Download review problems start of semester

Review Problems for Quality Class
1. The Boy Scouts are an organization of young males between the ages of 10 and
15 years old. The weight of these young males can be assumed to be normally
distributed with a mean of 100 pounds and standard deviation of 15 pounds.
a. Draw a picture of this probability distribution and show values along the xaxes so I can be sure you drew the correct distribution.
b. If you randomly select one of these young males, what is the probability that
they weigh more than 80 pounds?
c. If you randomly select one of these young males, what is the probability that
they weigh less than 80 pounds?
d. If you randomly select one of these young males, what is the probability that
they weigh exactly 80 pounds?
2. A supplier is producing ball bearings for a major auto producer who has provided
specs on the diameter to be .30 + .012 inches. The distribution is known to be
approximately normal with standard deviation 0.002 inches. It is also known
that larger diameter ball bearings fit better and last longer resulting in a product
that will satisfy the customer for many years. It is your job to decide on a target
value for the mean diameter such that (1) almost all of the ball bearing diameters
will be within spec and (2) the diameters of the ball bearings are as large as
possible to increase the life expectancy of the product. What would you
recommend for a target value for the mean diameter? Draw a picture
(distribution) explaining your answer.
3. Fill volumes of Koke are known to have a mean of 2.00 liters with standard
deviation of 0.10 liters. If you take a random sample of 25 Kokes,
a. What is the sampling distribution of the sample mean? List any assumptions
you make.
b. Draw a picture of the distribution of fill volume and the sampling distribution
of the sample mean on the same graph.
4. The manufacturer of balloons claims that the average pressure to bust a balloon is
90 p.s.i. and the standard deviation is 10 p.s.i.. The National Child Safety Council
is concerned about the possibility that these balloons explode in the face of a
child, who is blowing them up, and causing harm to their eyes. Studies show that
children have sufficient lung capacity to blow up a balloon to a maximum of 50
p.s.i.. Therefore, as long as the balloon will hold at least 50 p.s.i., you should
have no safety issues. Before you certify these balloons as “Child Safe”, you
decide to test the hypothesis that the average busting pressure is 90 p.s.i.
[manufacturer’s claim] against the alternative hypothesis that it is not 90 p.s.i. at
a 5% level of significance [two tailed test]. You sample 25 balloons and blow
them up until they bust. You may assume that the data is normally distributed and
the standard deviation is known and equal to 10 p.s.i. [means you can use zscores] . The sample of 25 baloons resulted in a sample mean (x-bar) of 96 p.s.i.
and sample standard deviation of 9.7 p.s.i. [not needed here since std. dev. is
know to be 10 p.s.i.]
a. Test the hypothesis (two tailed test) that the mean pressure to bust a
balloon is 90 p.s.i. at a 5% level of significance.
b. What is you managerial conclusion?
c. Sketch the power curve.
d. Is there anything else you want to tell me about these balloons? [are they
safe???]
5.
The Dean has accused us of grade inflation for this quality course. In order to
look at this issue, the class grade point average for the last 12 semesters was
collected as shown below.
a. Plot a scatter diagram for this data and comment on whether you believe
the Dean’s claim and why.
b.. Think about the nature of what occurs over 12 consecutive semesters of
teaching one specific course in a department, and comment on potential
pitfalls associated with this analysis.
GPA
SEMESTER
2.94
1
2.79
2
3.15
3
3.41
4
3.44
5
3.60
6
2.86
7
3.30
8
3.62
9
3.23
10
3.62
11
3.25
12
6. At the end of the semester, a professor wishes to possibly curve grades for a class
of 36 students. The class average grade is 75 with a standard deviation of 6. Assume
grades are normally distributed.
a. What percent of the students would you estimate have a passing grade (>
60)?
b. What percent of the students would you estimate have a failing grade (<
60)?
c. If you wanted to give approximately 20% of the students an “A”, what is
the lowest grade you could make and still get an “A”?
7. Historically, quality students average 75 with a standard deviation of 6 on the first
test.
a. For a class of 16 students, what is the sampling distribution of the class
average on the first test? What is the probability that the class average is
above 76.5? What assumptions do you have to make in order to work this
problem?
b. For a class of 100 students, what is the sampling distribution of the class
average on the first test? What is the probability that the class average is
above 76.5? What assumptions do you have to make to work this
problem?
8. Historically, the standard deviation of student grades on the first test is known to
be 6 (assume this is true). The professor tells the class that historically student
average 75 on the first test. After the professor hands back the first test, he states that
your class average for the 36 students who took the test was 73.5 . In order to harass
the professor, the students go to the Dean and claim that the professor lied to them
about class averages on the first test being 75 (since their class averaged 73.5).
Prepare the professor’s defense using hypothesis testing by answering the following
questions.
a. State the null and alternative hypothesis (assume a two tailed test).
b. At the 0.05 level of significance, is there evidence that the mean
grade is different from 75?
c. Compute the p-value and explain what this means (use pictures if
you wish)
d. Sketch the power curve for this hypothesis test.
9. Random Number Simulation Homework:
a. Generate 100 random numbers from a normal distribution with mean 20 and
standard deviation 0.5 . Plot a histogram of the data. Select your bin ranges
(cell boundaries) such that you have a pretty histogram.
b. Generate 100 random numbers from a normal distribution with mean 23 and
standard deviation 0.5 . Plot a histogram of the data. Select your bin ranges
(cell boundaries) such that you have a pretty histogram.
c. Now combine all the data (200 observations) in problems 1 and 2 above and
plot a histogram.
d. Put all 3 histograms on one sheet and comment on what you see.
10. Normal Distribution:
a. Given a normal distribution with mean 20 and standard deviation 0.5,
calculate the following probabilities.
a. P(X < 20.7)
b. P(X < 20.7)
c. P(X > 20.7)
d. P(18 < X < 20.7)
b. Assume fill volume of coke is approximately normally distributed with a
standard deviation of 0.5 fl.oz. If the bottle has a label value of 20 fl.oz. and
you want to set the mean fill at a value such that 99% of your bottles have at
least 20 fl.oz., what would you suggest your target be for the mean fill.
c. Assume the fill volume of coke is approximately normally distributed with
mean 20 fl.oz. and standard deviation 0.05 fl.oz. If your marketing people
wish to put an ad on TV that states “At least 95% of our cokes have at least
X(?) fl.oz., what should “X” be?