Motivation of athletes.A statistician keeps track of every serve that a player hits during the U.S. Open Tennis Championship. The statistician reports that the mean serve speed was 100 miles per hour (mph) and the standard deviation of the serve speeds was 15 mph.

a.Suppose the statistician also observes that the distribution of serve speeds was mound-shaped and symmetric. What percentage of the player’s serves was between 115 mph and 145 mph?

b.Consider the following serve speeds: 50 mph, 80 mph, and 105 mph. Using the z-score approach for detecting outliers, which of these would represent outliers in the distribution of the player’s serve speeds?

c.If nothing is known about the shape of the distribution, what percentage of the player’s serve speeds are less than 70 mph?

The question itself says that the distribution is mound-shaped and symmetric, so we will use the empirical rule to find the percentage of serves between 115 mph and 145 mph.

According to the empirical rule, 95% of the observations fall between $\overline{\mathrm{x}}\pm 2\mathrm{s}$, and 100% fall between $\overline{\mathrm{x}}\pm 3\mathrm{s}$. Because 115 and 145 are greater than 100, both will lie to the right of the mean. Therefore we will only look at the positive side.

If s = 15, $\overline{x}+15=115$ is the first standard deviation from the mean. 2^nd deviation from the mean would be $\overline{x}+2s=100+2\left(15\right)=130$ . And the 3^rd standard deviation from the mean would be $\overline{x}+3s=100+3\left(15\right)=145$.

The graph is given below:

Looking at the graph, we say that 33.32% of players’ serves were between 115 and 145.

$$\begin{gathered} \text{z-score=}\frac{\text{χ}-{\displaystyle \stackrel{-}{\chi }}}{\text{s}} \\ \text{Let'sfindthez-scorefor50mph,} \\ \text{z-score=}\frac{\text{50}-\text{100}}{\text{15}} \\ \text{=}-\text{3.33} \end{gathered}$$

Because the value of the z-score is beyond ±3, 50 mph is an outlier.

z-score of 80 mph,

$\begin{array}{c}\text{z-score=}\frac{\text{80}-\text{100}}{\text{15}}\\ \text{=}-\text{1.33}\end{array}$

80 mph is not an outlierbecause it falls between ±3.

z-score of 105 mph,

$\begin{array}{c}\text{z-score=}\frac{\text{105}-\text{100}}{\text{15}}\\ \text{=0.33}\end{array}$

105 mph is not an outlier.

If the shape of the distribution is not known, we use Chebyshev’s rule. According to which, 75% of observations lie between $\overline{x}\pm 2s$ , and 89% lie between $\overline{x}\pm 3s$. As 70 < 10, we will calculate the left side of the mean.

$\overline{x}-s=100-15=85$

$\overline{x}-2s=100-2\left(15\right)=70$

Again we will make a distribution graph to make it easier to understand.

The graph is given below:

Therefore, 50 – 37.5 = 12.5%.

12.5% of players’ serves lie below 70 mph.