Standard Normal Distribution. In Exercises 9–12, find the area...

Question

Standard Normal Distribution. In Exercises 9–12, find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

Standard normal distribution curve with shaded area between z = -0.84 and z = 1.28.

Accepted Answer

Hello. In this video, we are going to be determining the probability of being within a region on a standard normal distribution. So, the problem says that we have a graph that depicts the standard normal distribution of test scores with a mean of zero and a standard deviation of 1. We want to find the area of the shaded region. Now, in order to approach this problem, we're going to use the approach of using Z scores to determine the area of the region. Specifically, what we want to do is you want to determine the probability of having a Z score that is between the two, the defined area. Now, the end points of this area is given to us as Z is equal to 1.35 and Z is equal to -0.75. So in order to write this probability, we want to write the probability that the value of our Z score is between 1.35 and it is between -0.75. Now, since we are working with the Z scores, remember that Z scores only give us the probability that are to the left of a defined endpoint. And in order to get these probabilities, we are going to have to use the Z table. Now, in order to find the area, Of the shaded region, we need to take the probability or the area to the left of the curve starting at 1.75, and we wanted to subtract the area. To the left of the curve at -0.75, and that's going to leave us the shaded region. That we want to find. So one way we can rewrite this probability is we are going to find the probability that the Z scores to the left of 1.35 and subtract the probability of being to the left of negative 0.75. So what we need to determine these probabilities by using our Z table. Now, by using the Z table, The probability of our Z score being to the left of 1.35 is this decimal that is given to us as 0.9115. And for the probability of the Z score being to the left of negative 0.75, that probability is given to us as 0.2266 from the Z table. So now that we have the values of both of these probabilities, we are going to plug them back into the equation we created, and that is going to give us 0.9115 minus 0.2266. And by using a calculator, the difference of these two values is going to be 0.6849. Now, what this is, is that this is the probability that the Z score we have is going to be between 1.35 and 0.75. And in other words, this is going to be the area of the shaded region on the normal distribution. 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 we will go ahead and see you all in the next video.

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