Questions: A survey found that women's heights are normally distributed with mean 62.8 in and standard deviation 3.1 in. The survey also found that men's heights are normally distributed with mean 67.4 in and standard deviation 3.9 in. Most of the live characters employed at an amusement park have height requirements of a minimum of 55 in and a maximum of 64 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 19.09%. (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. b. If the height requirements are changed to exclude only the tallest 50% of men and the shortest 5% of men, what are the new height requirements? The new height requirements are a minimum of in. and a maximum of 67.4 in. (Round to one decimal place as needed.)

Solution Steps

To find the percentage of men who meet the height requirement of \( 55 \, \text{in} \) to \( 64 \, \text{in} \), we first calculate the Z-scores for the lower and upper bounds of the height requirement.

The Z-score for the lower bound \( (X = 55) \) is calculated as follows: \[ Z_{start} = \frac{X_{start} - \mu_{men}}{\sigma_{men}} = \frac{55 - 67.4}{3.9} \approx -3.1795 \]

The Z-score for the upper bound \( (X = 64) \) is calculated as: \[ Z_{end} = \frac{X_{end} - \mu_{men}}{\sigma_{men}} = \frac{64 - 67.4}{3.9} \approx -0.8718 \]

Using the cumulative distribution function \( \Phi \), we find: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-0.8718) - \Phi(-3.1795) \approx 0.1909 \]

Thus, the percentage of men meeting the height requirement is: \[ \text{Percentage} = P \times 100 \approx 19.09\% \]

Next, we need to find the new height requirements that exclude the tallest \( 50\% \) of men and the shortest \( 5\% \) of men.

To find the Z-score corresponding to the shortest \( 5\% \): \[ Z_{5\%} \approx -1.6449 \quad (\text{from Z-tables}) \]

Now, we calculate the new minimum height requirement: \[ \text{New Minimum Height} = \mu_{men} + Z_{5\%} \cdot \sigma_{men} = 67.4 + (-1.6449) \cdot 3.9 \approx 67.6 \, \text{in} \]

The maximum height requirement remains the mean height of men: \[ \text{New Maximum Height} = \mu_{men} = 67.4 \, \text{in} \]

Final Answer