Questions: A survey found that women's heights are normally distributed with mean 63.5 in. and standard deviation 3.5 in. The survey also found that men's heights are normally distributed with mean 67.5 in. and standard deviation 3.1 in. Most of the live characters employed at an amusement park have height requirements of a minimum of 56 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 7.32%. 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 in.

Solution Steps

To find the percentage of men who meet the height requirement of \(56 \, \text{in} \leq X \leq 63 \, \text{in}\), we use the normal distribution parameters for men's heights, where the mean \( \mu = 67.5 \, \text{in} \) and the standard deviation \( \sigma = 3.1 \, \text{in} \).

The probability \( P \) is calculated as follows: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi\left(\frac{63 - 67.5}{3.1}\right) - \Phi\left(\frac{56 - 67.5}{3.1}\right) \] Calculating the Z-scores: \[ Z_{end} = \frac{63 - 67.5}{3.1} \approx -1.4516 \] \[ Z_{start} = \frac{56 - 67.5}{3.1} \approx -3.7097 \] Thus, the probability is: \[ P \approx \Phi(-1.4516) - \Phi(-3.7097) \approx 0.0732 \] The percentage of men meeting the height requirement is: \[ \text{Percentage} = P \times 100 \approx 7.32\% \]

The new height requirements are set to exclude the tallest \(50\%\) and the shortest \(5\%\) of men. We need to find the corresponding Z-scores for these percentiles.

For the shortest \(5\%\): \[ z = \frac{0.05 - 0}{1} = -1.645 \] For the tallest \(50\%\): \[ z = \frac{0.5 - 0}{1} = 0 \]

Using these Z-scores, we can calculate the new height requirements: \[ \text{New Minimum Height} = \mu + z \cdot \sigma = 67.5 + (-1.645) \cdot 3.1 \approx 67.7 \, \text{in} \] \[ \text{New Maximum Height} = \mu + z \cdot \sigma = 67.5 + (0) \cdot 3.1 \approx 69.0 \, \text{in} \]

Final Answer