Stat 2470, Portfolio Problem Set #3, Fall 2014
Name ______________________________________
Instructions: Follow the directions for the portfolio carefully. Be sure to answer all parts of each
question as completely as possible, and provide explanations for your responses. Try to think about the
question in the context of a larger study.
1. Let 𝜇 denote the true average radioactivity level (in picocuries). The value 5 pCi/L is considered
the dividing line between safe and unsafe water. Would you recommend testing 𝐻0 : 𝜇 =
5, 𝐻𝑎 : 𝜇 > 5, or 𝐻0 : 𝜇 = 5, 𝐻𝑎 : 𝜇 < 5 ? Explain your reasoning. [Hint: Think about the
consequences of a Type I error and Type II error for each possibility.] What 𝛼 value would you
use for the case you decided on?
2. The calibration of a scale is to be checked by weighing a 10 kg test specimen 25 times. Suppose
the results of different weighings are independent of one another and that the weight on each
trial is normally distributed with 𝜎 = 0.200 𝑘𝑔. Let 𝜇 denote the true average weight reading of
the scale.
a. What hypotheses should be tested?
b. Suppose the scale is to be recalibrated if either 𝑥̅ ≥ 10.1032 𝑜𝑟 𝑥̅ ≤ 9.8969. What is the
probability that recalibration is carried out when it is actually unnecessary?
c. What is the probability that recalibration is judged unnecessary when in fact 𝜇 = 10.1?
When it’s 𝜇 = 9.8?
𝑥̅ −10
d. Let 𝑧 = 𝜎 . For what value 𝑐 is the rejection region of part (b) equivalent to the two√𝑛
tailed region of either 𝑧 ≤ −𝑐 𝑜𝑟 𝑧 ≥ 𝑐?
e. If the sample size were only 10 rather than 25, how should the procedure for part (d) be
altered so that 𝛼 = 0.05?
f. Using the test of part (e), what would you conclude from the following data:
9.981
10.006 9.857
10.107 9.888
9.728
10.439 10.214 10.190 9.793
g. Re-express the test procedure of part (b) in terms of the standardized test statistic 𝑍 =
𝑋̅−10
𝜎
√𝑛
.
h. Suppose the rejection region is {𝑥̅ : 𝑥̅ ≥ 10.1004 𝑜𝑟 𝑥̅ ≤ 9.8940} = {𝑧: 𝑧 ≥ 2.51, 𝑜𝑟 𝑧 ≤
−2.65}. What is 𝛼 for this procedure?
i. What is 𝛽 under the situation in part (h), when 𝜇 = 10.1? When 𝜇 = 9.9? Is this desirable?
j. Sketch the graph of the rejection regions for part (b) and (h).
3. Automatic identification of the boundaries of significant structures within a medical image is an
area of ongoing research. A paper from 2005 discussed a new technique for such identification.
A measure of the accuracy of the automatic region is the average linear displacement (ALD).
The paper gave the following ALD observations for a sample of 49 kidneys (in units of pixel
dimensions).
1.38
0.44
1.09
0.75
0.66
1.28
0.51
0.39
0.70
0.46
0.54
0.83
0.58
0.64
1.30
0.57
0.43
0.62
1.00
1.05
0.82
1.10
0.65
0.99
0.56
0.56
0.64
0.45
0.82
1.06
0.41
0.58
0.66
0.54
0.83
0.59
0.51
1.04
0.85
0.45
0.52
0.58
1.11
0.34
1.25
0.38
1.44
1.28
0.51
a. Summarize/describe the data.
b. Is it plausible that ALD is at least approximately normally distributed? Why or why not?
Must normality be assumed prior to calculating a confidence interval for true average ALD
or testing hypotheses about true average ALD? Explain.
c. The author of the article commented that in most cases the ALD is better than or of the
order of 1.0. Does the data in fact provide strong evidence for concluding the true average
ALD under these circumstances is less than 1.0? Carry out an appropriate test of
hypotheses.
d. Calculate an upper confidence bound for true average ALD using a confidence level of 95%
and interpret this bound.
