Exercise 1

During the summer of 2018, Coldstream Country Club in Cincinnati,Ohio, collected data on 443 rounds of golf played from its white tees. A portion of the data for each golfer’s score on the twelfth hole are shown in the following table (check the file “Coldstream12.csv” for the complete dataset).

##    Hole_12_Score
## 1              6
## 2              7
## 3              5
## 4              7
## 5              5
## 6              5
## 7              5
## 8              4
## 9              5
## 10             5

  1. Construct an empirical discrete probability mass function for the player scores on the twelfth hole.

  2. A par is the score that a good golfer is expected to get for the hole. For hole number 12, par is four. What is the probability of a player scoring less than or equal to par on hole number 12?

  3. What is the expected score for hole number 12?

  4. What is the variance for hole number 12?

  5. What is the standard deviation for hole number 12?

Exercise 2

In Gallup’s Annual Consumption Habits Poll, telephone interviews were conducted for a random sample of 1014 adults aged 18 and over. One of the questions was, “How many cups of coffee, if any, do you drink on an average day?” The following table shows the results obtained (Gallup website).

##   Number of Cups per Day Number of Responses
## 1                      0                 365
## 2                      1                 264
## 3                      2                 193
## 4                      3                  91
## 5                      4                 101

Define a random variable x = number of cups of coffee consumed on an average day.

  1. Develop a probability mass function (PMF) f(x) for x.
  2. Graph f(x)
  3. Develop a cumulative distribution function (CDF) F(x) for x.
  4. Graph F(x)
  5. Compute the expected value of x.
  6. Compute the variance of x.
  7. Compute the standard deviation of x.
  8. What is the probability a person drinks at least one cup of coffee on an average day?
  9. What is the probability a person drinks at less than 4 cups of coffee on an average day?
  10. What is the probability a person drinks between 1 and 3 cups of coffee on an average day, inclusive?

Exercise 3

The CDF of a random variable x is shown in the following table.

##   x F(x)
## 1 1  1/8
## 2 2  3/8
## 3 3  3/4
## 4 4    1

Determine:

  1. the PMF of x
  2. P(1 ≤ x ≤ 3)
  3. P(x ≥ 2)
  4. P(x < 3)
  5. P(x >1.4)

Exercise 4

Suppose a company produces t-shirts with logos of a football team.Each t-shirt has a unique number from 1 to 20 assigned to it. The number assigned to each t-shirt follows a discrete uniform distribution.

  1. What is the probability that a randomly selected t-shirt will have a number between 5 and 10, inclusive?

  2. Find the expected value of the number assigned to a t-shirt.

  3. If a customer purchases three t-shirts at random, what is the probability that all three t-shirts will have even numbers?

  4. If a customer purchases two t-shirts at random, what is the probability that at least one of the t-shirts will have a number less than or equal to 5?

Exercise 5

Suppose a sample of 20 Colombians is selected as part of a study of the state of the nation. Colombians in the sample are asked whether or not they are satisfied with the way things are going in Colombia.

  1. Compute the probability that exactly 4 of the 20 Colombians surveyed are satisfied with the way things are going in Colombia.

  2. Compute the probability that at least 2 of Colombians surveyed are satisfied with the way things are going in Colombia.

  3. For the sample of 20 Colombians, compute the expected number of Colombians who are satisfied with the way things are going in Colombia.

  4. For the sample of 20 Colombians, compute the variance and standard deviation of the number of Colombians who are satisfied with the way things are going in Colombia.

Exercise 6

A survey company asked to a representative sample of teenagers in England how have they listened to music in the past 12 months. Nearly 35% of the teenagers said they use Spotify. Suppose 10 English teenagers are selected randomly to be interviewed about how they listen to music.

  1. What is the probability that none of the 10 teenagers uses Spotify?

  2. What is the probability that 4 of the 10 teenagers use Spotify?

  3. What is the probability that at least 2 of the 10 teenagers use Spotify?

  4. What is the probability that less than 3 of the 10 teenagers use Spotify?

  5. What is the probability that between 2 and 8 (inclusive) of the 10 teenagers use Spotify?

Exercise 7

In R, the runif() function can be used to generate random numbers between a and b, with b > a. If we let x denote a random number generated using runif(), then x is a continuous random variable with the following probability density function (PDF).

f(x) = 1 for a ≤ x ≤ b f(x) = 0 elsewhere

Suppose that a = 0 and b = 1.

  1. Graph the probability density function (PDF).
  2. What is the probability of generating a random number between 0.25 and 0.75?
  3. What is the probability of generating a random number with a value less than or equal to 0.30?
  4. What is the probability of generating a random number with a value greater than 0.60?
  5. Generate 50 random numbers by using runif() and create a histogram.
  6. Compute the mean and standard deviation for the random numbers in part (e).

Exercise 8

A Gallup Daily Tracking Survey found that the mean daily discretionary spending by Americans earning over $90,000 per year was $136 per day. The discretionary spending excluded home purchases, vehicle purchases, and regular monthly bills. Let x = the discretionary spending per day and assume that a uniform probability density function applies with f(x) = 0.00625 for a ≤ x ≤ b.

  1. Find the values of a and b for the probability density function.
  2. What is the probability that consumers in this group have daily discretionary spending between $100 and $200?
  3. What is the probability that consumers in this group have daily discretionary spending of $150 or more?
  4. What is the probability that consumers in this group have daily discretionary spending of $80 or less?

Exercise 9

The American Automobile Association (AAA) reported that families planning to travel over the Labor Day weekend spend an average of $749. Assume that the amount spent is normally distributed with a standard deviation of $225.

  1. What is the probability of family expenses for the weekend being less that $400?
  2. What is the probability of family expenses for the weekend being $800 or more?
  3. What is the probability that family expenses for the weekend will be between $500 and $1000?
  4. What would the Labor Day weekend expenses have to be for the 5% of the families with the most expensive travel plans?

Exercise 10

Given that z is a standard normal random variable, find z for each situation.

  1. The area to the left of z is 0.9750.
  2. The area between 0 and z is 0.4750.
  3. The area to the left of z is 0.7291.
  4. The area to the right of z is 0.1314.
  5. The area to the left of z is 0.6700.
  6. The area to the right of z is 0.3300.

References

Anderson, D.R., Sweeney, D.J., Williams, T.A., Camm, J.D. & Cochran, J.J., 2016. Statistics for Business & Economics, 13th ed. Cengage Learning, Boston.