the quarterback rating (qbr) is a statistical measure used to evaluate a quarterbacks performance. here are…

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: $1$