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Problem 58 Hard Difficulty

On a given day, the number of square feet of office space available for lease in a small city is a normally distributed random variable with a mean of 750,000 square feet and a standard deviation of 60,000 square feet. The number of square feet available in a second small city is normally distributed with a mean of 800,000 square feet and a standard deviation of 60,000 square feet.
a. Sketch the distribution of leasable office space for both cities on the same graph.
b. What is the probability that the number of square feet available in the first city is less than $800,000 ?$
c. What is the probability that the number of square feet available in the second city is more than $750,000 ?$

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Video Transcript

right. This problem is about the number of square feet of office space available in small cities. We know it's normally distributed with a mean, uh, 7,750,000 square feet and a stand. Standard deviation have 60,000 square feet. Um, we have a second small city that has a mean of 800,000 square feet and a standard deviation, also of 60,000 square feet. It does say that the office space square footage is normally distributed. So therefore, when we complete part A, which is to draw a representation, a sketch of this distribution on the same graph. So we've got City one or I'll call it City A. I'll put in blue, and for City be we're gonna put it in pink. So we're going to start by drawing the bell shaped curve. And keep in mind in the center is where the, um, average is going to go for the blue. And so let's start with the blue curve. So I'm gonna actually I'm gonna actually get rid of the white. I'm only going to do blue. Here's the blue curve and thes center of it is at 750,000 and we count by standard deviations. So therefore, 6000 later, we're going to be at what we call our point of inflection. And our point of inflection is kind of where the curve starts turning from a scooping down to a scooping up. So right around here is going to be one standard deviation out. So we're adding 60,000 and will be at 810,000 and then we'll be about the same distance out will be at 870,000 and the same distance out. We're going to be at 930,000 and then we're gonna go back the other way. If we subtract 60,000 from the average where it 690,000 and this attracted again, we are at 600 30,000 and we subtract again, keeping our spacing uniform. We would be at 570,000. So there is our blue curve. That's the curve that is representing city A. Now, when it comes time to do city be now, the peak has to be a 800,000. So it looks like 800,000 would be like somewhere right in here. So our peak has to be right there, and then everything else is going to be uniform. So if you think about the distance from our peek out, we went over approximately that for and that far. So there's our curve scooping in that direction. And then the scoop down is just gonna parallel the other car because they had the same exact, um, standard deviation. The curves are going to look identical in shape and size. They're just going to be shifted a little bit further. One way or the other part. B Part B is asking us to determine the probability that the number of square feet available in the first city is less than 800,000, so X is less than 800,000. But keep in mind, we're using the first city. So we're using the blue numbers. And the first city, which we called City A, had an average of 750,000, and in this instance, we're trying to go where the probability is less than 800,000. So we will need a Z score and to brush up on your formula, Z is equal to X minus mu over stigma. So for the city number one or city A. It's going to be 800,000, minus 750,000 divided by the standard deviation, which was 60,000. So our Z score for the first city, at 800,000 would be 0.83 So we know that 0.83 is appear in our bell curve. So when we're talking about the office space being less than 800,000, it's no different than if we were to solve the problem where Z is less than 0.83 and Z being less than pointing three. You would then look into your standard normal table in the back of the book, and you're going to find a value of 30.7967 as the probability that in the first city we have office space less than 800,000 square feet. Let's look at Part C and in part C. We're using information from the second city, and we want to determine the probability that the square feet available in the second city is greater than 750,000. So in this case, are Bell Curve is going to have the peak. Since we're using the data from the second city, the peak is going to be a 800,000 and we're trying to find the probability of being greater than 700 50,000. So we will need our Z score. Z equals 750,000 minus 800,000 all over our standard deviation, which was 60,000 no matter which city you were in. So the Z score turns out to be a negative 0.83 So when we're talking about the square footage available in the second city to be greater than 750,000, it's no different than saying that Z would be greater than negative 0.83 Because this is taking us to the right. We're going to have to use one, minus the probability of Z being less than negative 10.83 And when you look in your standard normal table, you will arrive at an area to the left of negative 0.83 to be points to 033 resulting in an area to the right or probability of 0.7967 So if you've noticed Part B and Part C had the same result because they had the same spread and the value that we were, um, talking about the actual X minimum or maximum was the same distance away from our average. So part B. The probability that your square footage was less than 800,000 in the first city was 0.7967 And for part C, the probability that the square footage available in the second city was greater than 750,000 was also 0.7967