WEBVTT
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right in this problem. You were given information about
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medical hospital insurance benefits, and it says that the
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average yearly amount is 4006 or$4064 and that the
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standard deviation is$460 and that information is solely about
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the population. We're then going to select a sample
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off 20 patients. So we have to talk about
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the information about the sample and we're selecting 20 patients
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, so the sample size is 20. And if
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that's the case, we're going to talk about the
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fact that we need an average of the sample means
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and we're going to need the standard deviation of the
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sample means. And keep in mind that the average
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of the sample means is the same as the average
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of the population, which in this case is 4064
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And the standard deviation of the sample means is equivalent
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to the standard deviation of the population divided by the
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square root of N. So in this case is
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going to be 460 divided by the square root of
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20 and we have two questions. Part A is
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asking us to find the probability that the mean benefit
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for the random sample is less than 3800. So
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for part A, we're doing the probability that the
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mean is less than 3800. And then for part
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B, we're going to do that. The mean
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is more than 4100. So the first thing we're
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going to want to definitely do is talk about the
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bell shaped curve. We have 4064 right in the
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center. And for part, I were looking for
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3800. We're less than 3800. All right,
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so we're gonna want to switch the 3800 Intuit Z
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score, so Z equals 3800 minus 4064 divided by
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the standard error of the mean, which is for
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60 over the square root of 20. So the
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Z score associated with 3800 is negative 2.57 So on
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the bell, we can put negative 2.57 So being
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less than 3800 is the same as the probability that
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Z is less than negative 2.57 And if you look
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in your standard normal table, you're gonna find that
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the probability of less than the Z being less than
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negative 2.57 is 0.0 51 So that was for part
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a. Now we want to do part B and
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for part B, you're asked to find the probability
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that the average is greater than 4100. So 4100
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would be like somewhere here. So you're looking for
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this so again, we're gonna need a Z score
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associated with 4100. So we're going to do 4100
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minus the average of the sample means for 064 all
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over the standard deviation of the sample means which is
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460 divided by the square root of 20. So
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that Z score is going to be 0.35 So 0.35
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is equivalent to a 41 100. So being greater
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than 4100 is the same thing as the probability that
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Z is greater than 0.35 which can be written as
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one minus the probability that Z is less than 0.35
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And when you use the standard normal table the area
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to the left of 0.35 would be 0.6 368 And
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when you do one minus that, you get a
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probability of being point 36 three to. So just
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in summary, the probability that the 12 selected patients
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haven't average um, hospital insurance benefit less than 3800
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is 0.51 and the probability that the 20 patients selected
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have an average insurance benefit worth something greater than 4100
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would be 0.3632