Question

A survey found that​ women's heights are normally distributed with mean 62.5 in. and standard deviation 2.5 in. The survey also found that​ men's heights are normally distributed with mean 67.5 in. and standard deviation 3.2 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. Question content area top Part 1 A survey found that​ women's heights are normally distributed with mean 62.5 in. and standard deviation 2.5 in. The survey also found that​ men's heights are normally distributed with mean 67.5 in. and standard deviation 3.2 in.

Answer 1

The student's question pertains to using the normal distribution to calculate probabilities given the mean and standard deviation of women’s and men’s heights. The z-score calculation is essential in this scenario for determining the likelihood of amusement park character height requirements being met.

Step-by-step explanation:

The question involves calculating probabilities and making inferences about populations using the normal distribution. This requires knowledge of statistical concepts such as the normal curve, mean, standard deviation, and the use of z-scores to find probabilities and interpret data. The student is asked to use the given mean and standard deviation values for women's and men's heights to determine the likelihood of various height requirements being met.

Calculating Z-scores

To calculate the probability of someone falling within certain heights, first calculate the z-score, which is the number of standard deviations a value lies from the mean. The formula for the z-score is:

Z = (X - μ) / σ

where X is the value, μ is the mean, and σ is the standard deviation.

Using Z-scores to Find Probabilities

Once the z-score is calculated, use standard normal distribution tables or software to find the corresponding probability. For example, to find the probability that a randomly chosen female is shorter than 60 inches, calculate her z-score using the mean and standard deviation for women’s heights. Then look up this z-score in the standard normal distribution table to find her probability.