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Problem 7 Medium Difficulty

Using the Central Limit Theorem assume that females have pulse rates that are normally distributed with a mean of 74.0 beats per minute and a standard deviation of 12.5 beats per minute (based on Data Set 1 "Body Data" in Appendix $B$ ).
a. If 1 adult female is randomly selected, find the probability that her pulse rate is between 72 beats per minute and 76 beats per minute.
b. If 4 adult females are randomly selected, find the probability that they have pulse rates with a mean between 72 beats per minute and 76 beats per minute.
c. Why can the normal distribution be used in part (b), even though the sample size does not exceed $30 ?$

Answer

a. 0.1271 $\\$
b. 0.2510$\\$
c. Because the original population has a normal distribution, the
distribution of sample means is normal for any sample size.

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Video Transcript

alright in this problem, we're dealing with the population of female pulse rates, and we were informed that that population is normally distributed with a mean of 74 beats per minute and a standard deviation of 12.5 beats per minute. And this problem comes in three parts. So let's start with part A part A is we're going to select one adult female and we want to know what's the chances of the probability that her pulse rate is between 72 beats per minute and 76 beats per minute? So my recommendation is we start with an image of a normally distributed set of data or bell shaped curve. We know the average is 74 and we're trying to figure out the probability of being between 72 and 76. We will need Z scores in order to solve this. So as a refresher, the Z score formula is X minus mu over sigma. So we will need to find the Z score, associate it with 72 so we're gonna do 70 to minus 74 over 12.5 and you will get a Z score of negative 0.16 and a Z score associated with 76 is 76 minus 74 over 12.5 or positive 0.16 And I like to put those back in my picture, so I'm gonna put a negative 0.16 72 and I'm going to put a positive 0.16 at 76. So when I am discussing the chances of being between 72 76 I'm also talking about the probability that the Z score will be between negative 0.16 and positive 0.16 And because it's an expression discussing between this, you will have to separate it as the probability that Z is less than 0.16 minus the probability that Z is less than negative 0.16 And at this point, you would have to go to the table in the back of your book standard Normal table, which is a two in this book, and you would look up both values and the probability associated with a positive 0.16 would be 0.5636 And the probability associated with the Z being less than negative 0.16 is 0.4364 for an overall probability off 0.1272 So the probability that that one adult is selected and her pulse rate is between 72 76 beats per minute would be 760.1272 So it's to part B and in part B, we're going to select a sample, and in this case we're Onley selecting four adult females at random. So our sample sizes for and you're asked to find the probability that the average or the mean of those four females pulse rates is between 72 and 76. So notice the difference between the two problems in the part A. We're using X and in part B. We're using an X bar here in order to solve this part B. Then we will need to find out what the average of the sample means is, and we will need to determine this standard deviation of those sample means. And we'll use the Central Limit Theorem to do that. And the Central Limit Theorem says that the average of sample means is the same as the average of the population, and in this case that was 74 beats per minute, and the standard deviation of the sample means will be equivalent to the standard deviation of the population divided by the square root of your sample size. So that would be 12.5 divided by the square root of four. So again, we're going to draw the bell shaped curve again. We're gonna put the average in the center, and we want to find between 72 and 76. So again, we are going to have to find the Z score. But we will have to modify our Z score formula a little bit. So we're going to have to use X bar minus mu X sub X bar over Sigma sub X bar. So for 72 our Z score well read 70 to minus 74/12 700.5 over the square root of four. So that Z score is going to be negative. 0.32 and we'll do the Z score for 76 is well, so we're gonna do 76 minus 74 over 12.5 over the square root of four, and we get a Z score of positive 0.32 So again, we'll put them back on our picture. So we have negative point three to associate it with 72. We have positive 720.3 to associate it with 76. So when we're discussing the probability that the sample mean will be between 72 76 it's no different than asking you for the probability that the Z score is between negative 760.32 and positive 0.32 We will have to rewrite this expression as the probability that C is less than positive 0.32 minus the probability that C is less than negative 0.32 And again, you're going to use table to a or a two in the back of your book. And the probability associated with Z being less than positive 0.32 is 0.6 to 55 And the probability that Z is less than negative 0.32 is 0.374 five for an overall probability of 0.2510 So, just in summary, if we select four people at random, the probability that they're mean pulse rate is between 72 76 beats per minute would be 0.2510 So there's one final part to this problem. Part C. In part C is asking you. Why can the normal distribution be used in Part B, even though we had a sample size that did not exceed Besides 30? And the reason is because the original population of female pulse rates is normally distributed, so the distribution of sample means can be approximated by a normal distribution for any sample size, yeah.