Bone Density Test A bone mineral density test is used to ident...

Question

Bone Density Test A bone mineral density test is used to identify a bone disease. The result of a bone density test is commonly measured as a z score, and the population of z scores is normally distributed with a mean of 0 and a standard deviation of 1.

a. For a randomly selected subject, find the probability of a bone density test score greater than -1.37.

a. For a randomly selected subject, find the probability of a bone density test score greater than -1.37.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says in a bone mineral density test, the test scores are normally distributed with a mean of zero and a standard deviation of 1. Find the probability that a randomly selected subject has a bone density test score greater than -1.37. We're given four possible choices as the answers. For choice A, we have 0.867. For choice B, we have 0.915. For choice C, we have 0.047, and for choice D, we have 0.119. So we're asked to find the probability that a randomly selected subject has a bone density test score greater than -1.37. So instead of dealing with bone density test scores, we're going to actually want to use our Z score. So we call that our Z scores Z is going to be equal to the quantity of X minus mu in quantity divided by stigma. We're mu here is our mean, and sigma is our standard deviation, and we're told the problem that we have a mean of zero and a standard deviation of 1. So substituting in those values means Z is going to be equal to the quantity of X minus 0 in quantity divided by 1. Which is just equal to X. So that means that our Z score is going to be the same as our bone density test score. And if you recall that our Z score is for a normal distribution with a mean of zero and a standard deviation of 1, which is exactly what we have for our uh bone density tests. So now all we need to do is to find the probability that the Z score is greater than -1.37. And so to do this, we can use the Z tables to find the cumulative probability corresponding to a Z score of -1.37. So using our tables, we'll see that the probability. Of Z is less than -1.37. It's going to be approximately equal to. 0.0853. So this gives us the probability that Z is less than -1.37. However, the question is asking for the probability of having a test score of greater than -1.37. So, to find that probability, we just need to basically complement the cumulative probability that we already found. So that means that the probability P of the greater than -1.37. is going to be equal to 1 minus probability P of Z less than -1.37. Which is going to be equal to 1 minus. 0.0853. Which is going to be equal to. 0.915. And here we just run into 23 decimal places. So this is the answer that we're actually looking for for this problem. And if we compare our answer to our answer choices, we see that the correct answer choice 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

Z Score

Normal Distribution

Probability Calculation

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