AP Statistics 2.11 A Describe a Normal Distribution Questions

Practise assessing whether a distribution is approximately normal by checking symmetry, quantile spacing, standard-deviation tails, and possible multimodality.

Exam points

judge approximate normality from symmetry, mean-median agreement, quartile spacing, and tail distances
recognise that combining separated normal groups can create a bimodal mixture rather than one normal distribution

A professional sports team evaluates potential players for a certain position based on two main characteristics, speed and strength.

Speed is measured by the time required to run a distance of 40 yards, with smaller times indicating more desirable (faster) speeds. From previous speed data for all players in this position, the times to run 40 yards have a mean of 4.60 seconds and a standard deviation of 0.15 seconds, with a minimum time of 4.40 seconds, as shown in the table below.

Table for Question AP Statistics 2.11 A Describe a Normal Distribution Questions question 1 — AP Statistics

Based on the relationship between the mean, standard deviation, and minimum time, is it reasonable to believe that the distribution of 40-yard running times is approximately normal? Explain.

Part (a):
No, it is not reasonable to believe that the distribution of 40-yard running times is approximately normal, because the minimum time is only 1.33 standard deviations below the mean $\left(z=\frac{4.4-4.6}{0.15} \approx-1.33\right)$ . In a normal distribution, approximately 9.2 percent of the z-scores are below -1.33. However, there are no running times less than 4.4 seconds, which indicates that there are no running times with a z-score less than -1.33. Therefore, the distribution of 40 -yard running times is not approximately normal.

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AP Statistics 2.11 A Describe a Normal Distribution Questions question 1