.149
D - A -6.149 2.000 10.149
B - C -12.149 -4.000 4.149
B - D -9.149 -1.000 7.149
B - A -7.149 1.000 9.149
A - C -13.149 -5.000 3.149
A - D -10.149 -2.000 6.149
A - B -9.149 -1.000 7.149
SAS output G
DATA
560 1
546 1
547 1
548 1
559 1
559 1
544 1
477 2
468 2
523 2
484 2
524 2
527 2
457 2
455 3
481 3
506 3
492 3
468 3
450 3
448 3
460 4
503 4
482 4
526 4
462 4
545 4
534 4
data wattage;
infile 'erglep656q17.dat';
input watts company ;
proc sort data=wattage;
by company;
proc glm data=wattage;
class company;
model watts = company;
means company / tukey cldiff;
run;
OUTPUT
General Linear Models Procedure
Dependent Variable: WATTS
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 24136.678571 8045.559524 12.24 0.0001
Error 24 15779.428571 657.476190
Corrected Total 27 39916.107143
R-Square C.V. Root MSE WATTS Mean
0.604685 5.079281 25.641299 504.82143
Tukey's Studentized Range (HSD) Test for variable: WATTS
Alpha= 0.05 Confidence= 0.95 df= 24 MSE= 657.4762
Critical Value of Studentized Range= 3.901
Minimum Significant Difference= 37.809
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
COMPANY Confidence Between Confidence
Comparison Limit Means Limit
1 - 4 12.33 50.14 87.95 ***
1 - 2 19.76 57.57 95.38 ***
1 - 3 42.62 80.43 118.24 ***
4 - 1 -87.95 -50.14 -12.33 ***
4 - 2 -30.38 7.43 45.24
4 - 3 -7.52 30.29 68.09
2 - 1 -95.38 -57.57 -19.76 ***
2 - 4 -45.24 -7.43 30.38
2 - 3 -14.95 22.86 60.67
3 - 1 -118.24 -80.43 -42.62 ***
3 - 4 -68.09 -30.29 7.52
3 - 2 -60.67 -22.86 14.95
SAS output H
DATA
150 77.4
150 76.7
150 78.2
200 84.1
200 84.5
200 83.7
250 88.9
250 89.2
250 89.7
300 94.8
300 94.7
300 95.9
CODE
data yield;
infile 'yield.dat';
input temp yield ;
proc glm data=yield;
model yield = temp;
run;
OUTPUT
General Linear Models Procedure
Dependent Variable: YIELD
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 509.25066667 509.25066667 1317.25 0.0001
Error 10 3.86600000 0.38660000
Corrected Total 11 513.11666667
R-Square C.V. Root MSE YIELD Mean
0.992466 0.718950 0.6217717 86.483333
Source DF Type III SS Mean Square F Value Pr > F
TEMP 1 509.25066667 509.25066667 1317.25 0.0001
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 60.26333333 80.96 0.0001 0.74439685
TEMP 0.11653333 36.29 0.0001 0.00321082
Richard Lockhart
Tue Feb 10 20:08:13 PST 1998