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STAT 5325 Test 2 – Summer 2009 – Name:_______________________
1. Based on historical data collected over the last 36 years, the probability
distribution for the points taken off [X] for this type problem is given by the
following discrete probability distribution.
X
P(X)
0
0.1
5
0.4
10
0.5
a. Calculate the mean of X
b. Calculate the variance of X
c. What is the probability that a randomly selected student has more than 0
point taken off?
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2. One out of 5 students [20%] don’t buy a book for their statistics class. In a class
of 20 students [assume these students are a random selection from the population
of all students who might take this course], calculate the following probabilities.
a. P(4 or fewer students don’t buy a book) =
b. P(exactly 4 students don’t buy a book) =
c. P(exactly 16 students did buy a book) =
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3.
With infections running wild in many hospitals, the chief administrator of Local
Hospital decided to find out how Local Hospital stacks up against the national
norm [national norm states that the average number of bacteria per square
yard of surface area should be no more than 9 bacteria/square yard]. The
number of bacteria per square yard is assumed to be a poisson random variable.
Assume that your hospital is at the maximum of the norm [true average = 9
bacteria/square yard]
a. If you go into the hospital, randomly sample one square yard of surface
area, and count the number of bacteria found, calculate the probability of
finding 19 or fewer bacteria.
b. If you actually found 20 bacteria, what would you conclude about the state
of the hospital?
c. In order to continuous monitor the state of the hospital, it was decided to
randomly sample one square foot of surface area each day to insure that
the hospital is being cleaned properly [takes too much time to sample 1
square yard]. If you do this, what would the mean of the poisson be in this
case?
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4.
If the random variable X has a normal distribution with mean 40 and std.
deviation 5, calculate the following probabilities.
a. P(X > 43) =
b. P(X < 37) =
c. P(X = 37) =
d. P(X > 65) =
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5. The amount of fluid in a bottle of IV is known to be normally distributed with a
mean of 50 grams and standard deviation of 2 grams.
a. If 16 bottles of this fluid are randomly selected from the supply room,
what is the sampling distribution of the sample mean. [be sure to provide
complete information about the sampling distribution of the sample mean]
Draw a picture of your sampling distribution and show the + 3σ limits.
b. Suppose that the population is not normally distributed. Does this change
your answer [yes or no] and explain your answer.
6. The time customers, with critical gun shot wounds, wait for service in the ER is
important since “people might die before they are treated”. You decide to do a
quick study and record the waiting time [in minutes] for 3 randomly selected gun
shot victims resulting in 5, 7, and 9 minutes.
a. Estimate the mean waiting time for customers in the ER.
b. Estimate the standard deviation of the waiting time in the ER.
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7. The amount of medication in a pill is critical to the health of the patient taking the
pill. If the average amount of medication is 8 grams, the patient will be cured.
However, if the average amount of medication is 7.2 or lower, the patient will
die. Likewise, if the average amount is 8.8 or larger, the patient will also die.
Before prescribing the pills for a patient, you wish to test the hypothesis that the
mean is 8.0 against the alternative hypothesis that it is not 8.0 at a 5% level of
significance. Since measuring the amount of medication in a pill is both time
consuming and expensive, you boss tells you to randomly sample 9 pills from all
pills stored in the hospital and conduct the following two tailed hypothesis test
Ho: μx = 8
H1: μx “not equal to” 8
The random sample of 9 pills resulted in a sample mean [X-bar] of 8.5 grams.
The manufacturer assured us that the true standard deviation of the amount of
medication is exactly 1.5 grams.
a.
Conduct the hypothesis test and provide a managerial summary of what
you found. [Feel free to use the unstandardized test statistic approach
(preferred since you need this to answer part b below), standardized test
statistic approach, or p-value approach].
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b. Sketch the power curve for this hypothesis test. Be sure to label the X and
Y axis.
c. After conducting this hypothesis test you gave several patients these pills
and were shocked when several of them died [assuming you failed to
reject the hypothesis and used the pills]? A more extensive study of the
amount of medication in the pills was then conducted resulting in the
discovery that the true average amount of medication in the pills was
actually 9 grams. The district attorney ask you in court during the trial for
malpractice to explain why your statistical hypothesis test failed to “catch”
this lot of bad pills. What is your answer. Be specific and complete in
your answer.
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8. In the previous problem, a sample of 9 pills resulted in a sample mean of 8.5
grams [recall that the standard deviation was also known to be 1.5 grams].
a. Your boss tells you that he doesn’t understand this “hypothesis testing”
stuff. All he wants is a 95% confidence interval estimate for the mean
amount of medication in the pills. What is the answer?
b. After providing the answer above, you boss tells you that this confidence
interval is too wide and he wants you to do a new study and estimate the
mean amount of medication in the pills within + 0.1 gram. How large of a
sample would you need to take to insure that your estimate for the mean
amount of medication is within + 0.1 gram with 95% confidence?