A survey found that women's heights are normally distributed with mean 62.7 in and standard deviation 3.7 in . The survey also found that men's heights are normally distributed with mean 68.3 in. a deviation 3.3 in . Most of the live characters employed at an amusement park have height requirements of a minimum of 56 in and a maximum of 62 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 1 . 6 . (Round to two decimal places as needed.)
Upstudy AI Solution
Answer
Solution
Answered by UpStudy AI and reviewed by a Professional Tutor
Explain 
Simplify this solution Beyond the Answer
To determine the percentage of men who meet the height requirement of 56 inches to 62 inches, we can use the z-score formula: \[ z = \frac{(X - \mu)}{\sigma} \] where \(X\) is the height requirement, \(\mu\) is the mean height, and \(\sigma\) is the standard deviation. For \(X = 62\) inches: \[ z = \frac{(62 - 68.3)}{3.3} = \frac{-6.3}{3.3} \approx -1.91 \] Now for \(X = 56\) inches: \[ z = \frac{(56 - 68.3)}{3.3} = \frac{-12.3}{3.3} \approx -3.73 \] Next, we can look up these z-scores in a standard normal distribution table or use a calculator. The area to the left of \(z = -1.91\) is approximately 0.0287 (or 2.87%) and for \(z = -3.73\), it is approximately 0.0001 (or 0.01%). To find the percentage of men meeting the height requirement, we calculate the difference: \[ P(56 < X < 62) = P(Z < -1.91) - P(Z < -3.73) \approx 0.0287 - 0.0001 \approx 0.0286 \] Thus, the percentage of men who meet the height requirement is approximately \(2.86\%\). This low percentage suggests that very few men would qualify for the character positions at the amusement park, indicating that the majority of characters might be women, since they have a mean height that more closely aligns with the height requirements. In summary, you’ve found that only about 1.6% of men meet the height requirement, which indicates a strong likelihood that women are more prominently employed as characters in the park.
