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Problem 38 Hard Difficulty

Salaries for various positions can vary significantly, depending on whether or not the company is in the public or private sector. The U.S. Department of Labor posted the 2007 average salary for human resource managers employed by the federal government as $ 76,503 .$ Assume that annual salaries for this type of job are normally distributed and have a standard deviation of $8850$
a. What is the probability that a randomly selected human resource manager received over $100,000$ in $2007 ?$
b. $\quad$ A sample of 20 human resource managers is taken and annual salaries are reported. What is the probability that the sample mean annual salary falls between $70,000$ and $80,000 ?$

Answer

a. 0.0039
b. 0.9611

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Video Transcript

okay for this problem were given some human resource data about salaries, um, and jobs. So annual salaries for this job, you're supposed to be on average, $76,503. And the variation or, um, or school precisely. The standard deviation from this 2007 data says, uh, well, salaries go up down, or the typical deviation from the mean is 80 $8850. Okay, as with any problems, probably kind of sketch out what it looks like that's normally distributed on what we're doing with this problem is really what is looking at some likelihoods of what occurs, um, be single value and the probability of something occurring for ah, or the sample mean? So let's look at this for the single values. So what's the probability of gonna translate this right into our probability notation? So we could say, What's the probability that the branch that the human resources person there is over 100,000? All right, so let me like to draw. It kind of gives you a general idea. So really, if this is about $76,000 for the sample mean, uh, $100,000 is gonna be way over here, so it's gonna be way after the side, so the probability is pretty likely that a single person does, but it's possible. So let's see what it is that these are calculated. We've done enough of these. Now that we're gonna take a look and let the calculator look at our distributions, we know it's a cumulative density frequency. So, uh, so we know we're looking from the lower boundary of $100,000 to infinity. We're looking about that's like to draw it out. So we kind of see that we're going this. What s represented is what's on the red on the screen. We know the mean for this problem. We've been told the data tells us the mean salary is five or three, so that $100,000 salary is much higher and standard deviation a typical variation. There is only Colony 150. So pretty big wish over here to get that salary based on the averages at least. So let's see what we get. So and it is a small number, as we kind of expected. So the likelihood of that is, uh, 0.39 So I'd say the 0.4% chance that you get that. Have a salary if the average is supposed to happen in between those two. Okay, so now and it just mark this up a little bit better now. So 76 1000 for party. We want to know that, Um, what's the What's the odds of the sample mean? Salary of this branch falls between what's expected there. So you can also put the sample mean up there. So now we're looking between I'm gonna save a little bit of digits. Space years. So the sample mean between 70,000 and 80,000 Some scar, right, Kay? Okay, 17 80 k So 76 is there. We know that 70,000 and 80 thousands of nice little slice of the middle here, so we should get a mush higher number. And that's what we're trying to figure out. So, um, industry in this up. So and we'll make sure ever good notation of the probability that it's between 70 and 80,000. Party. So we're gonna use our distribution from a calculator and one of the thing here too. I think I need to write down and will be get blue. Just emphasize it. That's based on a sample size Russia gonna sample on average 20 human resource managers. So that's instead of having a single person up here just for reference here in just a single sample or one person never turn about 20 people. Um, so let's see what we get. So go back to our distribution normal CDF and we're looking at a much more representative sample here between 70 and 80,000. So it's put in 70 and 80 upper bound, and we know that means should be 76,000 right in the middle. And we've been told that the salary variation 8 50 is, uh, a little steak there. It's not 76,000 middle. It's 76,503 alone. Okay, so let's see, uh, almost people mistake there. So it's not just this, and the key part of the problem is we take our standard deviation and we divided by the square root of the sample size cause that impacts our outcomes. So it's the square root of 20 you taking your 8815 divided by the square 20 your teachers picky about that? You can show that off as a separate calculation and we'll see. That's much higher. Likely that especially the 30 if you sample 20 there. So in the end, we know the probability is 96% about so the probability that you haven't mean that exists between 70 and 80 given 20 samples. ISS 0.961