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A manager at a local bank analyzed the relationship betweenmonthly salary (y, in $) and length of service(x, measured in months) for 30 employees. She estimatesthe ...

Question

A manager at a local bank analyzed the relationship betweenmonthly salary (y, in $) and length of service(x, measured in months) for 30 employees. She estimatesthe model: Salary = β0+ β1Service + ε. Thefollowing ANOVA table summarizes a portion of the regressionresults. dfSSMSF Regression 1 555,420 555,4207.64 Residual 27 1,962,873 72,699 Total 28 2,518,293 CoefficientsStandardErrort-statp-value Intercept 784.92 322.25 2.44 0.02 Service9.19 3.20 2.87 0.01 Whichof the fol

A manager at a local bank analyzed the relationship between monthly salary (y, in $) and length of service (x, measured in months) for 30 employees. She estimates the model: Salary = β0 + β1Service + ε. The following ANOVA table summarizes a portion of the regression results. df SS MS F Regression 1 555,420 555,420 7.64 Residual 27 1,962,873 72,699 Total 28 2,518,293 Coefficients Standard Error t-stat p-value Intercept 784.92 322.25 2.44 0.02 Service 9.19 3.20 2.87 0.01 Which of the following is the standard error of the estimate? A) 72699 B) 269.63 C) 322.25 D) 3.20



Answers

The following data are the monthly salaries $y$ and the grade point averages $x$ for students
who obtained a bachelor's degree in business administration with a major in information
systems. The estimated regression equation for these data is $\hat{y}=1790.5+581.1 x .$
$$
\begin{array}{l}{\text { a. Compute SST, SSR, and SSE. }} \\ {\text { b. Compute the coefficient of determination } r^{2} \text { . Comment on the goodness of fit. }} \\ {\text { c. What is the value of the sample correlation coefficient? }}\end{array}
$$

