Question

directions: show all your work. indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. a professional sports team evaluates potential players for a certain position based on two main characteristics, speed and strength.
a. 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.

	mean	standard deviation	minimum

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.
b. strength is measured by the amount of weight lifted, with more weight indicating more desirable (greater) strength. from previous strength data for all players in this position, the amount of weight lifted has a mean of 310 pounds and a standard deviation of 25 pounds, as shown in the table below.

	mean	standard deviation

calculate and interpret the z - score for a player in this position who can lift a weight of 370 pounds.
c. the characteristics of speed and strength are considered to be of equal importance to the team in selecting a player for the position. based on the information about the means and standard deviations of the speed and strength data for all players and the measurements listed in the table below for players a and b, which player should the team select if the team can only select one of the two players? justify your answer.

	player a	player b
amount of weight lifted	370 pounds	375 pounds

Explanation:

Step1: Check normality for speed - Calculate z - score for minimum time

For a normal distribution, about 99.7% of the data lies within 3 standard - deviations of the mean. The z - score is calculated as $z=\frac{x-\mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. For the minimum time $x = 4.40$ seconds, $\mu=4.60$ seconds, and $\sigma = 0.15$ seconds.
$z=\frac{4.40 - 4.60}{0.15}=\frac{- 0.20}{0.15}\approx - 1.33$. In a normal distribution, the minimum value should be around $z=- 3$ or more extreme. Since the minimum value has a z - score of only approximately - 1.33, it is not reasonable to believe that the distribution of 40 - yard running times is approximately normal.

Step2: Calculate z - score for weight lifted

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$. Given $\mu = 310$ pounds, $\sigma=25$ pounds, and $x = 370$ pounds.
$z=\frac{370 - 310}{25}=\frac{60}{25}=2.4$. The z - score of 2.4 means that a player who can lift 370 pounds is 2.4 standard deviations above the mean amount of weight lifted by players in this position.

Step3: Calculate z - scores for speed and strength for both players and make a decision

For speed:
For Player A, $x_A=4.42$ seconds, $\mu = 4.60$ seconds, $\sigma = 0.15$ seconds. $z_{A - speed}=\frac{4.42 - 4.60}{0.15}=\frac{-0.18}{0.15}=-1.2$.
For Player B, $x_B = 4.57$ seconds, $z_{B - speed}=\frac{4.57 - 4.60}{0.15}=\frac{-0.03}{0.15}=-0.2$.
For strength:
We already know for a weight of 370 pounds, $z = 2.4$. For a weight of 375 pounds, $z_{B - strength}=\frac{375 - 310}{25}=\frac{65}{25}=2.6$.
Since speed and strength are of equal importance, we can consider the sum of z - scores.
For Player A: $S_A=z_{A - speed}+z_{A - strength}=-1.2 + 2.4 = 1.2$.
For Player B: $S_B=z_{B - speed}+z_{B - strength}=-0.2+2.6 = 2.4$.
Since $S_B>S_A$, the team should select Player B.

Answer:

a. It is not reasonable to believe that the distribution of 40 - yard running times is approximately normal because the z - score of the minimum time ($z\approx - 1.33$) is not extreme enough as expected in a normal distribution (where about 99.7% of data is within $z=-3$ to $z = 3$).
b. The z - score for a player who can lift 370 pounds is 2.4, which means the player is 2.4 standard deviations above the mean amount of weight lifted by players in this position.
c. The team should select Player B. Player A has a combined z - score of 1.2 (sum of speed and strength z - scores) and Player B has a combined z - score of 2.4. Since speed and strength are of equal importance and Player B has a higher combined z - score, Player B is a better choice.