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Additional Review Problems for Final.
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
0
1
2
3
4
5
P(X)
0.6
0.1
0.05
0.03
0.02
0.2
a. What is the probability that a randomly selected student has more than 2 point
taken off?
2. One out of 10 students [10%] don’t buy a book for their statistics class. In a class of 25
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(3 or fewer students don’t buy a book) =
b. P(exactly 3 students don’t buy a book) =
c. P(exactly 5 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 3 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 = 3 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 4 or fewer
bacteria.
b. If you actually found 5 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 20 and std. deviation 2,
calculate the following probabilities.
a. P(X > 43) =
b. P(X < 17) =
c. P(X = 17) =
d. P(X > 19) =
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5. The amount of fluid in a bottle of IV is known to be normally distributed with a mean of
100 grams and standard deviation of 4 grams.
a. If 4 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.
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6. 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 20 grams, the patient will be cured. However, if the
average amount of medication is 17.5 or lower, the patient will die. Likewise, if the
average amount is 22.5 or larger, the patient will also die. Before prescribing the pills
for a patient, you wish to test the hypothesis that the mean is 20.0 against the alternative
hypothesis that it is not 20.0 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 4 pills from all pills stored in the hospital and conduct the following
two tailed hypothesis test
Ho: μx = 20
H1: μx “not equal to” 20
The random sample of 4 pills resulted in a sample mean [X-bar] of 18.1 grams. The
manufacturer assured us that the true standard deviation of the amount of medication is
exactly 2.0 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 pvalue 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 17.5 grams. The district
attorney asks 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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7. In the previous problem, a sample of 4 pills resulted in a sample mean of 18.1 grams
[recall that the standard deviation was also known to be 2.0 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 + 0.2 gram. How large of a sample would you need
to take to insure that your estimate for the mean amount of medication is within +
0.2 gram with 95% confidence?
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8. A long debate has been going on for many years as to the need to prescribe heavy doses
of aspirin to relieve headaches. A recent study conducted a blind test where each patient
with a headache was given similar looking pills but with different amounts of aspirin in
each pill. The patients were asked to document the time it took for the headache to go
away. Detail records were kept depicting exactly how much aspirin (X mg.) was given to
each patient and the time (Y hours) until their headache went away. This data was given
to a statistics student at North Texas State to analyze but all they knew how to do was to
plug the data into a regression package but knew nothing about how to analyze or
interpret the results. The results of the regression Y = βo + β1*X is shown on the last
page. [you may tear off this page to assist you in answering the following questions].
a. What model was found. [Y = a + b*X] or [Time = a + b*(Amount of Aspirin)].
Write down the model you found which should help you predict the amount of
time until your headache goes away as a function of the amount of aspirin taken.
b. Test the hypothesis at a 5% level of significance that the amount of aspirin taken
is significant with respect to predicting the time until their headache goes away.
Provide complete justification for your answer using the attached output.
Ho: β1 = 0
Ha: β1 = 0
c. How much of the variability in “Time until headache goes away” is explained by
your model?
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Time to Cure Headache
d. Draw the linear regression fitted to this data [part a above] on the following
scatter diagram of the data.
Plot of Fitted Model
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2
1
0
0
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Amount of Aspirin
e. Based on the scatter diagram above, what advice would you give your patients as
to the maximum amount of aspirin needed to cure your headache?
f. Does this linear model, in your opinion, fit this data? Explain your answer.
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OUTPUT A _ Regression Output: