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A shoe store developed the following estimated regressionequation relating sales to inventory investment and advertisingexpenditures.Å· = 21 + 12x1 + 7x2wherex1 =...

Question

A shoe store developed the following estimated regressionequation relating sales to inventory investment and advertisingexpenditures.Å· = 21 + 12x1 + 7x2wherex1 = inventory investment ($1,000s)x2 = advertising expenditures ($1,000s)y = sales($1,000s).(a) Predict the sales (in dollars) resulting from a$15,000 investment in inventory and an advertising budget of$10,000._________$ (b)Interpret b1 and b2 inthis estimated regression equation.Sales can be expected to increase by $ ________ for every

A shoe store developed the following estimated regression equation relating sales to inventory investment and advertising expenditures. Å· = 21 + 12x1 + 7x2 where x1 = inventory investment ($1,000s) x2 = advertising expenditures ($1,000s) y = sales ($1,000s). (a) Predict the sales (in dollars) resulting from a $15,000 investment in inventory and an advertising budget of $10,000. _________$ (b) Interpret b1 and b2 in this estimated regression equation. Sales can be expected to increase by $ ________ for every dollar increase in inventory investment when advertising expenditure is held constant. Sales can be expected to increase by $ _________ for every dollar increase in advertising expenditure when inventory investment is held constant.



Answers

A shoe store developed the following estimated regression equation relating sales to
inventory investment and advertising expenditures.
$$\hat{y}=25+10 x_{1}+8 x_{2}$$
where
$$\begin{aligned} x_{1} &=\text { inventory investment }(\$ 1000 s) \\ x_{2} &=\text { advertising expenditures }(\$ 1000 s) \\ y &=\text { sales }(\$ 1000 s) \end{aligned}$$
$\begin{array}{l}{\text { a. Predict the sales resulting from a } \$ 15,000 \text { investment in inventory and an advertising }} \\ {\text { budget of } \$ 10,000 .} \\ {\text { b. Interpret } b_{1} \text { and } b_{2} \text { in this estimated regression equation. }}\end{array}$

Hi, folks. We're working on problem number 18. Number 18 gives us some information and asks us to perform a linear regression on the data. And I've been to take a look at the graph and to talk about it a little bit. Um, so the charts that they give us has three columns in it, and we have to take a second and say, What are we really looking at here? What are we entering? What are we using? Right when we enter this into a graphing utility, Um, the months 1234 those air. Not really What you need to be looking at. Those were just apparently randomly selected months. That's what it's described in the paragraph. Um, they do label, however, as X and y the advertising expenditures and the sales volume. That's actually what you want to type in to your columns. So don't do the month Thio. Start with the advertising expenditures and then do the sales volume. I have entered that over here in this calculator, and you could see how the information is set up. I'm gonna go out and perform my linear aggression right now so I can take a look at it. Okay, so here I choose linear regression. And in this one, I'm going to type C one because that's the cattle. That's the column that I've entered this information into. Right, So see one And the next one? That's not the one. That's a cue. See one and we'll get it. Here, Alfa load C one. There you go. On N C two. So Alfa Mood C two and then right here. It says, What do you want us to do with the equation? I want you to put it into the graph for so we can look at it that we won't have to re type it myself s. So I go ahead and select Why one and it's gonna put it in Y one, and we should be ready to run the regression, right? Well, these air our numbers in the UNR ound ID form. Okay, Um, let's go ahead and write those down here on the next page. Okay, so I got these two things written down. I got the correlation coefficient again from the calculator. It just says CEO har um And then there's the equation itself. And I did that by taking the A value in the B value and rounding them and putting them into the equation. Okay, so that's what the second part a and part B ask for. Okay, so now we need Thio. Look at both of them in the same window and interpret the slope of this model in the context of this will be fun. Okay, so we had this entered it, put it into the graph for for us because of that middle thing that I pressed and we should be able to take a look at the graph were zoomed in a weird level. If you ever get a graph, it looks like this. You can use my favorite zoom data number nine, and it should re center it essentially right simply Alright, What we've got here is the line that the calculator came up with and the data points that we plotted. Um, now, if you look at this, it pretty much conforms to align. It's not exact, it's not, But it is actually kind of sort of like that stretched out a lot of information, so it's pretty close. What they're asking to do, though, is to think about the slope of that line. What does that mean? What really What it means is that means as you spend more money on advertising, you get more revenue, right? So it's how quickly and at what rate that your expenditures get to turn into, you know, more sales. So let's go ahead and put that on the on the answer sheet on this page. Okay, So the way that I phrased it was the slope is representing the rate in which the advertising expenditures increase total sales. Okay. And I like to put rate in there because, really, that's that's what slopes are. All right. And then the last part D it asked us to use the model toe, figure out a number for if we spend $1500 on advertising. So we're gonna go ahead and put that in for the expenditures, which is X, and then we're going to run it through our equation. We're going to see what happens. Okay, so what we do is we substitute this in for X, and then we go ahead and run it through. I like thio. Use the if I have already entered the information in a graphing utility or calculator. Um, you could just look up. What number? Here's the X value. 1.5. And there's what? My result waas. Okay, that saves a little bit of time there, Um, does the math for you? In fact, you've already done the math just by, you know, entering in the calculator. So one important thing here, why did I put 1.5 and 1500 right? Didn't say $1500. Well, if you look at the original data, the original data was in thousands of dollars. Okay, so 1.5 would be in thousands of dollars. Okay, Now the sales volume, right? They never tell us what that is. What about what the volume is? Maybe it's units. Maybe it's I don't know. We don't know what it is. So we're not gonna have any units on this one. But this is for 1500. Okay. Uh huh. All right. Thank you.

