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.

Mickey Mouse Disney World requires that people employed as a Mickey Mouse character must have a height between 56 in. and 62 in.

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 Mickey Mouse characters?

Accepted Answer

Welcome back, everyone. In this problem, a coffee shop measures the time it takes to prepare a cup of coffee. The time is normally distributed with a mean of 3 minutes and a standard deviation of 0.5 minutes. The shop guarantees that a cup of coffee will be prepared in 2 to 4 minutes. What percentage of coffee cups can be expected to be prepared within this time frame? A says it's 90.08%, B 93.26%, C 94.11%, and D 95.44%. Now how can we figure out the percentage of coffee cups that can be expected to be prepared within this time frame? What do we know? Well, we were told that the time is normally distributed with a mean of 3 minutes and a standard deviation of 0.5 minutes. So let's say that mu the mean equals 3. And the standard deviation sigma is equal to 0.5. To find the the probability, or sorry, the percentage of coffee cups that can be expected within this time frame, we want the probability that the preparation time is between 2 and 4 minutes. In other words, we are solving for the probability. That X is between 2 and 4. Now how can we find that? Well, given that the time is normally distributed, we can convert the values to standard Z scores, find that probability for those Z scores within the table, and then use that to help us solve. What do we know about the Z square? Well, recall that the D score is equal to X minus mu, the mean divided by the standard deviation sigma. OK? So that means for X equals 2. Then Our first Z score will be 2 minus 3 divided by 0.5, which equals -2. And for a second Z score, where X equals 4, then Z2 will be equal to 4 minus 3 divided by 0.5, which equals 2. So here the probability that X is between 2 and 4 will be the same probability that X Z sorry, is between 2 and 2. Now what will those be? Well, if we think about our normal distribution, OK, if we were to just do a quick draw here. And we were to put 1 Z square here where. We have a value of Z2, here, where Z equals 2 and the other on the left here where Z equals -2. Then what, how we can find that probability. OK, from a probability table is to first find the probability that X is less than 2. All of this which I have highlighted there. And then find the probability that X is less than negative2, OK. And then the value that remains, the section that remains here will be the probability that X is negative, Z will be between 2 and 2. So in other words, We can say the probability that Z is between 2 and 2 will be equal the probability that Z is less than 2 minus the probability that Z is less than -2, and we can find both of these probabilities on or on a probability standard distribution, sorry, a normal distribution table. Now, using the table for negative Z squares, if you were to check your table, you should find that the probability that Z is less than -2 would be 0.0228, and the probability Z is less than 2 is 0.9772. And now when we subtract those, then we get our probability to be 0.9544 or 95.44%. What does this mean? Well, this tells us that 95.44% of coffee cups can be expected to be prepared within 2 to 4 minutes. If we look back at our answer choices, that means D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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Key Concepts

Normal Distribution

Z-Score

Percentile and Probability

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