Questions: A survey found that women's heights are normally distributed with mean 62.4 in. and standard deviation 3.1 in. The survey also found that men's heights are normally distributed with mean 68.7 in. and standard deviation 3.8 in. Most of the live characters employed at an amusement park have height requirements of a minimum of 57 in. and a maximum of 63 in. Complete parts (a) and (b) below. a. Find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as characters at the amusement park? The percentage of men who meet the height requirement is 6.58%. (Round to two decimal places as needed.) Since most men do not meet the height requirement, it is likely that most of the characters are women.

A survey found that women's heights are normally distributed with mean 62.4 in. and standard deviation 3.1 in. The survey also found that men's heights are normally distributed with mean 68.7 in. and standard deviation 3.8 in. Most of the live characters employed at an amusement park have height requirements of a minimum of 57 in. and a maximum of 63 in. Complete parts (a) and (b) below.
a. Find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as characters at the amusement park?

The percentage of men who meet the height requirement is 6.58%.
(Round to two decimal places as needed.)
Since most men do not meet the height requirement, it is likely that most of the characters are women.

Transcript text: A survey found that women's heights are normally distributed with mean 62.4 in. and standard deviation 3.1 in. The survey also found that men's heights are normally distributed with mean 68.7 in . and standard deviation 3.8 in. Most of the live characters employed at an amusement park have height requirements of a minimum of 57 in . and a maximum of 63 in. Complete parts (a) and (b) below. a. Find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as characters at the amusement park? The percentage of men who meet the height requirement is $6.58 \%$. (Round to two decimal places as needed.) Since most men $\square$ the height requirement, it is likely that most of the characters are $\square$

Solution

Solution Steps

Step 1: Standardize the Range

To standardize the range, we convert the minimum and maximum values into their equivalent z-scores using the formula $z = \frac{X - \mu}{\sigma}$. For the minimum value $X_{min} = 57$, the z-score is $z_{min} = \frac{57 - 68.7}{3.8} = -3.079$. For the maximum value $X_{max} = 63$, the z-score is $z_{max} = \frac{63 - 68.7}{3.8} = -1.500$.

Step 2: Calculate the Cumulative Distribution Function (CDF)

Using the standard normal distribution, we find the cumulative probabilities for $z_{min}$ and $z_{max}$. The cumulative probability for $z_{min} = -3.079$ is 0.00104. The cumulative probability for $z_{max} = -1.500$ is 0.0668.

Step 3: Find the Percentage within the Range

Subtract the cumulative probability of $z_{min}$ from that of $z_{max}$ and multiply by 100 to find the percentage of the population within the specified range. The percentage of the population within the range is 6.58%.