Graphical Analysis In Exercises 17–22, find the indicated z-sc...

Question

Graphical Analysis In Exercises 17–22, find the indicated z-score(s) shown in the graph.

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Accepted Answer

Below there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the value of the indicated Z scores shown in the following graph. So in our following graph, we are given a curve represented in dark blue, and under the curve, we have two shaded areas that are shaded in green, and they both have an area of 0.011. And for the left side of our curve, we have Z1 equals some value we do not know. And then we have the peak of our curve is at 0, and then on the right side of our curve we have Z2 is equal to some value that we do not know. And this is following the Z axis, it appears, which is a horizontal axis. Awesome. So now that we have a visualization of what's happening for this particular prom, let us know once again that we're ultimately trying to determine the value of the indicated Z scores that are shown in this graph that is given to us by the prom itself. And so that means we're solving for two separate answers. We're trying to find the value of Z1 and Z2. So we're trying to find these two different Z score values. So with that in mind, Let us first off note that the two areas are given and they are each equal to once again 0.011 and 1 in the left tail and one in the right tail of our standard normal distribution. And this area of 0.011 in the left tail, as we should recall, means that the cumulative probability from the far left up to that Z score will be 0.011. So therefore, we need to note that we are looking for the Z score when the probability P of Z is less than Z1 is equal to 0.011. Awesome. So, with that in mind, we need to use a standard normal distribution table, which you should be able to find one in your textbook. And we need to note that the Z score that corresponds to the cumulative area of 0.011 will be -2.29 as we could see. So I've gone ahead and borrowed a standard normal distribution table, and I've gone ahead and copied and pasted it here and as you could see, When we follow along, we have our. Negative. 2.29, so we can see. When we follow our Z score down, we have -2.2, and then if we add 0.09 and we follow it down, we will find the value of our area, which is 0.0110. Fantastic. So moving right along here, so therefore, without a doubt, we could safely say that Z1 is equal to -2.29. And As we should recall, we can use symmetry to help us solve for the right tail, because the normal distribution is symmetric. The Z square that leaves an area of 0.011 to the right of it will be the positive version of the left tail result, meaning that Z2 will be equal to 2.29. So that will mean, without a doubt, our final two answers undoubtedly will have to be the following. To recap here, so Z1 is equal to -2.29 and Z2 is equal to 2.29. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Graphical Analysis In Exercises 17–22, find the indicated z-score(s) shown in the graph.

Key Concepts

Z-Score

Standard Normal Distribution

Area Under the Curve

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