Question

In Exercises 13–16, use z scores to compare the given values.

Oscars In the 87th Academy Awards, Eddie Redmayne won for best actor at the age of 33 and Julianne Moore won for best actress at the age of 54. For all best actors, the mean age is 44.1 years and the standard deviation is 8.9 years. For all best actresses, the mean age is 36.2 years and the standard deviation is 11.5 years. (All ages are determined at the time of the awards ceremony.) Relative to their genders, who had the more extreme age when winning the Oscar: Eddie Redmayne or Julianne Moore? Explain.

Short answer

The z-score for Julianne Morre is 1.55.

The z-score for Eddie Redmayne is -1.25.

The age of Julianne Morre, 54 years, seems to be more extreme than the age of Eddie Redmayne, 33 years.

Step-by-step solution

Given information 

In the set of all best actors, the mean age is 44.1 years, and the standard deviation of the ages is 8.9 years.

The age of the actorEddie Redmayne is 33.

For all best actresses, the mean age is 36.2 years, and the standard deviation of the ages is 11.5 years.

The age of the actress Julianne Moore is 54.

Define the variables and the z-score 

Define M as the age of actors and F as the age of actresses.

Then,

$$\begin{gathered} {\overline{x}}_M=44.1 \\ {s}_M=8.9 \\ m=33 \end{gathered}$$

$$\begin{gathered} {\overline{x}}_F=36.2 \\ {s}_F=11.5 \\ f=54 \end{gathered}$$

A z-score is a standard value is computed to know how far and in what direction from the mean is the raw value.

It has the following expression:

$Z=\frac{x-\overline{x}}{s}$

When the absolute z-score is large (in absolute terms), it suggests a larger difference from the mean value in terms of standard deviations.

Z-score for the actors and actress

The z-score for the actor is as follows:

$$\begin{gathered} Z=\frac{m-{\overline{x}}_M}{s_M} \\ =\frac{33-44.1}{8.9} \\ =-1.25 \end{gathered}$$

The z-score for the actress is as follows:

$$\begin{gathered} Z=\frac{f-{\overline{x}}_F}{s_F} \\ =\frac{54-36.2}{11.5} \\ =1.55 \end{gathered}$$

Therefore, -1.25is thez-score for Eddie Redmayne’s age, and 1.55 is theZ-score for Julianne Moore’s age.

Step 4: Comparison of z-scores 

Julianne’s age equal to 54 years is 1.55 standard deviations above the mean, and Eddie’s age equal to 33 years is 1.25 standard deviations below the mean.

The z-score of 1.55 is farther from the mean as compared to 1.25. Thus,Julianne Moore’s age is more extreme than Eddie Redmayne’s age.