Southwest Airlines currently has a seat width of 17 in. Men have hip breadths that are normally distributed with a mean of 14.4 in. and a standard deviation
of 1.0 in. (based on anthropometric survey data from Gordon, Churchill, et al.).
a. Find the probability that if an individual man is randomly selected, his hip breadth will be greater than 17 in.
b. Southwest Airlines uses a Boeing 737 for some of its flights, and that aircraft seats 122 passengers. If the plane is full with 122 randomly selected men, find the probability that these men have a mean hip breadth greater than 17 in.
c. Which result should be considered for any changes in seat design: the result from part (a) or part (b)?
a. 0.0047 $\\$
b. 0.0000 (Table: 0.0001)$\\$
c. The result from part (a) is relevant because the seats are occupied by individuals
this problem is about the seat with on Southwest airline flights, and we're given some information. So I recommend starting by writing out what you know what is given to you and in this particular one were given that men's hip breaths are normally distributed. You were also given that the average man's hip breath is 14.4 inches and the standard deviation of that population is 1.0 inches. And in part a, you are asked to determine if a single man is randomly selected. What's the probability that his hip breath is greater than 17 inches? Because they told you about the normal distribution? We're going to draw ourselves a bell curve. We're gonna put the average in the center of that bell to get an image of what's going on, and we want the probability of being greater than 17. So we will need Z scores to solve this and to refresh your memory. The formula for a Z score is X minus mu over sigma, so we want the Z score for 17. So we're going to do 17 minus 14.4 over 1.0, and our Z score turns out to be 2.6. Now. I like to always put our Z score onto our picture, so we're gonna have to 0.6 here. So when we're talking about being ah ah, hip breath greater than 17 we're also talking about having a Z score greater than 2.6. Now, if you go to the table in the back of your book table A two, which is your standard normal table, the table is designed to talk about areas or probabilities that extend into the left tail of the bell curve. And as you can see, our picture is extending into the right tail. So in order to handle that, we're going to have to rewrite our probability statement to read one, minus the probability that Z is less than 2.6. When you look up 2.6 in the table, you will find an area or a probability off 0.9953 And when you subtract that from one, we get an overall probability of 10.47 So recapping part A. The probability that one randomly selected man has a hip breath that is greater than 17. This 0.47 in Part B. We're going to select a sample from the flight, and that sample is going to have a sample size of 122 passengers, and you are asked to determine if the plane is full with 122 randomly selected man Herman. What's the probability that they're mean? Hip breath is greater than 17? And in order to tackle this, we are going to have to determine the average of our sample needs and the standard deviation of our sample means, and we're going to use the central limit the room to determine these two statistics. The Central Limit Theorem says that the average of the sample means is equivalent to the average of your population, and in this case it was 14.4. And the standard deviation of our sample means what equal the standard deviation of our population divided by the square root of our sample size. So in this case, it's going to be 1.0, divided by the square root of 122. Again we're going to draw are bell shaped curve. We're going to put this average in the center, and we're trying to determine the probability that the average is greater than 17. In doing so, we are going to have to use the Z score formula, but there's going to be a slight modification on it because we are working with sample means. So we're going to use e equals X bar minus the average of our sample means over the standard deviation of our sample needs. So in this instance, Z equals 17 minus 14.4 divided by we're going to use this expression as the value for the standard deviation of our sample meat. When you calculate this out, the Z score ends up being 28.72 Yes, it is very ugly. I like to take that Z score and put it back onto our picture, so we have a 28.72 So when we're talking about the probability that the average was greater than 17 it's no different than talking about the probability that are Z score is greater than 28.72 And just like earlier, when we are dealing with the standard normal table in the back of the textbook, the standard normal table goes into your left tail are pictures going into the right tail. So we are going to have to rewrite this as one, minus the probability that C is less than 28 0.72 And when we're looking at that table at the very, very bottom of the positive side, it says that a Z score that is less than 3.5 or greater um, is going to be a 0.9999 So we've got to do one minus 10.9999 and we get 0.1 So again recapping Part B, we're going to select 122 male passengers, and the probability that they're mean hip breath is greater than 17 is going to be 170.1 Now we want to go into part C and in part C. The question is asking you which result should be considered for any changes in seat designs? Should we use part A or part B? And the answer is we should use part a so part A should be used when considering hot seat design changes. And the reason will be because seats are occupied by individual people, not groups of people