Okay, so this problem is talking about the average salaries off of business administration agrees, and it tells us that there between 30,000 and 45,000 and then it says we can use a 95% confidence interval to find the planning value number one and number two. To find out how many people we needed our sample toe have our margin of error be $500.200 dollars and then $100. Okay, So the first thing I'm going to use that 95% confidence interval they gave us confidence intervals are constructed using the sample mean, which is what expire represents. And then you add or subtract the Z value. Um, sorry, that on that's busy value, which you get from how confident you are. So in this case will use those e value for 95% confident, um, which in this case, it will all cases for 95% confident. There's a 1.96 and you should have a table. Or that should be pretty known because that's a pretty common one. Um, and then we have the population standard deviation right up there, and then skirt of Ed. That's what we want to know in this case, right? How the number of people we want in our sample. So I'm gonna say that's what we want to know. Okay, Now, since we don't have the sample mean that should kind of give us a clue that for planning value, we're going to be using this piece of the equation right there. Okay, that's what we call our margin of error. And since it gives us those values for A B and C, I know that's the piece I want to use. So let's go ahead and write that down. That that's our margin of error. And now there's one little problem here that that population standard deviation. I'm on top of the fraction. We don't know that. So we're gonna have to estimate that using s okay, which is the sample standard deviation. Hence the s for sample. Um, and we can estimate this by using our range, and then we divide that by four. So in this case, you take the high number, which is 45,000. So tracked the loan, which is 30,000 and divide by four. So I got 15,000 divided by four. Which excuse me, is 37 50. Okay, so that's what we're gonna be using right up there for our population standard deviation. So let's put this all together and all kind of right in red's. We remember that's an emergent of air. Um, so that's right. Margin of error. Because that's the piece we're using is going to be that the value of 1.96 we're going to times it by that 37 50 we just found, Oops, 37 50 that we found. And then we're gonna have the divide by squared of end, which is what we're going to end up solving for, um So if we want to simplify this a little bit, um, I would make the 1.96 into a fraction causes multiplying. Fractions is easy. You just to talk time's top and bottom times bottom. So we actually end up getting right over here that our margin of error his equal to know when you multiply the tops. I got the number of excuse me 73 50 like so and then you just say okay. And that's over the square root of in. So that's pretty simple. And now we start plugging in our values for a so that first or not for a for emergency error. So we're going to use part eight and say, That's going to be our 500 equals R 73 50 divided by Squared of Ben. And we're just gonna do, um, reverse algebra to get Oregon algebra but working backwards to get end by itself. So the first thing I need to do is I need thio times by the square root of And so we'll do that in blue so you don't get confused so times by scored event and whatever you do to one side, you've got to do to the other. So it's gonna have to be won'th quite by the square to bed as well. So now I get that I have 500 square root of end on the left and 73 50 on the right, cause word of it in on top and squared event on bottom. Make it one. And now I say okay to get and by itself, I've gotta move that 500. It's being multiplied. Soto undo that. I divide right 505 100 gives me a one squared event equals and then 73 50. Divided by 500 was 14.7. Okay, it's kind of unlikely. 14. Let's try that again. 14.7. And now the last part is you need to undo that square root. So we're going to square because older in verses, whatever you do to one side, you do the other and you get that end is 216 point. Oh, night. Now, you can't have 00.9 of a person, so you've gotta round up and say you need 217 people. Okay, So for part, a tow, have a $500 margin of error, which means how much you want to be above and below your sample. Mean you would need 217 people or call it graduates with business administration degrees. Okay, so I'm gonna leave that up because that's how you get part a. And I'll write that down real quick just so we know where we're looking. So it's part eight. Now let's do part beat, and it's gonna follow a very similar thing. But instead of 500 we're going to use 200. So let's go ahead and do. I'm sorry. 200 equals that 73 50 divided by the square root event. Now, I want to point something out here and all kind of pointed out an orange so that we can see it. See how we started with a 500 then the animal bottom. But when we did our algebra notice, our end came to the other side and our 500 was on the bottom. You see that? So they kind of switch spots. That's always gonna happen because you're always gonna have toe multiply and then divide in this case. So I'm actually just gonna skip to that step and say, OK, the switch spots? No. If you're not understanding where that comes from, you can always go into a long way. I'm just trying to save you a little bit of time and I get to the square root of in is 73 50 over 200 right? If you look to the left, you can see I got squared then by itself, by 73 50 and the 500 on bottom last time. So now the 200 some bottom here. So when I divide 73 50 by 200 I get that. The square root of end is 36.75 Okay? And so now I just have to square and square on both sides, and I get the end is 135 serial 1350.56 Now, again, you can have 0.56 of a person. So I'm going to say we need at least through 13 51 or 1351 people. Okay, so let's part b. So once you kind of figure out what you have from above, you'll be able to just do the algebra in part A, B and C. So the hardest part is just figuring out, um, to use s and to get that value. So now let's go over here will do part, see, and we're going to the exact same thing. But now we want the margin of error to be within $100 of our sample means So we're gonna soak a 100 equals. You still use that 73 50 divided by the square root event. And I hope you've noticed that it doesn't do to switch. So I'm going to just go ahead and do that shortcut and go back and say, Okay, now that I've done that, I have that the square root of N is equal. This 73 50 divided by 100. That should hopefully be a pretty easy one. Gets the squirt event as 73.5. Okay. And then Carol, zoom in a little bit. Oh, get rid of that. So then we square each side and I get that and is 5402 0.25 Now you can't have 1/4 of a person, right? I said that every time. So I said, OK, I really need 5000 403 people to be within $100 of that sample mean Okay, so now it asks, would you recommend using that now? I personally would not. So I'm gonna write. No. And the reason for that is because that means I need basically 5400 graduates to come. Thio, give me their salaries toe work that all out. And it only gives me $100 from the average and $100 when you're dealing with salaries between 30,045 is not that big of a deal. So I'm gonna say no, right? It's hard to get that many people. And ah, $100 isn't a big deal per se when dealing with salaries on. That would be the whole solution to this problem. So you just have to figure out what it wants you to use, which is the margin of error. Then you figure out your green values, right, and you have to approximate your population start deviation with s, you go ahead, put clothes in, and then a B and C to actually get the answers is just algebra s, so I hope that helps.


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