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.
e. If the mean bone density test score is found for 9 randomly selected subjects, find the probability that the mean is greater than 0.23.

e. If the mean bone density test score is found for 9 randomly selected subjects, find the probability that the mean is greater than 0.23.

Accepted Answer

Hello, in this video, we are going to be determining the probability of being to the right of a certain sample size on a normal distribution. So, we are told that in a study on sleep patterns, researchers measure the amount of REM sleep individuals get per night. The REM sleep scores are normally distributed with a mean of 0 and a standard deviation of 1. If the mean RAM sleep score is found for 25 randomly selected individuals, we want to find the probability that the mean is greater than 0.1. So, before we approach this problem, let's just go ahead and list out the given information. We are told that the REM sleep study is normally distributed. Now, for this normal distribution, we are given a mean of 0 and a standard deviation of 1. We are going to represent this mean with the variable new, and again, we are setting new equal to 0. Now, for the standard deviation, We are going to represent standard deviation with the variable sigma, and this is going to equal to one. Now, we are told that we have a sample size of 25 randomly selected individuals. We are going to represent this sample size. With the variable N, and we are setting this equal to 25. Now, our goal in this case is to find the probability that the mean of the sample size is greater than 0.1, and we're going to represent that with P of X greater than 0.1. Now, what this means for the standard distribution is that on the standard distribution, We want to find the probability that is to the right of 0.1. But what we need to do is we need to find this probability with respect to the fact that we are working with the sample size. So what we need to do is we need to find the standard deviation of the sample size. We need to convert this this into a Z score with respect to the sample size, and we are going to need to find the mean of the sample size. So first, let's go ahead and start by finding the mean of the sample size, which is referred to as new sub X. Now, the mean of the sample size, new sub X is equal to the mean of the overall distribution. And since the mean of the overall distribution is 0, that means that the mean of the sample size is going to be 0. Now let's go ahead and find the standard deviation. With respect to the sample size. Now, in order to find the standard deviation of the sample size, the standard deviation of the sample size is written as sigma sub X. This is equal to sigma divided by the square root of N. So what this means is that if we want to find the standard deviation with respect to the sample, we need to take the standard deviation of the overall normal distribution and divide that by the square root of the sample size. This is going to give us 1 divided by the square root of 25, and if we plug this into a calculator, this is going to simplify to be 0.2. Now what we need to do is we need to go ahead and convert our our probability of the sample size into a Z score. So, we're going to change P of X greater than 0.1 into a Z score. With respect to the sample size. Now, since we are working with the sample size, in order to convert this into a Z score, this is going to be defined as X minus sigma sub X or new sub X, divided by sigma sub X. So this is saying that we are going to take the value of the score or of the standard. What is it? We are going to take the value. Of the probability that we are looking for minus the mean of the sample size and divide that by the standard deviation of the sample size. So that is going to give us 0.1 minus 0 divided by 0.2. And when we plug this into a calculator, that is going to give us the value of 0.5. So that's going to allow us to rewrite this probability to be the probability that Z is greater than 0.1. Now, we are going to find this probability by using the Z table, but there's one thing that we need to remember about the Z table. The Z table gives us the probabilities to the left. Of the standard distribution or the normal distribution. So if we want to find the probability of Z greater than 0.1, what we can go ahead and do is we can take 100% of the normal distribution, which is 1, and subtract the probability that is to the left of 0.5. And I apologize, this should be Z greater than 0.5. So, in other words, in order to find the probability to the right of 0.5 on the normal distribution, this is going to be 1 minus the probability to the left of 0.5 on the normal distribution. Now, by using the Z table, the probability that is to the left of 0.5 is going to be 0.6915. So, we have 1 minus 0.6915, and when we plug this into a calculator, we get to 0.3085, which is going to be the solution to this problem, and the overall solution 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.

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Key Concepts

Z Score

Normal Distribution

Central Limit Theorem

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