In Exercises 25–28, use these parameters (based on Data Set 1 ...

Question

In Exercises 25–28, use these parameters (based on Data Set 1 “Body Data” in Appendix B):

Men’s heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in.

Women’s heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in.

Snow White Disney World requires that women employed as a Snow White character must have a height between 64 in. and 67 in.

a. Find the percentage of women meeting the height requirement.

Accepted Answer

Hello, everyone, let's take a look at this question together. A fashion modeling agency requires female models to have a height between 67 inches and 70 inches. Women's heights are normally distributed with a mean of 64.5 inches and a standard deviation of 2.7 inches. What percentage of women meet the modeling agency's height requirement? Is it answer choice A 9.36%? Answer choice B, 15.64%, answer choice C 21.28%, or answer choice D 29.93%. So from the information provided in the question, we note that the women's heights are normally distributed with a mean of 64.5 inches and a standard deviation of 2.7 inches. And we want to determine the probability that the height X is between 67 and 70 inches, and so we compute the two Z scores using the formula Z is equal to x minus the mean divided by the standard deviation. So plugging in the values for Z1, we get 67 minus 64.5 divided by 2. 7, which is equal to 0.9259, as the value for Z1 and plugging in the values into Z2, we get 70.64.5 divided by 2.7, giving us 2.0370 as the value for Z2. And so we are looking for the probability that Z is between Z1 and Z2, which we can find by subtracting the probability that Z is less than Z1 from the probability that Z is less than Z2. So then we use a table for positive Z scores to find the probability that Z is less than 0.9259, and since 0.9259 is between 0.92 and 0.93, We should perform linear interpolation. So we let P1 equal the probability that Z is less than 0.9259. And plugging in the values, we get 0.8227407407 as the value for P1, and then we calculate the same for P2, which refers to the probability that Z is less than 2.0370 using the table for positive. scores to find that value, we get 0.9791518519 as the value for P2. Therefore, the probability that the height is between 67 and 70 inches is 0.1564, which tells us that about 0.15 64 or 15.64% of the women meet the modeling agency's height requirement. And looking at our answer choices, we can identify answer choice B as the correct answer, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

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Key Concepts

Normal Distribution

Z-scores

Percentile and Area Under the Curve

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