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
Best Supporting Actor
muequals42.0
muequals49.0
sigmaequals7.3
sigmaequals15
In a particular​ year, the Best Actor was 59 years old and the Best Supporting Actor was 45 years old.
Determine the​ z-scores for each.
Best Actor:
z
equals


Best Supporting Actor:
z
equals


​(Round to two decimal places as​ needed.)
Interpret the​ z-scores.
The Best Actor was

(more than 2 standard deviations above
more than 1 standard deviation above
less than 1 standard deviation above
less than 2 standard deviations below)
the​ mean, which

(is not, is)
unusual. The Best Supporting Actor was

(less than 1 standard deviation below
more than 1 standard deviation above
more than 2 standard deviations below
more than 1 standard deviation below)
the​ mean, which
(is
is not)
unusual.

Answer 1

Solution:

$Z score =\frac{X-\sigma}{\mu}$

1. For Best Actor

$X_{1}$= 59 years

$\sigma=7.3, \mu=42.0$

Z, Score for best actor named, $Z_{1}$

$Z_{1}=\frac{59-7.3}{42}\\\\Z_{1}= \frac{51.7}{42}\\\\Z_{1}=1.23095\\\\ Z_{1}=1.24$

Z-Score for best actor = 1.24

2. Z , Score for best supporting actor , called $Z_{2}$

$X_{2}$=49 years

$\sigma=15, \mu=49.0$

 $Z_{2}=\frac{49-15}{49}\\\\Z_{2}= \frac{34}{49}\\\\Z_{2}=0.6938\\\\ Z_{2}=0.70$      

Z-Score for best supporting actor = 0.70

Z-Score is usually , the number of standard deviations from the mean a point in the data set is.

3. As, $Z_{1}=1.24$

So, we can say that,Option (B) The Best Actor was more than 1 standard deviation above is not unusual.

4.As, $Z_{2}=0.70$

So, we can say that,Option(A) The Best Supporting Actor was less than 1 standard deviation below, is not unusual.