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Homework #9 [due : April 17, 2008]
Math 241
NAME :
LAB SEC # :
1. A can of lemon pie mix contains an average of 30 oz with a standard
deviation of 2 oz. The contents of the cans follow a Normal model.
Let T be the total amount of lemon pie mix in four randomly chosen
cans. So T is a sum!
(a) Compute E(T).
(b) Calculate SD(T). [assume independence]
(c) Find the probability that four cans contain more than 125 oz
lemon pie mix total, or P(T > 125).
of
2. In the 1990s, the US Justice Dept and other groups studied possible
abuse by Philadelphia police officers in their treatment of minorities.
One study conducted by the American Civil Liberties Union(ACLU),
analyzed whether African American drivers were more likely than others
in the population to be targeted by police for traffic stops. In this
study, they had 262 police stops during one week of which 207 were
African Americans. At the time, Philadelphia’s population was 42.2%
African American. Does this suggest possible bias? Is it higher
than we’d expect if we take into account ordinary random variation ?
Assume p = 0.422 and treat the 262 police car stops as independent
trials. Note that they wouldn’t be if once a car was stopped, the
police followed that car and stopped it repeatedly.
(a) Compute the mean and standard deviation for this binomial dist.
[refer to page 384]
 = __________
 = __________
(b) Check the Success/Failure condition. Is it satisfied? ________
(c) Using the 68-95-99.7 Rule, find the bounds for the three standard
deviations case. We expect 99.7% of the observations to be within
three standard deviations of the mean. For this problem, we
would not be surprised if between
  3 = _____________
and
 + 3 = ____________
of the 262 people stopped were African Americans.
(d) What about the 207 in the sample ?
Your conclusion ?
3. According to a survey completed by the Pew Internet & American Life
Project, about 28% of Internet users say they download either music
or video files. Assuming the percentage is correct, use the normal
approximation to the binomial to find the probability that in a survey
of 2000 American Internet users at most 570 download music/video files.
 = ________
 = ___________
P(X < 570) = ?
4. SIMULATION WITH SPSS !!
The American Red Cross says that about 45% of the US population has
Type O blood. The experiment is to select a random sample of n = 20
people and determine whether they are Type O or not. Our statistic
of interest is p̂ , the proportion in the sample who are Type O.
(a) Refer to the Instructions for running the simulations. You will be
constructing three histograms of p̂ values to copy(paste) here.
In addition to n = 20, you are simulating for n = 40 and n = 100.
**[be sure to shrink your histograms, but not too small !!]**
(b) Write a sentence or two describing what you observe about shapes,
centers and spreads as n increases from 20 to 100.
5. This problem is a modification of Exercise #18, p.425. It’s believed
that 4% of children have a gene that may be linked to juvenile
diabetes. Researchers hoping to track twenty of these children for
several years test 752 newborns for the presence of this gene.
(a) Let p̂ represent the proportion of children in the sample of 752
newborns who have juvenile diabetes. What is the appropriate
model for the sampling distribution of p̂ ? Specify the name,
the mean and the standard deviation.
(b) Use your model to approximate the probability that the researchers
find enough subjects for their study, or P( p̂ > 20/752) .
6. The weight of potato chips in a medium size bag is stated to be ten
ounces. The amount that the packaging machine puts in these bags
follows the N(10.2,0.12) model with units of ounces.
(a) What fraction of bags sold are underweight or less than 10 ounces ?
(b) Some of the chips are sold in “bargain packs” of 3 bags. What’s the
probability none of the three is underweight ?
(c) Specify the sampling distribution models for y when n = 3 and when
n = 24. [name, center, spread]
(d) What is the probability that the mean weight of three bags is below
the stated amount, or P( y < 10) ? Also, compute the same probability
for a 24-bag case of potato chips.
7. TRUCKING ALONG : On many highways, state police conduct inspections of
driving log books from large trucks to see if the trucker has driven
too many hours in a day. At one truck inspection station they issued
citations to 49 of 348 truckers that they checked.
(a) Check the various conditions to verify the sampling distribution
for p̂ can be modeled by the Normal. [independence; random condition;
10% condition; success/failure]
(b) Construct and interpret a 95% confidence interval for p, the true
proportion of truckers who have driven too many hours in a day.