Standard Normal Distribution. In Exercises 17–36, assume that ...

Question

Standard Normal Distribution. In Exercises 17–36, assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.

Less than -2.00

Less than -2.00

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says assume that the scores on a standardized anxiety assessment are normally distributed with a mean of zero and a standard deviation of 1. Draw a graph of the standard normal curve and answer the following question. If a person is randomly selected and given this assessment, what is the probability that their score is less than -1.50? We're given four possible choices as our answers. For choice A, we have 0.1336. For choice B, we have 0.0668. For choice C, we have 0.8664, and for choice D, we have 0.9332. Now, for the first part of this problem, we need to draw a graph of the standard normal curve. And so, when we draw this, we get the following graph. So we will have a bell-shaped curve with its peak at X equal to 0, and we need to Highlight or shade the region that corresponds to the area that is less than -1.50, and we have actually done that on our graph. So, Z equal to -1.50, and we, we've marked that on our graph and just shaded the region to the left of that. And so, what we need to do is to find the area underneath the curve, which is going to give us the probability that a randomly selected person scores less than -1.50 on this test. Now, since our test scores are normally distributed with a mean of 0 and a standard deviation 1, that means that our test score is also going to be our Z score. So to find the probability that we're looking for, we're just going to use our Z table and look up a Z score of -1.5 in our Z table. And we'll see that this corresponds to a row, which has a value of -1.5 and a column or 0.00. And the intersection of these two, this row and column gives us the value of our probability. And looking this up, we see that the area, in this case, is going to be equal to 0.0668. So that means the probability P that Z is going to be less than minus 1.50. is going to be equal to 0.0668. And so this here is the value that we're looking for for this problem. And if we look at our answer choices, this actually corresponds to answer choice B. So, I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Standard Normal Distribution

Z-scores

Probability and Area Under the Curve

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