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Numerade Educator

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Problem 12 Easy Difficulty

The average teacher's salary in North Dakota is $\$ 37,764$. Assume a normal distribution with $\sigma=\$ 5100 .$
a. What is the probability that a randomly selected teacher's salary is greater than $\$ 45,000 ?$
b. For a sample of 75 teachers, what is the probability that the sample mean is greater than $\$ 38,000 ?$

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Video Transcript

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