So I put my data into my calculator and graft list one and list to and then had it do a linear regression and found that this was the model that I was getting cool and where this was in putting these shoe size for these males. And this was out putting the height in terms of inches. And we know that our data is set up, that the lowest is 8.5 for a size and the highest is 12.5. So anything outside there is going to be extrapolating. So if we look at finding using this model to predict for 11.5 or 11 and size 11.5, it is it is fine to use this model. We would end up finding That we'd estimate 72 points 865 is what we would estimate for the height. However, for part B, if we input eight, that would be extrapolating. So we'll say that that is not meaningful because that is extrapolating outside of the domain that we had. And likewise, if we try to predict for that Bigfoot fifth size 15.5, Uh that's not going to be meaningful either, because the highest shoe we had was 12.5, so that would be extrapolating. And so this value would not be meaningful. And then our last one. If we predict for a size 10 that's within the domain, so that would be interpreting and we get 70.06 as what our equation our regression equation would estimate.

Okay for this problem were given some information about cost production volume for some manufacturing. So I'm gonna write out what we want to find here first, and we're going to work our way and use Excel to figure these things out. So for part a just a plan ahead we want to find with a regression equation. So we're gonna find out a regression equation when you use our excel to do that, he could also you the graphing calculator. Um, so the regression equation Ah, I used the use like was a plus B x on a part B. You want to find out what is the cost per unit produced? And really, that's the average cost for human part C. We want to by what's called the coefficient of determination. Provisional determination is the r squared value. And then finally you were going to say whatever situation we make regression equations to figure out what happens if we have a certain number of units that will have ah ah model. And then we're going to say, Well, what if we have X equals 2 500 units? What will the cost be? So let's go over here So the first thing have to put in some data over to my excel. And if you have timing, kind of give it a nice title, I'm just going to list this out. You see it in your pocket, But production volume unit says 404 50. It's nicely going up. It wasn't. Doesn't matter. Excel can still do the best fit line and the regression equation, regardless of it being in order. Um, so the total cost is 4 4004 units, 5000 and so on. So we're gonna do is we're gonna let excelled a little. The work for us and Europe, the regression equation. So we can answer all these questions on the average increase and look at the coefficient of determination. So there's different ways to get this to happen. But good news is I have my values from a textbook here, um, going to make a table, you see, insert gold school on this, I go insert, and I look for all these different drafts. Well, in our when I use the most going to like a scatter plot. So there's your scatter plot. We don't want to connect the lines so since it made a table for us, you might as well give it the title. So this is production volume. Keep it simple. Constructs production alive. Volume that's what we want to do is like the ruler just ruler between these guys here, Um, you would excel. There's different ways to do it But I click on the data and then I just do. It's called out of trend lines. I click right click on the data Do add a trend line. We want a linear fit. And so the biggest thing here, the one thing to remember is we want to. The equation is what we care about. Your answer. That question in this r squared value. There's one thing remember from this video, the R