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Numerade Educator

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Problem 22 Hard Difficulty

Assume that the mean systolic blood pressure of normal adults is 120 millimeters of mercury (mm Hg) and the standard deviation is 5.6. Assume the variable is normally distributed.
a. If an individual is selected, find the probability that the individual’s pressure will be between 120 and 121.8 mm Hg.
b. If a sample of 30 adults is randomly selected, find the probability that the sample mean will be between 120 and 121.8 mm Hg.
c. Why is the answer to part a so much smaller than the answer to part b?

Answer

a .1255 $\\$
b .4608$\\$
c. Individual values are more variable than the sample means.

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Video Transcript

we're going to start with what we know in this problem. We know that the mean systolic blood pressure of normal adults is 120 millimeters of mercury, and we know that standard deviation is 5.6. We also know that we're going to assume normal distribution. So again, we're going to use our bell shaped curve part A. If an individual is selected, so only one individualised selected. We want the probability that that individuals systolic blood pressure is between 1 20 and 1. 21.8. So we're gonna start by placing the 1 20 the mean on our bell, and we want to be between 1 20 and 1 21.8. So we will need our Z score for both, because 1 20 is right at the peak of the bell. We know that 1 20 would correspond with the Z score of zero and to calculate Z score for 1 28 we'll do 1 21 0.8 minus 120 divided by the standard deviation, which was 5.6, and we will get a Z score of 0.32 So when we talk about the X value between 1 21 21.8 were also talking about the Z value being between zero and 00.3 to. So we're going to solve that by changing this into its equivalents attraction problem. So the probability that Zuse less than 3.2, minus the probability that he's less than zero We'll use our standard normal table from the back of your textbook, and we find the probability that Z's list in 3.2 is 0.6255 We know that the probability that C is less than zero is 00.5000 because that is half the bell, so therefore we get a value of 0.1255 So the probability of us selecting an individual, um, with a pressure between 121 21.8 will be 0.1255 Now, let's go on to Part B. We're just gonna scoop this up a little bit and far for Part B. It's asking us now we're going to select a sample of 30. So let's still talk about what the population waas. The population had a mean systolic blood pressure of 1 20 with a standard deviation 5.6. And now we want to talk about the sample and the sample size. In this case, we're going to select 30 adults at random, so we now need the average of the sample means, and we need the standard deviation of the sample means, and the average of the sample means is the same as the average of the population, which is 1 20. And the standard deviation of the sample means is equal to the standard deviation of population divided by the square root event. So in this case, it would be 5.6 divided by the square root of 30. So again we're going to set up our bell shaped curve. We're going to put the average in the center and the average was 1 20. And let's look at the problem that we're trying to solve. And the problem we're trying to solve is a sample of 30 adults is selected at random. What's the probability that the sample mean is between 1 20? So this time we're gonna have an X bar in our statement and 1 21.8 so we're gonna need the 1 21.8 on our bill, and we want to be an average of the 30 adults to be between 1. 21 21.8. So, again, we're going to need to calculate the Z scores for each of those. And this time the formula is gonna change slightly. It's gonna be X minus or Sorry, X bar minus Musa of X bar over Cygnus X bar. So here we have an average of 1 21.8, minus 1 20 divided by 5.6 over the square root of 30. So the Z score turns out to be 1.76 again, we know that the Z score associated with 1 20 is zero because it's at the peak of the bell. And if you think about the fact that being between 1 21 21.8 is the same as when our Z score is between zero in 1.76 so we could rewrite that as the probability that Z is less than 1.76 minus the probability that is less than zero, and we would rely on our standard normal table and we would find that the probability that C is less than 1.76 is 0.9608 and the probability of Z being less than zero is 00.5000 and we get an answer of point for six zero eight. So for Part B, the question was, What's the probability that when we select 30 people at random there, blood pressure is between 1 21 21.8? And I should say their average is between 1 21 21.8 and that probability would be point for 608 and then finally, for part, C Part C is asking us why is the answer to part a so much smaller than part B? So let's just recap with the part A and part B were in part A. We were talking about an individual persons blood pressure, and we had an answer of 0.1255 and in part B. We were talking about taking a sample of 30 adults and their average blood pressure being between 1 21 21.8, and we found it to be 0.4608 So the question is saying, Why is this one so much smaller? And the reason would be that a sample mean will be closer to the true mean than an individual value, so the true mean is more likely to be the 0.46 several eight.