Questions: 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=46.0 σ=9.3 Best Supporting Actor μ=52.0 σ=12 In a particular year, the Best Actor was 68 years old and the Best Supporting Actor was 49 years old. Determine the z-scores for each. Best Actor: z= Best Supporting Actor: z= (Round to two decimal places as needed.)

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=46.0 σ=9.3 Best Supporting Actor μ=52.0 σ=12 In a particular year, the Best Actor was 68 years old and the Best Supporting Actor was 49 years old. Determine the z-scores for each. Best Actor: z= Best Supporting Actor: z= (Round to two decimal places as needed.)
Transcript text: 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 $\mu=46.0$ $\sigma=9.3$ Best Supporting Actor \[ \begin{array}{l} \mu=52.0 \\ \sigma=12 \end{array} \] In a particular year, the Best Actor was 68 years old and the Best Supporting Actor was 49 years old. Determine the z -scores for each. Best Actor: $z=\square$ $\square$ Best Supporting Actor: $z=$ $\square$ (Round to two decimal places as needed.)
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Solution

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Solution Steps

Step 1: Subtract the Population Mean from the Individual's Value

For the first individual, we calculate $z_1 = \frac{(68 - 46)}{9.3} = 2.37$.

For the second individual, we calculate $z_2 = \frac{(49 - 52)}{12} = -0.25$.

Step 2: Divide the Result by the Population's Standard Deviation

This step has already been incorporated into the calculations above, where the subtraction result is directly divided by the standard deviation.

Final Answer:

The $z$-score for the first individual is 2.37, and for the second individual is -0.25. These $z$-scores allow us to compare the positions of these individuals within their respective populations, relative to the population mean and standard deviation.

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