Assignment 6 Economics 31 Fall 1999
Confidence Intervals and Hypothesis Tests
Reading: Mirer Chtr 18, Beals Chtr.8 (for additional help - read ahead
Mirer Chtr. 11,12,13, Beals Chtr. 9,10,11)
1. A small internet start up company is considering buying adspace
on a
major newspaper's website. Before purchasing this ad, however, the
company
wants an estimate of the average yearly computer software purchases
of the
households patronizing this newspaper's website. The company
contracts with
a market survey firm to randomly sample 100 family heads whom own
their own
computers and patronize this website daily and estimate, at the 95%
level
of confidence, the mean expenditure in the population. The contract
with
the market survey firm provides that the start up company will pay
$25 per
completed interview. Assume that the population distribution of
expenditures is normal and that the population standard deviation is
known
to be $68. Suppose the report from the market survey firm states that
the
95% confidence interval has a lower limit of $56.15 and an upper
limit of
$93.85. The report is accompanied by a bill for 100 interviews.
Explain why
the above confidence limits might lead one to suspect that the
market
survey firm did not complete 100 interviews. How many interviews
were
actually made?
2. A certain multinational tobacco firm is considering replacing
the type
of tobacco used in one of its namesake brands from blended American
style,
to an all-Bright, Virginia tobacco, labeled as stronger and less
sweet.
(Richard Kluger 116 Ashes to Ashes) They sampled 200 randomly
selected
consumers who regularly smoke cigarettes and gave each of them one
box of
each type in a blind taste test. The result was that 160 preferred
the
blended American style. Construct a confidence interval at the 95%
level of
confidence for the population proportion of cigarette consumers who
prefer
American style.
3. The Gallup Poll periodically takes a random sample of about
1500|
Americans. The percentage that favor the acceptance
(decriminalization) of
marijuana possession increased from 23% in 1979 to 29% in 1999.
a. Construct a 95% confidence interval for the population
percentage in
favor, each year
b. Find a 95% confidence interval for the change in this
percentage from
1979 to 1999.
4. To see what difference class attendance made, Professor
Hollister
sampled grades from his large statistics class of 530 students. From
the
220 students who attended class less than half the time, he took a
random
sample of 5 grades. From the remaining 310 students who attended at
least
half the time, he took an independent random sample of 5 other
grades:
irregulars regulars
41 69
81 56
52 83
69 70
62 92
a. Construct a 95% confidence interval for the mean difference
between the
two groups of students.
b. Explain, without necessarily doing the calculations, whether
the
interval would be wider or narrower under each of the following
conditions:
i. each sample contained thirty students.
ii. an 80% confidence interval was required, with everything else
the same
as in part a.
c. To what extent does part a support the contention that ''it is
worth 18
points to come to class regularly''?
5. In a random sample of 100 Democrats, 27 were in favor of
marijuana|
legalization. Another independent sample of 100 Republicans, found 22
in
favor of drug legalization. The confidence interval .02<Px -
Py<.08 was
calculated for the difference between the population proportions (for
more
info see http://www.gallup.com).
What is the probability content of this
interval?
Part 2
1. A professional photographer uses a battery pack to power his
strobe
lights when he is doing a photo session. He has purchased a new type
of
battery pack that is supposed to average 100 flashes before it
needs
recharging. He has recharged the pack 10 times. The mean number of
flashes
was 115 with a simple standard deviation of 8. Assume that the
distribution
of flashes per charge is normal. Test at the 5% significance level
the
hypothesis that the mean number of flashes per charge is equal to
100.
2. The Economic Policy Institute published data in 1999 suggesting
that
17.8 % of all households in America are headed by single females
(Cornell
University).Out of 300 households in Philadelphia randomly sampled by
a
statistician, the number of households headed by single females is
45. Test
at the 5% significance level the hypothesis that the true proportion
of
houses headed by single females is equal to 0.18.
3. There are two kinds of avocado, black and green; Person A wants
to buy the
type that has the lowest cost per ounce. However, the vendor quotes
her a
price on a per avocado basis. Suppose green avocados had a known
standard
deviation of 2 ounces and black avocados had a known standard
deviation of
3 ounces. If the mean weight of the green avocados is more than 5
ounces
above the mean weight of the black avocados, it will be cheaper to
buy the
green ones. Person A takes a sample of both types. The sample mean of
30
green avocados is 22 ounces. The sample mean of 20 black avocados is
16
ounces. At the 5% significance level, can she reject the null
hypothesis
that the mean difference is equal to 5 ounces? Assume that both
population
distributions are normal.
4. Leo Burnett is in charge of a new promotional campaign for
Philip
Morris. Philip Morris introduced a new package in 1954. According
to
Burnett's agency, ''the lowercase m at the beginning of the Marlboro
logo
served to trivialize the product. Similarly, the diagonal
peppermint
stripes that formed the chevrons on the red ''roof'' of the pack were
too
fussy distract[ing] from the clean, strong basic design''
appealing to
youth and adolescents (Kluger 181). Philip Morris's head, Joseph
Cullman,
disputes this analysis and orders a test. To test the effect of the
new
package, the company conducted a test in which the 100 outlets in a
chain
of tobacco stores were divided into two groups of equal size. The
first
group retained the old package and the second group used the new
package.
The shelf space for each design was the same at every outlet. The
average
sale per store during the test period was 350 cases for the old
design and
360 for the new design, with sample standard deviations of 35 and
40,
respectively. If the 100 outlets were considered a representative
sample of
all stores that carry Marlboro's, should the hypothesis that the new
design
will outsell the old design be rejected at the 5% significance level?
Why
might the observed gain of 10 cases per store be an unreliable
predictor of
the potential benefit of the new design?