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Probability and Sampling Distributions
1 In each part, indicate, (1) whether the variable is discrete or continuous AND (2) whether it is
binomial or not AND (3) if it is binomial, give values for n and p.
a. Number of times a “head” is flipped in 10 flips of a coin
b. Time to complete a 30 question multiple choice test
c. Number of correct answers on a 30 question multiple choice test for somebody who randomly
guesses at every question. There are four answer choices for each question,
d. A woman buys a lottery ticket every week for which the probability of winning anything at all
is 1/10. She continues to buy them until she has won 3 times. X = the number of tickets she buys.
2 In Stat 200 last year, students were asked to rate how much they liked various kinds of music on
a scale of 1 (don’t like at all) to 6 (like very much). Following is a probability distribution (with
one probability not given) for females’ rating of Top 40 music.
X=Rating
1 2 3 4
Probability .04 .05 .09 ??
5 6
.32 .26
a. Is X = music rating a discrete variable or a continuous variable? Explain.
b. What must be the value of the probability for X = 4 (the probability that rating equals 4)?
Explain how you determined this.
c. Determine E(X), the mean rating given by females. {HINT: Find (rating*probability) for each
rating, and then add up those values.}
d. Find the probability that the rating given is 3 or less. Note: When we find the probability that a
variable is less than or equal to some value it is called a cumulative probability. To find it, we add
probabilities for all values up to and including that point.
e. For each rating value, determine the cumulative probability.
X = Rating
1
2
3
4
5
6
Cumulative
Probability
f. Use the cumulative probabilities just found as an aid in finding the probability that the rating of
a randomly selected student would be greater than 4. Show work.
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3 Suppose somebody randomly guesses at every one of 20 True-False questions.
a. The number of correct guesses is a binomial random variable. What are the values of the
parameters n and p?
n=
p=
b. What is the expected number and standard deviation of correct guesses at the n = 20 answers?
Show how you determined this.
Use Minitab to find the probability for parts c through g. Instructions on how to do this are
given in this week’s Lab Activity folder.
c. Assuming random guessing, what is the probability that the number of correct guesses is 13 or
fewer?
d. What is the probability that somebody just guessing could get 14 or more correct? Show work
(Hint: This is the complement of the previous part.)
e. What is the probability that somebody just guessing would get 18 or more correct? (Hint:
You’ll have to deal with the probability of 17 or less first.
f. What is the probability that somebody just guessing gets exactly 10 correct? (You’ll have to
click on Probability rather than Cumulative Probability in the Minitab dialog box.)
g. What is the probability that the number correct for somebody just guessing is one of 9, 10, or
11 correct? Hint: First find separate probabilities for each of 9, 10, and 11. Show any work.
4 Suppose the amount students at PSU spent on textbooks this semester is a normal random
variable with mean μ= $360 and standard deviation σ = $90.
a. Use the empirical rule for bell-shaped data to determine intervals that will contain about 68%,
95% and 99.7% of amounts spent on textbooks by PSU students. Show work for each
interval.
b. With Minitab, find the cumulative probability for $300. Use Calc>Probability
Distributions>Normal, enter the mean and standard deviation, click on “Input Constant” and
enter 300 in the adjacent box. [Note: The notation for the probability you’re finding is P(X ≤
$300).]
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c. Write a sentence interpreting the value found in the previous part that would be understood
by somebody with no training in statistics.
d. What proportion of students spent more than $535? (Start like you did for part b, but then
you’ll have to do a “by hand” calculation using the complement rule to determine the final
answer. [Note: The notation for the probability you’re finding is P(X > $535).]
e. What is the probability that a randomly selected student will have spent between $300 and
$535? Show work. [Note: You’ll be able to utilize information from parts b and d.]
f.
Calculate a z-score for $400.
g. Use the Standard Normal Table in the text (or that linked to in the notes) to determine the
cumulative probability for the z-score you calculated in the previous part.
h. Write a sentence interpreting the value found in the previous part that would be understood
by somebody with no training in statistics.
i.
Use Minitab to find the probability a randomly selected student spent less than $400. If you
didn’t get the same answer as you did in part g, figure out why. The answers should be the
same.
j.
What proportion of students spent more than $400 on textbooks? Show any work.
k. Find the probability an amount spent is less than or equal to $240. Solve the problem using
Standard Normal Table. Show calculation of the z-score as part of your solution. (You can
verify your answer using Minitab if you wish.)
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l.
With Minitab, find the 75th percentile of amounts spent on textbooks. Toward the top of the
dialog box click “Inverse Cumulative Probability” and in the Input Constant box enter
0.75 (decimal fraction version of 75%).
m. Write a sentence interpreting the value found in the previous part that would be understood
by somebody with no training in statistics.
n. In the Standard Normal Table search for the cumulative probability most near 0.75. You’re
looking “inside” the table where the probabilities are for the value closest to 0.7500. What is
the z-score with this cumulative probability?
o. Refer to the previous part. Determine the amount spent on textbooks having the z-score found
in the previous part. Show work. [Note: Compare your answer to the answer found in part l.
If the answers differ by a lot, something’s not right.]
5 The term sampling frame refers to the group that actually had a chance to get into the sample.
Ideally, this is the same as the population of interest, but sometimes it isn’t. In the following
situation, describe the population, the sampling frame, the sample, the parameter of interest, and
the statistic.
A Gallup Poll is done using random digit dialing to reach individuals in households with
land-line telephones. The purpose is to estimate the proportion of U.S. adults who favor stronger
gun control laws. One-thousand persons are sampled, and 63% favor stronger gun control.
a. Population =
b. Sampling frame =
c. Parameter =
d. Sample =
e. Statistic =
6 To get a better understanding of the Central Limit Theorem as discussed in the lecture notes
you can visit and review a simulation program a (this program is also the one in the lecture
notes):
http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html
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