Practice 2 Solutions

3. Calculate the standard deviation.
   $\left\{18, 20, 20, 21, 23\right\}$

$$\begin{gathered} \mu =\frac{18+20+20+21+23 }{5}=\frac{102}{5}=20.4 \\ {\left(x_i–\mu \right)}^2 \\ {(18-20.4)}^2=5.76 \\ {\left(20–20.4\right)}^2=0.16 \\ {\left(20–20.4\right)}^2=0.16 \\ {\left(21–20.4\right)}^2=0.36 \\ {\left(23–20.4\right)}^2=6.76 \\ \sum _{}{\left(x_i–\mu \right)}^2=13.2 \\ \sigma =\sqrt{\frac{13.2}{5}}=1.62 \end{gathered}$$

$1.6$

4. Name the mean and standard deviation.

$$\begin{gathered} 55–51=4 \\ 145–135=10 \end{gathered}$$

$$\begin{gathered} \mathrm{Resting}: \mu =55, \sigma =4 \\ \mathrm{Post}‐\mathrm{Workout}: \mu =145, \sigma =10 \end{gathered}$$

5. What percentage of athletes’ heart rates were higher than the mean?

50% of the data is always greater than the mean.

In both resting and post-workout, 50% of athletes’ heart rates are greater than the mean.

6. If 125 athletes were monitored, approximately how many had a post-workout heart rate between 155 and 165 bpm?

$\begin{array}{rcl}\mu +2\sigma & =& 13.5\%\\ 0.135\left(125\right)& =& 16.875\approx 17\end{array}$

A total of 17 athletes have a post-workout heart rate between 155 and 165 bpm.

7. If 125 athletes were monitored, how many have a resting heart rate less than 51 bpm?

$\begin{array}{rcl}1–\left(50\%+34\%\right)& =& 16\%\\ 0.16\left(125\right)& =& 20\end{array}$

20 athletes have a resting heart rate less than 51 bpm.

8. If 150 athletes were monitored, how many fell between one standard deviation from the mean?

$\begin{array}{rcl}\mu \pm \sigma & =& 68\%\\ 0.68\left(150\right)& =& 102\end{array}$

102 athletes

9. Write an inequality that represents 34% of the post-workout data above the mean (i.e., $\_\_< x< \_\_$).

See the Post-Workout graph.

$145<x<155$

10. The coach created a new workout for the athletes. It increased their average heart rate by 5 bpm, but decreased the variance by 3 bpm. Sketch a graph for the new workout (Workout 2).

11. The team coach tells the athletic trainer that during a game, the heart rates of the athletes vary greatly. Which workout should the trainer recommend to the coach?

$$\begin{gathered} \mathrm{Post}‐\mathrm{Workout} \mathrm{Heart} \mathrm{Rate} \left(\mathrm{first} \mathrm{workout}\right): \mu =145, \sigma =10 \\ \mathrm{P}\mathrm{ost}‐\mathrm{W}\mathrm{orkout} \mathrm{Heart} \mathrm{Rate}, \mathrm{Workout} 2: \mu =150, \sigma =7 \end{gathered}$$

The trainer should recommend the first workout because the athletes had a greater variability of heart rates.

12. Calculate the standard deviation. $\left\{12, 16, 18, 22, 24, 28\right\}$

$$\begin{gathered} \mu =\frac{12+16+18+22+24+28 }{6}=\frac{120}{6}=20 \\ {\left(x_i–\mu \right)}^2 \\ {(12–20)}^2=64 \\ {\left(16–20\right)}^2=16 \\ {\left(18–20\right)}^2=4 \\ {\left(22–20\right)}^2=4 \\ {\left(24–20\right)}^2=16 \\ {\left(28–20\right)}^2=64 \\ \sum _{}{\left(x_i–\mu \right)}^2=168 \\ \sigma =\sqrt{\frac{168}{6}}=5.3 \end{gathered}$$

$5.3$