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ECON 2100
PROBLEM SET 7
Textbook questions
Chapter 7: 47, 49. Chapter 8: 55, 67, 73, 77, 83, 89, 91
Statistical investigation
These questions again use data from a survey of Albers students conducted in Fall 2014,
regarding their housing situation. The data are found in the file “housing survey data fall 2014.”
a. Use the data given to create a 95% confidence interval for the difference between the average
number of people living together on-campus, and the average number of people living together
off-campus.
b. Interpret the confidence interval created in part a. Exactly what does it tell you?
c. Use the data given to create a 95% confidence interval for the difference between the
proportion of on-campus students that own cars, and the proportion of off-campus students that
owns cars.
d. Interpret the confidence interval created in part c. Exactly what does it tell you?
Everyday statistics
Please answer the following questions based on the article “Certain Cognitive Behavior Therapy
Appears Beneficial for Female Veterans with Post Traumatic Stress Disorder” (the article
follows the questions).
1. Create a 95% confidence interval for the difference between the two treatment groups
(prolonged exposure and present-centered therapy) in the proportion of women who no longer
meet the criteria for diagnosis of PTSD. Fully interpret the interval. Does the confidence
interval show evidence of a difference between the two treatments?
2. Create a 95% confidence interval for the difference between the two treatment groups in the
proportion of women who achieved total remission. Fully interpret the interval.
3. According to this evidence, which treatment is more effective?
4. Based on the evidence, would you recommend that one of these treatments be adopted as the
standard treatment for all female veterans suffering from PTSD?
1
CERTAIN COGNITIVE BEHAVIORAL THERAPY APPEARS BENEFICIAL FOR
FEMALE VETERANS WITH POSTTRAUMATIC STRESS DISORDER
CHICAGO—Using a cognitive behavioral therapy called “prolonged exposure”
appears more effective than “present-centered” therapy, a supportive
intervention to treat female military veterans and active duty women with
posttraumatic stress disorder, according to a study in the February 28 issue
of JAMA.
“Events such as the terrorist attacks on September 11, 2001, the war in
Iraq, and hurricane Katrina have focused attention on posttraumatic stress
disorder (PTSD), an anxiety disorder that can result from exposure to
traumatic events like combat, rape, assault, and disaster. Posttraumatic
stress disorder is characterized by symptoms of re-experiencing the
traumatic event, avoiding reminders of the event or feeling emotionally
numb, and a state of increased psychological and physiological tension. The
disorder is associated with psychiatric and physical illnesses, reduced quality
of life, and substantial economic costs to society”, according to background
information in the article. “Lifetime prevalence in U.S. adults is higher in
women (9.7 percent) than in men (3.6 percent) and is especially high
among women who have served in the military.” There has been no prior
study to evaluate treatment for PTSD in this group.
Paula P. Schnurr, Ph.D., of the National Center for PTSD, White River
Junction, Vt., and colleagues conducted a study to compare the effectiveness
of two types of treatments for PTSD, prolonged exposure and presentcentered therapy. Prolonged exposure is a cognitive behavioral therapy
(CBT) in which a patient is asked to vividly recount a traumatic event
repeatedly until the patient’s emotional response decreases and to gradually
confront safe but fear-evoking trauma reminders. Present-centered therapy,
a supportive intervention which is typically used by Department of Veterans
Affairs (VA) clinicians to address the problems of female veterans with PTSD,
includes discussing and reviewing general daily difficulties that may be
manifestations of PTSD.
The randomized controlled trial included female veterans (n = 277) and
active-duty personnel (n = 7) with PTSD who were recruited from nine VA
medical centers, two VA readjustment counseling centers, and one military
hospital, from August 2002 through October 2005. Participants were
randomly assigned to receive prolonged exposure (n = 141) or presentcentered therapy (n = 143), delivered in 10 weekly 90-minute sessions.
PTSD symptom severity data were collected before and after treatment and
at 3- and 6-month follow-up.
2
The researchers found that women who received the prolonged exposure
therapy were more likely than women who received the present-centered
therapy to no longer meet criteria for the diagnosis of PTSD (41.0 percent
vs. 27.8 percent) and were more than twice as likely to achieve total
remission (15.2 percent vs. 6.9 percent). Self-reported PTSD, depression,
and overall mental health improved from pretreatment to post-treatment in
both groups. Anxiety decreased and quality of life improved with prolonged
exposure.
“Practice guidelines for PTSD recommend prolonged exposure and other
CBT, but the treatments are not widely used. Along with recent findings, our
study demonstrates the feasibility of implementing CBT across a range of
clinical settings. With the high prevalence of PTSD among military personnel
returning from service in Iraq and Afghanistan, the challenge for large health
care systems like those of the VA and the Department of Defense is to find
efficient ways to train personnel to promote dissemination of these effective
treatments,” the authors conclude.
