Aardvark Auto Rental rents vehicles. Each day Aardvark Automotive Rental keeps track of whether its SUVs were returned in “failing” condition. Aardvark must use more of its resources to restore each SUV with a “failing” returned condition. Aardvark rents an average of five SUVs each day. The condition of returned SUVs is recorded for many days in which five SUVs were rented.

Download the file titled SUV Failures. It contains a scatter plot of the number of failures versus frequency. To compare the results to the binomial distribution, complete the following:

- Explain why SUV rental condition scenario can be a binomial experiment.
- Using the SUV Failure scatter plot, construct a frequency distribution for the number of failures.
- Compute the mean number of failures. The formula for the mean is as follows:

The terms x represent the total number of failures (0, 1, 2, 3, 4, 5) and f is the corresponding frequency (number of days where xᵢ failures occurred).

Explain what the numerical result means.

- From the frequency distribution, construct the corresponding relative frequency distribution.Explain why the relative frequency distribution table is a probability distribution.Then, use Excel to create a scatter plot of the probability distribution:Select the two columns of the probability distribution. Click on INSERT, and then go to the Charts area and select Scatter. Then choose the first Scatter chart (the one without lines connecting).
- Using the frequency distribution, what is the SUV failure average? In part 3, note that the numerator in the formula for the mean is the total number of failures. The total number of trials is the denominator of the formula for the mean multiplied by 5. What does this average mean?
- The Binomial Distribution is uniquely determined by n, the number of trials, and p, the probability of “success” on each trial. Using Excel, construct the Binomial Probability Distribution for five trials, n, and probability of success, p, as the SUV failure average in part 5. Here is an explanation of the BINOM.DIST function in Excel. (Links to an external site.)

For example, In Excel

=BINOM.DIST(7,15,0.7, FALSE)

represents the probability of 7 successes out of 15 (n) trials. The 0.7 is the probability of success, p.

Using the above value of n=5 with probability of success, p, as the SUV failure average in part 5, what is the probability of at least three SUV failures?

- Using the formula for the mean of the Binomial Distribution, what is the mean number of failures in part 6 above?
- In Excel, create a scatter plot for the Binomial Distribution. The instructions for creating a scatter plot are in part 4 above.
- Use the results above to compare the probability distribution of SUV failures and the Binomial Distribution. Compare the means in parts 4 and 6, too.
- If the probability distribution of SUV failures and the Binomial Distribution differ, explain why that is so.
- Do you think the Binomial Distribution is a good model for the SUV failure scenario? Why or why not?
- How can Aardvark Auto Rental use the binomial distribution to approximate SUV failures?
- In what other scenarios can Aardvark Auto Rental use the Binomial Distribution? Explain.

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