Using the Central Limit Theorem. In Exercises 5–8, assume that...

Question

Using the Central Limit Theorem. In Exercises 5–8, assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of 1.2 kg and a standard deviation of 4.9 kg (based on Data Set 13 “Freshman 15” in Appendix B).

a. If 1 male college student is randomly selected, find the probability that he gains between 0.5 kg and 2.5 kg during freshman year.

a. If 1 male college student is randomly selected, find the probability that he gains between 0.5 kg and 2.5 kg during freshman year.

Accepted Answer

Although, in this video we are told that the weight of apples in a large orchard are normally distributed with a mean of 150 g and a standard deviation of 20 g. What is the probability that a randomly chosen apple weighs between 140 g and 170 g? In order to approach this problem, let's just go ahead and list out our given information. First, we are given a mean which is 150 g, and we are given a standard deviation of 20 g. We want to find the probability that if we pick an Apple X. Its weight is going to be between 140 g and 170 g. So, in order to approach this problem, the first thing we need to do is we need to convert these endpoints of the probability into Z scores. Now, the Z score is the is defined as the following. It is the end point minus the mean. Which is new Divided by our standard deviation. Now, for X is equal to 140, this Z score is going to be Z is equal to 140 minus 150 divided by 20, and this is going to give us -0.5. Furthermore, for the endpoint of 170, the Z score for this value is going to be 170 minus 150 divided by 20, and this is going to give us a value of 1. So, we can convert our probability into a Z score probability, and now what we want to do is we want to find the probability that our Z score is between -0.5 and 1. This is equivalent to saying that we want to find the probability that Z is less than 1 minus the probability that Z is less than -1.5. And by using the Z table in the appendix, we get that these probabilities are going to be 0.8413 minus 0.3085, and this is going to simplify to be 0.5328. So what this means is that the probability of the apple's weight being between 140 and 170 is about 53.28%. And with that being said, the solution to this problem is going to be D. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

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Key Concepts

Normal Distribution

Central Limit Theorem (CLT)

Probability Calculation

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