Question

the quarterback rating (qbr) is a statistical measure used to evaluate a quarterbacks performance. here are the top 30 quarterback ratings for a recent season in the national football league (nfl).

30 35 40 45 50 55 60 65 70
quarterback rating

n | mean | sd | min | q1 | med | q3 | max
30 | 54.37 | 11.55 | 30.6 | 46.2 | 56.95 | 63 | 72.8

a. calculate and interpret the z - score for patrick mahomes, with a quarterback rating of 63.
b. describe the mathematical operations used to calculate the z - score for patrick mahomes.
c. suppose that we converted all 30 of the quarterback ratings into z - scores. what would be the shape, mean, and standard deviation of the z - scores?

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the data - point, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Identify values

We are given that $x = 63$, $\mu=54.37$, and $\sigma = 11.55$.

Step3: Calculate z - score

$z=\frac{63 - 54.37}{11.55}=\frac{8.63}{11.55}\approx0.75$.
The interpretation of a z - score of $0.75$ is that Patrick Mahomes' quarterback rating is approximately $0.75$ standard deviations above the mean quarterback rating of the top 30 quarterbacks in the NFL for the recent season.

Step4: Describe operations for part b

To calculate the z - score, we first subtract the mean ($54.37$) from the value of Patrick Mahomes' rating ($63$) to get the deviation from the mean ($63 - 54.37=8.63$). Then we divide this deviation by the standard deviation ($11.55$).

Step5: Answer part c

If we convert all data points to z - scores, the shape of the distribution of z - scores is the same as the shape of the original distribution. The mean of the z - scores is $0$ and the standard deviation of the z - scores is $1$.

Answer:

a. The z - score is approximately $0.75$. It means Patrick Mahomes' rating is about $0.75$ standard deviations above the mean.
b. First, subtract the mean from the data - point. Then, divide the result by the standard deviation.
c. Shape: same as the original distribution; Mean: $0$; Standard deviation: $1$