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Chapter 6: Q 12. (page 369)

Skee BallAna is a dedicated Skee Ball player (see photo in Exercise 4) who always rolls for the 50-point slot. The probability distribution of Ana’s score Xon a randomly selected roll of the ball is shown here. From Exercise 8, μX=23.8.

(a) Find the median of X.

(b) Compare the mean and median. Explain why this relationship makes sense based on the probability distribution.

Short Answer

Expert verified

Part (a) 20

Part (b) Distribution is right-skewed.

Step by step solution

01

Part (a) Step 1. Given information.

The given information is:

02

Part (a) Step 2. Find the median of X.

ScoreProbabilityCumulative Probability
100.320.32
200.270.59
300.190.78
400.150.93
500.071

The median will be the score for the category with a cumulative probability of at least 0.5 and a probability of less than 0.5 for the prior category.

The category 20 has a cumulative probability of 0.59, while the category 10 has a cumulative probability of 0.32.

As a result, the median is set at 20.

03

Part (b) Step 1. Compare the mean and median.

The given mean is 23.8. 20 is the median that we calculated.

We can see that the mean is more than the median, indicating that the distribution is skewed to the right. This is because the mean is influenced by unusual values more strongly than the median, and thus there appear to be unusually large values in the distribution which are affecting the mean.

This is consistent with the conclusion we reached in the previous exercise about the form of the histogram.

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Most popular questions from this chapter

Airlines typically accept more reservations for a flight than the number of seats on the plane. Suppose that for a certain route, an airline accepts 40reservations on a plane that carries 38passengers. Based on experience, the probability distribution of Y=the number of passengers who actually show up for a randomly selected flight is given in the following table. You can check that μY=37.4andσY=1.24.

There is also a crew of two flight attendants and two pilots on each flight. Let X=the total number of people (passengers plus crew) on a randomly selected flight.

a. Make a graph of the probability distribution of X. Describe its shape.

b. Find and interpret role="math" μX.

c. Calculate and interpret σX.

Benford’s law and fraud

(a) Using the graph from Exercise 21, calculate the standard deviation σY. This gives us an idea of how much variation we’d expect in the employee’s expense records if he assumed that first digits from 1 to 9 were equally likely.

(b) The standard deviation of the first digits of randomly selected expense amounts that follow Benford’s law is σX=2.46. Would using standard deviations be a good way to detect fraud? Explain your answer.

Benford’s law Faked numbers in tax returns, invoices, or expense account claims often display patterns that aren’t present in legitimate records. Some patterns, like too many round numbers, are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the first digits of numbers in legitimate records often follow a model known as Benford’s law. 4 Call the first digit of a randomly chosen legitimate record X for short. The probability distribution for X is shown here (note that a first digit cannot be 0).

Part (a.) A histogram of the probability distribution is shown. Describe its shape.

Part (b). Calculate and interpret the expected value of X.

Ms. Hall gave her class a 10-question multiple-choice quiz.

Let X=the number of questions that a randomly selected student in the class answered correctly. The computer output gives information about the probability distribution of X. To determine each student’s grade on the quiz (out of 100), Ms. Hall will multiply his or her number of correct answers by 5and then add 50.Let G=the grade of a randomly chosen student in the class.

Easy quiz

a. Find the median of G.

b. Find the interquartile range (IQR) of G.

A company’s single-serving cereal boxes advertise 1.63ounces of cereal. In fact, the amount of cereal X in a randomly selected box can be modeled by a Normal distribution with a mean of 1.70ounces and a standard deviation of 0.03ounce. Let Y=the excess amount of cereal beyond what’s advertised in a randomly selected box, measured in grams (1ounce=28.35grams).

a. Find the mean of Y.

b. Calculate and interpret the standard deviation of Y.

c. Find the probability of getting at least 1grammore cereal than advertised.
See all solutions

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