squared is the coefficient of determination. It's gonna help us answer. I believe it's question part C. Okay, so there we've put it in. Um, I don't think you have to do this because we're not submitting our graph. But just for the knowing that we can kind of extend beyond our data points some periods is what that calls what they call it there. All right, so let's look and see close this so I can see it. And then now we can see our trendline. All right, so basically, excels Answer that question for us. So let's go over here. Um, statisticians like to do I equals a plus B X. So let's look at this. So we want to do I equals, uh 12. 46.7, 7.6 secs. And that's the rate of change. Okay. And your stats class, you wanna go out? White hat prediction. Right. So that seem sort of our one cost per unit is really just this. Okay, so on average, it's $7.60 per unit. Your slope and your r squared value is the coefficient determination that tells you how much of the cost could be attributed to the volumes that we'll look back at our r squared value. So the R squared is point 9587 If you're answering a sentence, you want to say 95.87% of the cost can be attributed to that. All right? And finally, we're doing a little bit of algebra here. So this is what if the X is 500? So what we're gonna do for the last part is just plug it in here, plugging into our regression creation cancer in this year s equation, we're gonna plug it in. So then we know that it is 12. 46.7 less, 7.6 times X. They say the X we care about What if it costs 500 engulfed inside and your calculator here, So I'm going to take 7.6 times 500 plus 1246.7. If you calculate that out, you're going to get 5000 46 dollars in some of the sense way determination of mixing predictions. You're good to go.

So in this question were given the following multiple regression equation that has two independent variables and the data comes from 10. Observations were also given the SST and the S S R. In part A we are asked to compute three values s S e MSC and M s r. Okay, so let's go ahead and compute these values. So the s s e it's basically SST minus s s car. So that's 16,000 minus 12,000, which gives us 4000. So we have our first value over here. Now let's compute the second value. So now we have m s knees were going to write that down. So MSC from the formula is SSE over and minus P minus one, which is 4000 over n minus P minus one. So our mean squared error MSC is 571.43 We rounded it to to the CMO places. Now the final calculation is for m s are So let's go ahead and write our formula down. So m s R is basically s s are over p. So SSR is 12,000 over pink, which is to that gives us 6000 so we have our three values for part thank you Now in part B were asked to use an F test at a 0.5 level of significance to determine whether there is a relationship. So, in part B, write this down there has to use an F test point off five level significance to determine whether the parameters theta one theta two Sequels 20 or which one we reject on our hypothesis. So let's go ahead and compute our F statistic. So our F statistic is basically given by the formula M s r over M S e. So we have MSR over here. We have a messy over here, so let's calculate that. So we have 6000 over 571 point for three, which gives us the value off 10.5. So this is our statistic and were asked to use that to determine whether the relationship is significant. So let's look at the table. So we see in our table that for the degrees of freedom in the numerator off to and in the denominator off seven with 0.5 significance, we see that the F statistic is 4.74 so we see that our value here is higher than this value. And so the question asks us to determine where whether there is a relationship, so there is a significant relationship between the variables, so this significant and that is our answer to part.


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