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Numerade Educator

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Problem 21 Hard Difficulty

An important application of regression analysis in accounting is in the estimation of cost.
By collecting data on volume and cost and using the least squares method to develop an
estimated regression equation relating volume and cost, an accountant can estimate the cost associated with a particular manufacturing volume. Consider the following sample of pro-
duction volumes and total cost data for a manufacturing operation.
$$
\begin{array}{ccc}{\text { Production Volume (units)}} & {\text { Total } \cos t(S)} \\ {400} & {4000} \\ {450} & {5000} \\ {550} & {5400} \\ {600} & {5900} \\ {700} & {6400} \\ {750} & {7000}\end{array}
$$
$$
\begin{array}{l}{\text { a. Use these data to develop an estimated regression equation that could be used to }} \\ {\text { predict the total cost for a given production volume. }} \\ {\text { b. What is the variable cost per unit produced? }} \\ {\text { c. Compute the coefficient of determination. What percentage of the variation in total }} \\ {\text { cost can be explained by production volume? }}\end{array}
$$
$$
\begin{array}{l}{\text { d. The company's production schedule shows } 500 \text { units must be produced next month. }} \\ {\text { Predict the total cost for this operation. }}\end{array}
$$

Answer

a. $\hat{y}=1246.667+7.6 x$
b. $\$ 7.6$ per unit
c. 95.87$\%$
d. $\$ 5046.667$

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Video Transcript

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.