Question

the table shows population statistics for the ages of best actor and best supporting actor winners at an awards ceremony. the distributions of the ages are approximately bell - shaped. compare the z - scores for the actors in the following situation. best actor: μ = 45.0, σ = 8.7; best supporting actor: μ = 54.0, σ = 14. in a particular year, the best actor was 65 years old and the best supporting actor was 50 years old. determine the z - scores for each. best actor: z = ; best supporting actor: z = (round to two decimal places as needed.) interpret the z - scores. the best actor was the mean, which unusual. the best supporting actor was the mean, which unusual.

Explanation:

Step1: Recall z - score formula

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

Step2: Calculate z - score for Best Actor

For the Best Actor, $x = 65$, $\mu=45.0$, and $\sigma = 8.7$. Substitute these values into the formula: $z_{1}=\frac{65 - 45.0}{8.7}=\frac{20}{8.7}\approx2.30$.

Step3: Calculate z - score for Best Supporting Actor

For the Best Supporting Actor, $x = 50$, $\mu = 54.0$, and $\sigma=14$. Substitute these values into the formula: $z_{2}=\frac{50 - 54.0}{14}=\frac{- 4}{14}\approx - 0.29$.

Step4: Interpret z - scores

A z - score between - 2 and 2 is considered usual. A z - score outside this range is considered unusual. The Best Actor's z - score of 2.30 is greater than 2, so the Best Actor's age is above the mean and is unusual. The Best Supporting Actor's z - score of - 0.29 is between - 2 and 2, so the Best Supporting Actor's age is below the mean and is usual.

Answer:

Best Actor: $z\approx2.30$
Best Supporting Actor: $z\approx - 0.29$
The Best Actor was above the mean, which is unusual. The Best Supporting Actor was below the mean, which is usual.