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# Probability and Random Variables Cumulative Review Problems for AP Statistics

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Review the following concepts if necessary:

### Problems

1. Consider the following histogram:
2. Which of the following statements is true and why?

1. The mean and median are approximately the same value.
2. The mean is probably greater than the median.
3. The median is probably greater than the mean.
3. You are going to do an opinion survey in your school. You can sample 100 students and desire that the sample accurately reflects the ethnic composition of your school. The school data clerk tells you that the student body is 25% Asian, 8% African American, 12% Latino, and 55% Caucasian. How could you sample the student body so that your sample of 100 would reflect this composition and what is such a sample called?
4. The following data represent the scores on a 50-point AP Statistics quiz:
5. 46, 36, 50, 42, 46, 30, 46, 32, 50, 32, 40, 42, 20, 47, 39, 32, 22, 43, 42, 46, 48, 34, 47, 46, 27, 50, 46, 42, 20, 23, 42

Determine the five-number summary for the quiz and draw a box plot of the data.

6. The following represents some computer output that can be used to predict the number of manatee deaths from the number of powerboats registered in Florida.
1. Write the least-square regression line for predicting the number of manatee deaths from the number of powerboat registrations.
2. Interpret the slope of the line in the context of the problem.
7. Use the empirical rule to state whether it seems reasonable that the following sample data could have been drawn from a normal distribution: 12.3, 6.6, 10.6, 9.4, 9.1, 13.7, 12.2, 9, 9.4, 9.2, 8.8, 10.1, 7.0, 10.9, 7.8, 6.5, 10.3, 8.6, 10.6, 13, 11.5, 8.1, 13.0, 10.7, 8.8.

### Solutions

1. II is true: the mean is most likely greater than the median. This is because the mean, being nonresistant, is pulled in the direction of outliers or skewness. Because the given histogram is clearly skewed to the right, the mean is likely to be to the right of (that is, greater than) the median.
2. The kind of sample you want is a stratified random sample. The sample should have 25 Asian students, 8 African American students, 12 Latino students, and 55 Caucasian students. You could get a list of all Asian students from the data clerk and randomly select 25 students from the list. Repeat this process for percentages of African American, Latino, and Caucasian students. Now the proportion of each ethnic group in your sample is the same as its proportion in the population.
3. The five-number summary is 20 – 32 – 42 – 46 – 50. The box plot looks like this:
4.
1. The LSRL line is: #Manatee deaths = – 41.430 + 0.12486(#boats).
2. For each additional registered powerboat, the number of manatee deaths is predicted to increase by 0.12. You could also say that the number increases, on average, by 0.12.
5. For this set of data, = 9.9 and s = 2.0. Examination of the 25 points in the dataset yields the following.
6. The actual values in the dataset (16, 25, 25) are quite close to the expected values (17, 23.8, 24.9) if this truly were data from a normal population. Hence, it seems reasonable that the data could have been a random sample drawn from a population that is approximately normally distributed.

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