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 normal distribution of IQ scores, shaded area to the left of score 118, mean 100, SD 15.

Accepted Answer

Hello. In this video, we are going to be determining the area of the shaded region of a null distributed graph. So 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, in order to determine the area of the shaded region, we are going to use the method of using Z scores to determine this area. So, before we do this, let's just go ahead and write down the given information. First, we are told that this graph has a mean of $150. We are going to represent this mean with the variable new, and we are going to set this equal to 150. Next, we are also told that we have a standard deviation of $20. We are going to represent this standard deviation. With the variable sigma, and we are setting this to $20. Now, in order to use the method of finding the Z score, we want to find the Z score that's to the left of 130, but in order to do this, we have to standardize this graph. So in order to find the Z score, the first thing we need to do is we need to convert this endpoint 130 into a Z score. In order to do this, we are going to use the formula, Z is equal to X minus U, divided by sigma. Now, what this means is that if we want to convert 130 into a Z score, we are going to take the value of the endpoint, subtract its mean, and divide this value by the standard deviation. So by plugging in the given information, we will have 130 minus 150 divided by 20. This is going to simplify to be -20 divided by 20, which is equal to -1. And so what this means is that in order to find the area to the left of the curve, we want to find the probability that our Z score is to the left of -1. Now, in order to find this probability, we are going to need to use our Z table. And by reading the Z table, the probability that the Z score is to the left of -1 is going to be 0.1587, and this is going to be the area to the left of the normal distribution at 130. 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.