4. Hexavalent chromium has been identified as an inhalation carcinogen and an air toxin of
concern in a number of different locales. An article gave the accompanying data on both indoor
and outdoor concentrations (nanograms/m2) for a sample of houses selected from a certain
region.
1
2
3
4
5
6
7
8
9
Indoor
0.07
0.08
0.09
0.12
0.12
0.12
0.13
0.14
0.15
Outdoor
0.29
0.68
0.47
0.54
0.97
0.35
0.49
0.84
0.86
10
11
12
13
14
15
16
17
18
Indoor
0.15
0.17
0.17
0.18
0.18
0.18
0.18
0.19
0.20
Outdoor
0.28
0.32
0.32
1.55
0.66
0.29
0.21
1.02
1.59
19
20
21
22
23
24
25
26
27
Indoor
0.22
0.22
0.23
0.23
0.25
0.26
0.28
0.28
0.29
Outdoor
0.90
0.52
0.12
0.54
0.88
0.49
1.24
0.48
0.27
28
29
30
31
32
33
Indoor
0.34
0.39
0.40
0.45
0.54
0.62
Outdoor
0.37
1.26
0.70
0.76
0.99
0.36
a. Calculate a confidence interval for the population mean difference between indoor and
outdoor concentrations using a confidence level of 95%, and interpret the resulting interval.
b. If a 34th house were to be randomly selected from the population, between what values
would you predict the difference in concentrations to lie?
c. Explain why it would be incorrect to conduct a two-sample t-test (or z-test) on this data set.
𝑝
5. In medical investigations, the ratio 𝜃 = 𝑝1 is often of more interest than the difference 𝑝1 − 𝑝2 .
2
𝑝̂
Let 𝜃̂ = 𝑝̂1 . When m and n are both large, the statistic ln(𝜃̂) has approximately normal
2
distribution with approximate mean value ln(𝜃) and approximate standard deviation
1
𝑚−𝑥
( 𝑚𝑥 +
𝑛−𝑦 2
).
𝑛𝑦
a. Use these facts to obtain a large-sample 95% confidence interval for estimating ln(𝜃), and
then a confidence interval for 𝜃 itself.
Treatment
Sample Size
Successes
Aspirin
730
141
Non-Aspirin
549
81
b. Using the data provided on the effectiveness of aspirin in colorectal cancer treatment,
calculate an interval of plausible values for 𝜃 at the 95% confidence level. What does this
interval suggest about the efficacy of the aspirin treatment?
c. Compare this result to the more traditional way of comparing proportion differences.
d. Explain why medical investigators might find one approach gives them more information
than the other?
6. In an experiment to compare bearing strengths of pegs inserted in two different types of
mounts, a sample of 14 observations on stress limit for red oak mounts resulted in a sample
mean and sample standard deviation of 8.48 MPa and 0.79 MPa respectively, whereas a sample
of 12 observations when Douglas fir mounts were used gave a mean of 9.36 MPa and a standard
deviation of 1.52. Test whether or not true average stress limits are identical for the two types
of mounts. Compare degrees of freedom and P-values for the unpooled and pooled t-tests.
Explain why you prefer once analysis over the other in terms of what it tells you about the
underlying data.
7. An article reports the following data on total Fe (iron) for four types of iron formation
(1=carbonate, 2=silicate, 3=magnetite, 4=hematite).
1.
20.5
28.1
27.8
27.0
28.0
25.2
25.3
27.1
20.5
31.3
2.
26.3
24.0
26.2
20.2
23.7
34.0
17.1
26.8
23.7
24.9
3.
29.5
34.0
27.5
29.4
27.9
26.2
29.9
29.5
30.0
35.6
4.
36.5
44.2
34.1
30.3
31.4
33.1
34.1
32.9
36.3
25.5
Carry out an analysis of variance F test at significance level 0.01 and summarize the results in an
ANOVA table. Would the results change if you used a significance level of 0.05? Carry out
Tukey’s procedure to determine how the means of the iron formations are grouped. Use the
underscoring method and then add a paragraph to summarize the results.