AP Stats Random Variables Practice

Which of the following is the best estimate of the standard deviation of the distribution in the figure?

• 18

• 9

• 54

• 97

• 106

A company determines the mean and standard deviation of the employees' salaries in one year. What is the best description of the standard deviation?

• Approximately the median distance between the individual salaries of employees and the mean of every employee's salary.

• The amount of money separating the highest salary from the lowest salary when considering the middle 50% of the distribution.

• The amount of money separating the highest salary from the lowest salary when considering all employees.

• The distance between the salary of an employee and the mean salary of all the employees.

• Approximately the mean distance between the individual salaries of employees and the mean of all employees' salaries.

The heights of adult women are approximately normally distributed about a mean of 65 inches with a standard deviation of 2 inches. If Bella is at the 87th percentile in height for adult women, then her height, in inches, is closest to

• 63

• 67

• 66

• 72

• 70

In a certain school, students can choose whether to eat in the school’s cafeteria. A reporter working for the school’s newspaper polled students on their reactions to changes in the menu at the cafeteria. For each student leaving the cafeteria in one 20-minute time period, the reporter used a die to determine whether to stop the student and ask how he or she felt about the new menu. In the reporter’s article it was stated that a random sample of the students showed that 23% of the school’s student population was happy with the new menu. Which of the following statements is true?

• Because each student leaving the cafeteria was randomly selected and could choose to answer or not, this is a random sample of the student population, and the 23% is an accurate measurement of the school population’s view of the new menu.

• Because students self-selected whether to eat in the cafeteria, the sampling method might be biased and the sample might not be representative of all students in the school.

• The survey would have been more effective if the reporter had collected the data in one 10-minute time period rather than in one 20-minute time period.

• The survey would have been more effective if students who cared about the food could have called the reporter to tell how they felt about the new menu, so that only students with opinions on the subject would have been surveyed.

• Because no treatment was imposed on the students eating in the cafeteria, one cannot make any conclusions about the new menu.

The boys and girls basketball teams at a high school had their heights measured at practice. The following data was recorded for their heights. What can you conclude from the distributions?

• The girls have a larger median and a larger range in heights.

• The boys had a smaller median height than the girls; however, the boys had a larger range of heights.

• The median height for the boys and the girls is the same; however, the girls have a larger range.

• The median height for the boys is larger; however, the girls have a larger range.

• No information can be concluded from the distribution since there is no key included.

The boxplots summarize two data sets, 1 and 2. Which of the following must be true?

• I only

• I and III only

• I and II only

• I, II, and III

• III only

A psychologist studied the number of puzzles that subjects were able to solve in a 5-minute period while listening to soothing music. Let 𝑋 be the number of puzzles completed successfully by a randomly chosen subject. The psychologist found that 𝑋 had the following probability distribution. The standard deviation of 𝑋 is 0.9. Let 𝐷 be the difference in the number of puzzles solved by two randomly selected subjects in a 5-minute period. What is the standard deviation of 𝐷?

• 1.8

• 0.81

• 0

• 1.27

• 0.9

A certain vending machine offers 20-ounce bottles of soda for $1.50. The number of bottles 𝑋 bought from the machine on any day is a random variable with mean 50 and standard deviation 15. Let the random variable 𝑌 equal the total revenue from this machine on a randomly selected day. Assume that the machine works properly and that no sodas are stolen from the machine. What are the mean and standard deviation of 𝑌?

• 𝜇𝑌 = $75,

  𝜎𝑌 = $18.37

• 𝜇𝑌 = $75,

  𝜎𝑌 = $22.50

• 𝜇𝑌 = $1.50,

  𝜎𝑌 = $22.50

• 𝜇𝑌 = $75,

  𝜎𝑌 = $33.50

• 𝜇𝑌 = $1.50,

  𝜎𝑌 = $33.75

Answer explanation

Y=1.5X and we multiply the mean and standard deviation by 1.5 to obtain the mean and standard deviation of 𝑌.

The weight of tomatoes chosen at random from a bin at the farmer’s market follows a Normal distribution with mean 𝜇 = 10 ounces and standard deviation 𝜎 = 1 ounce. Suppose we pick four tomatoes at random from the bin and find their total weight 𝑇. The random variable 𝑇 is

• Normal, with mean 10 ounces and standard deviation 1 ounce.

• Normal, with mean 40 ounces and standard deviation 4 ounces.

• Binomial, with mean 40 ounces and standard deviation 2 ounces.

• Normal, with mean 40 ounces and standard deviation 2 ounces.

• Binomial, with mean 40 ounces and standard deviation 4 ounces.

Answer explanation

Remember that you must find the square root of the variance after adding the variances!

The figure shows the probability distribution of a discrete random variable 𝑋. Which of the following best describes this random variable?

• Binomial with 𝑛 = 8, 𝑝 = 0.8

• Binomial with 𝑛 = 8, 𝑝 = 0.1

• Binomial with 𝑛 = 8, 𝑝 = 0.3

• Geometric with 𝑝 = 0.1

• Geometric with 𝑝 = 0.2

Answer explanation

We can see the probability of X = 0 is 0.05, but X = 0 is impossible in a geometric distribution.

Also, we know this must be binomial because 𝑋 = 1 is not the tallest bar.

Also, we are given that 𝑛 = 8. We can estimate the mean at about 2.4 and using 𝑛𝑝 = 8𝑝= 2.4, we find that 𝑝 = 0.3.