Aircraft Seat Width Engineers want to design seats in commerci...

Question

Aircraft Seat Width Engineers want to design seats in commercial aircraft so that they are wide enough to fit 99% of all adults. (Accommodating 100% of adults would require very wide seats that would be much too expensive.) Assume adults have hip widths that are normally distributed with a mean of 14.3 in. and a standard deviation of 0.9 in. (based on data from Applied Ergonomics). Find P99. That is, find the hip width for adults that separates the smallest 99% from the largest 1%.

Accepted Answer

Welcome back, everyone. In this problem, a shoe company wants to ensure that their new line of running shoes fits 95% of all athletes. To avoid producing too many sizes, they decide not to accommodate the largest 5% in foot length. Assuming foot lengths are normally distributed with a mean of 25.5 centimeters and a standard deviation of 1.2 centimeters, find the maximum foot length that fits within this 95%. A says it's 26.9 centimeters, B 27.5 centimeters, C, 28.8 centimeters, and the D 30.2 centimeters. Now, here, what are we really trying to figure out? Well, we're trying to find the maximum foot length that fits within this 95% interval. In other words, we were given the actual percentage, OK, we, we know what the interval is but now we want to use it to find the value. Recall that normally. We would take the value, OK, use it to find the Z score using the formula that Z Z equals x minus the mean mu divided by the standard deviation and then refer to our standard normal distribution table to find which probability corresponds to this Z score. But no, we're going to do the opposite. We're going to take this probability, find which uh Z square it corresponds to, and then use that to figure out our maximum foot length or X value. In other words, we are going to solve for X being equal to mu plus Z sigma. Now how can we do that? Well, so far we know that we were given a mean meal of 25.5 centimeters. Standard deviation sigma of 1.2 centimeters. So all we need to do now is to find the Z square. Since our probability, OK, since our probability. Let me spell that properly here. For a probability of 95%, OK. To find the corresponding Z score, we can go to the standard distribution, the standard normal distribution table. OK. So from the standard normal distribution table. OK. And we can use the table for positive Z scores to find that 0.95, 95% is found in the intersection of 1.6 and 0.45. So from the standard normal distribution table, the Z score. I 1.645. Now that we have the Z square, we can solve for our maximum foot length because no, that means our maximum foot length X. It's gonna be equal to 25.5. Plus 1.645 multiplied by 1.2. We know 1.645 is our standard deviation or one of these values is an approximation. So this is gonna be approximately equal to 27.5 centimeters. Thus, by designing shoes up to about 27.5 centimeters, the company will accommodate approximately 95% of all athletes' foot lengths. Therefore, B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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Key Concepts

Normal Distribution

Percentiles

Z-Score

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