IQ Scores. In Exercises 5–8, find the area of the shaded regio...

Question

IQ Scores. In Exercises 5–8, find the area of the shaded region. The graphs depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler IQ test).

Bell curve showing IQ scores with shaded area between 112 and 124, representing a section of the normal distribution.

Accepted Answer

Hello. In this video, we are going to be finding the area of a shaded region under a normally distributed curve. We are told that in a city, the monthly utility bills are normally distributed with a mean of $150 and a standard deviation of $20. We want to find the area of the shaded region. Now, before we approach this problem, let's just go ahead and write down the given information. First, we are told that this normal distribution has a mean of $150. We are going to represent this mean with the variable new. And we're going to set new equal to 150. Next, we are told that we have a standard deviation of $20. Now, we are going to represent the standard deviation with the variable sigma, and that is going to equal to 20. Now, in order to find the area of the shaded region, what we want to do is we want to use the method of the Z score. Specifically, what we are going to do is we are going to convert 128 and 75 into Z scores, and we are going to take the difference of these probabilities in order to find the Z score between the region. So, in order to approach this problem, let's first convert 128 and 75 into Z scores. Now, in order to do this, we are going to use the following equation. Z is equal to X minus U, divided by sigma. Now this is saying that if we want to convert a value into a Z score, we take the value from the normal distribution, subtract the mean of the normal distribution, and divide this by the standard deviation from the normal distribution. So, for 128, We are going to end up with Z is equal to 128 minus 150 divided by 20. And this is going to simplify to leave us with negative 22. Divided by 20, which will further simplify to be negative 1.10. So now what we can do is we can go and represent 128 as -1.10 on the Z score for the normal distribution. And we are going to do the same thing with 75. For 75, we have Z is equal to 75 minus 150 divided by 20. This is going to leave us with -75 divided by 20, which further simplifies to -3.75. So now we are representing 75 as -3.75. Now, what we want to do is you want to write the probability of the Z score being between these two values. So that probability will be represented as the following. We want to find the probability that our Z score is between -1.10. And -3.75. Now, in order to do this, we are going to take the probability to the left of -1.10, so we can write the probability that Z is to the left of -1.10, and we are going to subtract the probability that Z is to the left of -3.75. Now what we want to do is we want to use our Z table. In order to find the value of the probabilities. From the Z table, we know that the probability to the left of -1.10 is going to be 0.1357. And by the same Z table, the probability to the left of 3.75 is a super small value, which will be 0.0001. And this is going to give us a difference that is 0.1356, which is going to be the area of the shaded region. And with that being said, the solution to this problem is going to be C. So, I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.