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