On a given day, the number of square feet of office space...

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 $750,000 ?$

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 $750,000 ?$

Key Concepts

Probability Calculation

Probability calculation in the context of a normal distribution involves determining the area under the curve corresponding to an event of interest. This often requires converting values to their corresponding Z-scores and then using standard normal distribution tables or software to find the cumulative probabilities. It provides a measure of how likely an event is to occur within a normally distributed variable.

Graphical Representation of Probability Distributions

Graphical representation involves visualizing probability distributions, such as plotting the bell-shaped curve of a normal distribution. This aids in understanding the behavior of the distribution, such as central tendency, variability, and the areas that represent cumulative probabilities. Overlaying multiple normal distributions on the same graph can be useful for comparing different data sets or scenarios.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by a bell-shaped curve that is symmetric about its mean. It is defined by two parameters, the mean (which indicates the center of the distribution) and the standard deviation (which measures the spread or dispersion). Many real-world phenomena naturally approximate a normal distribution, making it a key concept in probabilistic analysis and statistical inference.

Standardization (Z-score Transformation)

Standardization is the process of converting a normal random variable to a standard normal variable using the Z-score transformation. The Z-score is calculated by subtracting the mean from the observed value and then dividing the result by the standard deviation. This transformation allows comparisons between different normal distributions and simplifies the process of finding probabilities using standard normal distribution tables.

Transcript

00:01 This problem is about the number of square feet of office space available in small cities.

00:07 We know it's normally distributed with a mean of 750 ,000 square feet and a standard deviation of 60 ,000 square feet.

00:26 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.

00:48 It does say that the office space square footage is normally distributed, so therefore when we complete part a, which is, is to draw a representation, a sketch of this distribution, on the same graph.

01:10 So we've got city 1, or i'll call it city a, i'll put in blue, and for city b we're going to put it in pink.

01:25 So we're going to start by drawing the bell -shaped curve, and keep in mind in the center is where the average is going to go for the blue one.

01:43 So let's start with the blue curve.

01:46 So i'm going to actually get rid of the white, and i'm only going to do blue.

01:56 Here's the blue curve, and the center of it is at 750 ,000.

02:06 And we count by standard deviations.

02:09 So therefore, 6 ,000 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.

02:23 So right around here is going to be one standard deviation out.

02:30 So we're adding 60 ,000 and we'll be at 810 ,000.

02:38 And then we'll be about the same distance out.

02:42 We'll be at 870 ,000.

02:47 And the same distance out, we're going to be at 930 ,000.

02:56 And then we're going to go back the other way.

02:59 If we subtract 60 ,000 from the average, we're at 690 ,000.

03:06 And then subtract it again.

03:10 We are at 630 ,000.

03:15 And we subtract again, keeping our spacing uniform.

03:19 We would be at 570 ,000.

03:23 So there is our blue curve.