STATISTICS 401B
Fall 2014
Laboratory Assignment 7
1. The following data are tensile strength measurements on wire that is used in industrial equipment. The
wire is manufactured by drawing the base metal through a die that has a specified diameter. The data
consist of tensile strengths of 18 samples from large spools of wire made with each of three dies. In
manufacturing processes, attention is often focussed on variabiliy. The variability of the tensile strengths
may be just as important as the mean strentghs of the wire produced by the dies. A testing agency
wanted to test the hypothesis that there are differences among the three dies with respect to the variability in the tensile strength of the wire produced by each. Use the text file die.txt downloaded from
the web page to create an appropriate JMP data table.
Tensile Strength (103 psi)
Die 1
85.769
86.725
86.725
84.292
87.168
84.513
84.513
86.725
84.513
84.513
83.628
82.692
82.912
83.407
84.734
84.070
82.964
85.337
Die
79.424
81.628
82.692
86.946
86.725
85.619
84.070
85.398
86.946
2
82.912
83.185
86.725
84.070
86.460
83.628
84.513
84.292
85.619
Die
82.423
81.941
81.331
82.205
81.986
82.860
82.641
82.592
83.026
3
81.768
83.078
83.515
82.423
83.078
82.641
82.860
82.592
81.507
(a) Examine evidence that the assumptions about the population distributions required to use Hartley’s test for the above purpose are satisfied or not.
(b) Use the statistics output from executing a JMP analysis to conduct Hartley’s test by hand using
α = .05. State the null and alternative hypotheses, test statistic, and the rejection region clearly.
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(c) Extract from the JMP output the results of performing Levene’s test for testing the above hypotheses. State the value of the test statistic, the p-value and your decision based on α = .05
2. A construction company performed a study of the daily gas consumption of 20 homes heated with gas,
installed with a new foam insulation material. They set up instruments to record the temperature both
inside and outside of the homes over a six-month period of time (Oct-Mar). The average difference
in these values over this period and the average daily gas consumption (in killowatt hours, kWh) are
recorded. The gas usage and average temperature difference data were used to build a simple linear
regression model to predict gas usage with the new insulation.
Temperature Difference (◦ F ), x
Gas Consumption(kWh), y
x (Continued . . .)
y (Continued . . .)
20.1
65.3
28.8
94.9
21.1
66.5
29.2
93.9
21.9
67.8
30.6
87.1
22.6
73.2
30.8
84.2
23.4
75.3
32.6
106.6
24.2
81.1
32.4
111.3
24.9
82.2
34.8
100.9
25.1
85.7
35.9
101.9
26.0
90.9
36.0
110.1
27.2
87.4
36.5
119.1
Perform the following computations by hand calculation. Must show work for all your numerical
answers. Attach Excel sheets to show your computations for parts b), f), g), and h). Answers to
some of the questions below must be provided on separate sheets when necessary. Finally, execute a
JMP program to obtain all quantities needed and confirm that all numbers you computed by hand can
be obtained from the JMP output by hi-liting and labeling them on the JMP output. Use the
insulation.jmp JMP data file and the text file insulation.txt provided.
a) Construct a plot (using software known to you or using JMP Graph Builder) which shows the
scatter of (x, y) data points with y on the vertical axis.
b) Compute the following using Excel:
X
xi =
,
X
x2i =
,
X
yi =
,
X
yi2 =
,
X
x i yi =
c) Use the method of least squares to obtain estimates β̂0 and β̂1 of the parameters in the model
y = β0 + β1 x + using the results of part (b).
d) Give the least squares prediction equation (which is the equation of the line that yields minimum
sum of squared residuals).
e) According to this model, what is the expected increase in the average gas consumption (in
kWh) associated with a 5◦ F increase in average difference in temperature?
f) Compute a table of predicted values ŷ and residuals y − ŷ corresponding to the observed values y.
(You may use Excel for this and the next two parts).
g) Extend the table in part(f) to include columns y − ȳ and ŷ − ȳ.
h) Compute the sums of squares (y − ȳ)2 , (ŷ − ȳ)2 and (y − ŷ)2 using the table in part (g).
Explain how these give a decomposition of total variability in gas consumption into two parts and
identify the parts.
P
P
P
i) What proportion of the total variability in the gas consumption is accounted for by the explanatory
variable x (i.e., by the linear regression model)? What does this say about the role of x in predicting
y?
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j) Give the point estimate s2 of σ2
k) Compute the estimated standard error of β̂1 .
`) Construct 95% confidence interval for β1 .
m) Test the hypothesis H0 : β1 = 0 vs. Ha : β1 6= 0 using a t-statistic. Give your conclusion using
α = .05.
n) Construct an analysis of variance table using quantities computed in parts (b),(c), and Syy . Include
a column for computing the F-statistic to test the hypothesis in part (m). Use the F-tables to
determine the rejection region and state your decision.
o) Construct a 95% confidence interval for the mean gas consumption for the population of months
with an average temperature difference of 25◦ F .
p) Construct a 95% prediction interval for the prediction of gas consumption for a month recording a
average temperature difference of 25 ◦ F .
q) (JMP analysis only) Save columns of predicted values, residuals, and confidence and prediction
intervals into the JMP data table. Re-format numbers so that the output is more readable. Journal
this table and save as a Word file. Obtain a printed copy to turn in along with the printed copy of
the output from the JMP analysis. Make sure the JMP analysis contains the following plots and
analyses (in addition to the standard output):
i) A plot of the data superimposed by the fitted regression line, the confidence bands, and the
prediction interval bands.
ii) A plot of the residuals against the corresponding temperature difference (x) values.
iii) A plot of the residuals against the corresponding predicted (ŷ) values.
iv) A normal probability plot of the residuals.
r) Study the plots obtained in (ii), (iii), and (iv) of part (q) above. Use these to comment on the
adequacy of the model you fitted to this data. Are the model assumptions you made supported by
these plots?
s) The manufacturer of the insulation contends that for an average temperature difference of 25 ◦ F ,
the new insulation will keep the average monthly gas consumption below 80 kWh. Do the data
support this contention?
Due Thursday, November 6, 2014 (turn in the first 20 minutes during lab)
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