Probability Distributions-Discrete

No Data Set Needed

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 5 6

Probability .04 .05 .09 ?? .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.

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.

Instructions on how to use Minitab or SPSS to calculate binomial probabilities to answer the following can be found in the online notes for this lesson by clicking the tab: Binomial Random Variable. Having this portion of the notes open the following: if interested in a single value (e.g. P(X=2)) then this is a simple PROBABILITY and select the probability button in minitab or select the PDF Non-central PDF in SPSS. If interested in less than equal to some value (e.g. P(X <= 2)) then this is a CUMULATIVE probability and select the cumulative probability button in minitab or the CDF and Non-central CDF in SPSS.

c. Assuming random guessing, what is the probability that the number of correct guesses is 13 or fewer (in minitab you will select Cumulative probability and in SPSS use CDF and Non-central CDF)?

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 or use the PDF and Non-central PDF option in SPSS.)

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.

Probability Distributions – Continuous SPSS users: Note that the method for the following is similar to that above with Binomials except if you want to find a probability these will involve using CDF and Non-central CDF but selecting cdf.normal in the functions list

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 or SPSS, find the cumulative probability for $300. [Note: The notation for the probability you’re finding is P(X ≤ $300).]

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 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. With Minitab or SPSS, find the 75th percentile of amounts spent on textbooks. {This requires you to find the inverse probability. How to do this is in the notes under finding cumulative probabilities}

j. Write a sentence interpreting the value found in the previous part that would be understood by somebody with no training in statistics.

k. 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?

l. 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 i. If the answers differ by a lot, something’s not right.]

No Data Set Needed

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 5 6

Probability .04 .05 .09 ?? .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.

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.

Instructions on how to use Minitab or SPSS to calculate binomial probabilities to answer the following can be found in the online notes for this lesson by clicking the tab: Binomial Random Variable. Having this portion of the notes open the following: if interested in a single value (e.g. P(X=2)) then this is a simple PROBABILITY and select the probability button in minitab or select the PDF Non-central PDF in SPSS. If interested in less than equal to some value (e.g. P(X <= 2)) then this is a CUMULATIVE probability and select the cumulative probability button in minitab or the CDF and Non-central CDF in SPSS.

c. Assuming random guessing, what is the probability that the number of correct guesses is 13 or fewer (in minitab you will select Cumulative probability and in SPSS use CDF and Non-central CDF)?

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 or use the PDF and Non-central PDF option in SPSS.)

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.

Probability Distributions – Continuous SPSS users: Note that the method for the following is similar to that above with Binomials except if you want to find a probability these will involve using CDF and Non-central CDF but selecting cdf.normal in the functions list

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 or SPSS, find the cumulative probability for $300. [Note: The notation for the probability you’re finding is P(X ≤ $300).]

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 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. With Minitab or SPSS, find the 75th percentile of amounts spent on textbooks. {This requires you to find the inverse probability. How to do this is in the notes under finding cumulative probabilities}

j. Write a sentence interpreting the value found in the previous part that would be understood by somebody with no training in statistics.

k. 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?

l. 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 i. If the answers differ by a lot, something’s not right.]

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