WEBVTT
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So this problem has two parts and we're talking about
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the population of teachers in North Dakota. So with
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this problem, we have teacherssalaries in North Dakota and
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the average teacher salary is 37,000 764 and the standard
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deviation is$5100. And we're going to use that
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information for both parts a part A and part B
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of this problem. And in part A. The
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question is asking you what is the probability that a
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randomly selected teacher salary is going to be greater than
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$45,000? So we're going to want to draw a
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bell shaped curve to represent the scenario going on.
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And in the center of the bell is always our
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average. We'd have 3000 or 37,000 764 and we're
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talking about the average being greater than 45,000. So
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we're going to need to do is we're gonna need
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to calculate the Z score associated with the 45,000.
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So we're going to do 45,000 minus the average,
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which is 37,764 all over the standard deviation of 5100
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and we're going to get a Z score of approximately
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ah, 1.42 So if we go back to our
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picture, we can put a 1.42 corresponding with the
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45,000. So now we can rewrite our problem instead
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of a saying the probability that X is greater than
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45,000. We could say the probability that the Z
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score is greater than positive 1.42 And because this is
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talking about to the right in terms of the probability
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distribution and when we looking the standard normal table in
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the back of your book, it always refers to
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the area to the left. We're going to have
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to then rewrite this statement as one minus the probability
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that Z is less than 1.42 We're then going to
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use the back of the book to gain the area
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to the left of 1.42 And when you do that
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, you're going to get a 0.9222 And when we
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complete the subtraction here, we're going to get a
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value of point 077 eight. So to summarize the
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part a of this, the probability that a randomly
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selected teacher in North Dakota we'll have a salary that
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is greater than 45,000 is 0.778 Now let's go to
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part B of this, and to do part B
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, we're going to have to reiterate the information again
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. So in part B, we had information about
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the population of North Dakota teachers, and the average
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was 37,000 764 and the standard deviation was 5100.
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But in part B, we're going to take a
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sample from that population, and the sample is going
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to be 75 teachers. And then Part B is
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asking us for that sample of 75 teachers. What
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is the probability that the sample mean just X bar
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is greater than 38,000? Again highly recommend drawing the
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bell shaped curve, and we put the average in
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the center. So we've got to now think about
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the concept of the fact that we took 75 teachers
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in this sample. So now the central limit theorem
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is going to apply, so the average of the
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sample means will be equivalent toothy average of the population
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, and in this case that's 37,764. So we
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could put 37,764 into the center of our bell.
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And the standard error of the mean or the standard
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deviation of the means is equivalent to the standard deviation
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of the population divided by the square root of end
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. And in this case, it would be 5100
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divided by the square root of 75. So we're
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trying to calculate the probability that the average is greater
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than 38,000. So 38,000 would be to the right
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of the average 37,764. And we're talking greater than
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so. We're going to need to calculate this thescore
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associate it with 38,000. So the Z score associated
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with 38,000 is gonna be 38,000. Subtract the average
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, which is 37,764 divided by the standard deviation,
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which is 5100 divided by the square root of 75
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. And that turns out to be approximately 0.40 so
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we could go back to our bell shaped curve and
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we can put a 0.40 where the 38,000 is.
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So by doing that, we can now rewrite our
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problem instead of our problem being the probability that the
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average is greater than 38,000. We could say that's
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the same thing as the probability that disease score is
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greater than 0.40 and the probability that Z is greater
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than point for zero. Because standard normal tape normal
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table reflects the area to the left or less than
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we have to rewrite this statement as one minus the
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probability that Z is less than 0.40 So you're then
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going to go to your standard normal table, and
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the probability that Z is less than 0.40 is going
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to be 0.6554 And when you subtract that from one
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, we get 10.34 46 So to summarize Part B
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, the probability that the average of the 75 teachers
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having a salary of more than 38,000 is going to
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be 0.34 46