MATH 221 Final Exam
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1. Explain the difference between a
population and a sample. In
which of these is it important to distinguish between the two in
order
to use the correct formula? mean; median; mode; range; quartiles;
variance; standard deviation.
2. The following numbers represent the
weights in pounds of six 7-
year old children in Mrs. Jones' 2nd grade class.
{25, 60, 51, 47, 49, 45}
Find the mean; median; mode; range; quartiles; variance; standard
deviation.
3. If the variance is 846, what is the
standard deviation?
4. If we have the following data
34, 38, 22, 21, 29, 37, 40, 41, 22, 20, 49, 47, 20, 31, 34, 66
Draw a stem and leaf. Discuss the shape of the distribution.
5. What type of relationship is shown by
this scatter plot?
6. What values can r take in linear
regression? Select 4 values in this
interval and describe how they would be interpreted.
7. Does correlation imply causation?
8. What do we call the r value.
9. To predict the annual rice yield in
pounds we use the equation
yˆ = 859 + 5.76x1 + 3.82x2 , where x1
represents the number of acres
planted (in thousands) and where x2 represents the number of acres
harvested (in thousands) and where r2 = .94.
a) Predict the annual yield when 3200 acres are planted and 3000
are harvested.
b) Interpret the results of this r2 value.
c) What do we call the r2 value?
10. The Student Services office did a
survey of 500 students in which
they asked if the student is part-time or full-time. Another
question
asked whether the student was a transfer student. The results
follow.
|
Transfer
|
Non-Transfer
|
Row
|
Totals
|
|
Part-Time
|
100
|
110
|
210
|
|
Full-Time
|
170
|
120
|
290
|
|
Column
Totals
|
270
|
230
|
500
|
a) If a student is selected at random
(from this group of 500
students), find the probability that the student is a transfer
student. P
(Transfer)
b) If a student is selected at random
(from this group of 500
students), find the probability that the student is a part time
student.
P (Part Time)
c) If a student is selected at random
(from this group of 500
students), find the probability that the student is a transfer
student
and a part time student. P(transfer ∩ part time).
d) If a student is selected at random
(from this group of 500
students), find the probability that the student is a transfer
student if
we know he is a part time student. P(transfer | part time).
e) If a student is selected at random
(from this group of 500
students), find the probability that the student is a part time
given he
is a transfer student. P(part time | transfer)
f) Are the events part time and transfer
independent? Explain
mathematically.
g) Are the events part time and transfer
mutually exclusive. Explain
mathematically.
Solution:
11. A shipment of 40 television sets
contains 3 defective units. How
many ways can a vending company can buy five of these units and
receive no defective units?
12. How do you recognize a discrete
distribution?
13. The random variable X represents the
annual salaries in dollars of
a group of teachers. Find the expected value E(X).
X = {$35,000; $45,000; $55,000}.
P(35,000) = .4; P(45,000) = .3; P(55,000) = .3
14. How do you recognize a binomial
experiment?
15. An advertising agency is hired to
introduce a new product. The
agency claims that after its campaign 61% of all consumers are
familiar with the product. We ask 7 randomly selected customers
whether or not they are familiar with the product.
a) Is this a binomial experiment?
Explain how you know.
b) Use the correct formula to find the
probability that, out of 7
customers, exactly 4 are familiar with the product. Show your
calculations.
16. How do you recognize a Poisson
experiment?
17. The mean number of cars per minute
going through the
Eisenhower turnpike automatic toll is about 7. Find the
probability that
exactly 3 will go through in a given minute using the correct
table,
formula, or Excel function.
18. How do you recognize a normal
distribution?
19. Label the following as continuous or
discrete distributions.
a) The lengths of fish in a certain lake.
b) The number of fish in a certain lake.
c) The diameter of 15 trees in a forest.
d) How many trees are on a farmer's acre.
20. Jack weighs 160 pounds and his sister
weighs 110 pounds. If the
mean weight for men his age is 175 with a standard deviation of 14
pounds and the mean weight for women is 145 with a standard
deviation of 10 pounds, determine whose weight is closer to
"average." Write your answer in terms of z-scores and
areas under
the normal curve.
21. On a dry surface, the braking
distance (in meters) of a certain car
is a normal distribution with mu = μ = 45.1 m and
sigma = σ = 0.5
(a) Find the braking distance that
corresponds to z = 1.8
(b) Find the braking distance that
represents the 91st percentile.
(c) Find the z-score for a braking
distance of 46.1 m
(d) Find the probability that the braking
distance is less than or
equal to 45 m
(e) Find the probability that the braking
distance is greater than
46.8 m
(f) Find the probability that the braking
distance is between 45 m
and 46.8 m.
22. A drug manufacturer wants to estimate
the mean heart rate for
patients with a certain heart condition. Because the condition is
rare,
the manufacturer can only find 14 people with the condition
currently
untreated. From this small sample, the mean heart rate is 101
beats
per minute with a standard deviation of 8.
(a) Find a 99% confidence interval for
the true mean heart rate of all
people with this untreated condition. Show your calculations.
(b) Interpret this confidence interval
and write a sentence that
explains it.
23. Determine the minimum required sample
size if you want to be
80% confident that the sample mean is within 2 units of the
population
mean given sigma = 9.4. Assume the population is normally
distributed.
24. A social service worker wants to
estimate the true proportion of
pregnant teenagers who miss at least one day of school per week on
average. The social worker wants to be within 5% of the true
proportion when using a 90% confidence interval. A previous study
estimated the population proportion at 0.21.
(a) Using this previous study as an
estimate for p, what sample size
should be used?
(b) If the previous study was not
available, what estimate for p
should be used?
25. Suppose you are performing a
hypothesis test on a claim about a
population proportion. Using an alpha = .04 and n = 90, what two
critical values determine the rejection region if the null
hypothesis is
Ho: p = 0.54?
(a) ± 1.96 (b) =± 2.05 (c) ± 2.33 (d) none of these
26. A restaurant claims that its speed of
service time is less than 15
minutes. A random selection of 49 service times was collected, and
their mean was calculated to be 14.5 minutes. Their standard
deviation is 2.7 minutes. Is there enough evidence to support the
claim at alpha = .07. Perform an appropriate hypothesis test,
showing
each important step. (Note: 1st Step: Write Ho and Ha; 2nd Step:
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