(JAMA. 2007;297:820-830. Available pre-embargo to the media at
www.jamamedia.org)
Editor’s Note: This study was conducted with support from the VA
Cooperative Studies Program and the Department of Defense.
For More Information: Contact the JAMA/Archives Media Relations
Department at 312/464-JAMA (5262) or email: mediarelations@jamaarchives.org .
3
47. In telephone interviews with a random sample of 1100 adults nationwide conducted
by the American Research Group, shoppers said they were planning to spend an average
of $976 for gifts this holiday season (source: American Research Group,
Inc., americanresearch group.com/holiday/).
1.
If the standard deviation of planned spending for the sample was $180, show the
95% confidence interval estimate of the average planned spending for the population of
shoppers represented by the sample.
2. Suppose you wanted to reduce the margin of error in the interval in part a by 20%
without sacrificing the 95% confidence level. How big a sample would be
required?
49. You plan to take a random sample from a population of 500 items and build a 95%
confidence interval estimate of the population mean.
a. You want a margin of error no bigger than ±5. You estimate the population
standard deviation to be 100. How large a sample is needed?
b. Suppose you want the margin of error to be no bigger than ±4. How large a
sample would be needed?
55. You want to estimate the proportion of gas stations in the state that have
stopped accepting cash. You take a simple random sample of 150 stations. Twelve
stations in the sample indicate that they have stopped accepting cash.
1.
Build a 95% interval estimate of the overall station proportion based on
your sample findings.
2. Build a 99% interval estimate of the overall station proportion based on
your sample findings.
67. The administration wants to estimate the proportion of students at Pitcairn
College (enrollment: 12,000) who favor a change in the pass/fail grading system.
A random sample of 200 students is selected without replacement. 85 students in
the sample express support for the change.
1.
Construct a 90% confidence interval estimate of the population proportion
here.
2. Construct a 90% confidence interval estimate of the total number of
students at Pitcairn who favor the change.
73. You are planning to construct a 90% confidence interval estimate of a
population proportion. You want your estimate to have a margin of error of .03.
What is the minimum required sample size if the population proportion is
expected to be approximately
1.
.05
2. .85
3. .55
77. A survey of wage scales for workers of comparable skill levels was
conducted in different regions of the country. In the sample of 500 office
workers in the Northeast, the average hourly wage was $11.84, with standard
deviation of $1.23. In the Southeast, a sample of 400 office workers showed
an average of $9.48, with standard deviation of $2.04.
Construct the appropriate 90% confidence interval estimate of the overall
difference in average hourly wage for the population of office workers in the two
regions.
83. The effect of store layout on sales is of strong interest to retailers. In an
experiment involving the layout of non-food sections of supermarkets in the
United Kingdom, researchers compared the impact of switching to alternative
layouts—including the boutique layout and the free-flow layout—for a number of
similar-sized, randomly selected supermarkets that had been using the standard
grid (parallel aisles) layout (source: The Location and Merchandising of non-food
in Supermarkets, C. Hart, M. Davies, utdt.edu). Assume that the data below
shows the change in sales per square foot (converted to US$) for six stores that
switched to the boutique layout in their nonfood sections and six stores that
switched to the free-flow layout.
1-Compute the sample means and sample standard deviations (s1 and s2) here.
1. Compute the pooled estimate of population standard deviation.
2. Using the pooled estimate in part b, construct and interpret the 95%
confidence interval estimate of the difference between the average change
in sales for all similar-sized stores that would switch to the boutique layout
and the average change in sales for all similar-sized stores that would
switch to the free-flow layout. Be sure to use the appropriate t-score for
your interval. For the two populations represented, assume that the salesper-square-foot values are normally distributed, with equal standard
deviations.
89. Vannex Inc. has received two large shipments of blu-ray amplifiers,
one from supplier A and one from supplier B. You have tested a sample of
14 amplifiers from each shipment to determine if there is a difference in
the average maximum volume without distortion between the units in the
two shipments. The average maximum volume for the sampled units in
shipment A is 260 decibels, with a standard deviation of 18 decibels; for
the sample of units from shipment B, the sample average is 228 decibels,
with a standard deviation of 21 decibels. Show the 90% confidence interval
estimate for the difference in average maximum volume without distortion
for the two shipments. (Assume the two population distributions are
normal, with equal standard deviations.)
91. FGI Research reports that 47% of Hispanic-Americans and 46% of AfricanAmericans own their homes (source: fgiresearch.com/Hispanics.htm). If this
result is based on a random sample of 2000 Hispanic-American adults and 2000
African-American adults, build and interpret the 95% confidence interval
estimate of the difference in the proportion of home owners in the two
populations represented.
…
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