Ergonomics. Exercises 9–16 involve applications to ergonomics,...

Question

Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.

Doorway Height The Boeing 757-200 ER airliner carries 200 passengers and has doors with a height of 72 in. Heights of men are normally distributed with a mean of 68.6 in. and a standard deviation of 2.8 in. (based on Data Set 1 “Body Data” in Appendix B).

a. If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A class of military submarines contains hatches with a clearance height of 60 inches. The heights of the naval personnel are normally distributed with a mean of 66 inches and a standard deviation of 3 inches. What is the probability that a randomly selected naval officer can pass through the hatch without bending over? Awesome. So it appears for this particular problem we're ultimately trying to determine whether or not. This naval officer can pass through the hatch without bending over. What is the probability of that happening that a randomly selected naval officer can pass through the hatch without having to bend over to duck to get into the hatch or to get through the hatch. So we're trying to figure out what is this probability value. So with that said, let's read off our multiple choice answers to see what our final answer might be. A is 0.9772. B is 0.0228. C is 0.0183. And finally D is 0.9817. Awesome. So first off, in order to find the probability that a randomly selected naval officer can pass through the pass through this hatch without bending over, We need to calculate the probability that a naval officer is 60 inches tall or shorter given that the heights are normally distributed with a mean of 66 inches and a standard deviation of 3 inches. So with that said, our first step is we need to standardize the value. So we need to convert the height of interest, which is 60 inches, into a Z score value, which is the number. of standard deviations away from the mean. So that means we need to recall and use the formula for the Z score, which that formula states that the Z score Z is equal to X. Multiple so it's gonna be capital X minus m. Divided by sigma, where sigma is our standard deviation, X is the value that we're interested in solving for, so that's our value that we're trying to solve for our target variable, and mu is the mean. So now we need to calculate the Z score. So I said X is the value we're interested in. So now we need to substitute in the given values into our formula, so we need to note that Z is going to be equal to 60 minus 66 divided by 3, which will give us an answer of -2. So now we need to use our Z score value to help us determine the probability. So we need to recall and use the standard normal distribution table, and when we use that table, we will find that P of Z is less than -2, will be equal to 0.0228. And that is our final answer. Hooray, we did it. So therefore, the probability that a randomly selected naval officer can pass through the hatch without bending over is approximately 0.0228, and this corresponds with multiple choice answer letter B, which once again is 0.0228. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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Key Concepts

Normal Distribution

Z-Score

Probability Calculation

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