Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Refer to exercise $21,$ where data on production volume and cost were used to develop an estimated regression equation relating production volume and cost for a particular manufacturing operation. Use $\alpha=.05$ to test whether the production volume is significantly related to the total cost. Show the ANOVA table. What is your conclusion?

Reject the null hypothesis $H_{0}$ or significant

Intro Stats / AP Statistics

Chapter 12

Simple Linear Regression

Linear Regression and Correlation

Piedmont College

Cairn University

Boston College

Lectures

0:00

10:31

Refer to exercise $21,$ wh…

02:31

For Exercises 9 through $1…

01:49

01:08

A sample of parts provided…

07:18

05:28

07:58

Use the data in WAGE2 for …

08:04

Refer to exercise 29 .…

06:25

An important application o…

02:04

03:55

For Exercises 7 through $2…

08:53

For Exercises 9 through $2…

08:33

For Exercises $9-14,$ use …

07:36

(Continuation of Exercise …

04:06

Consider the hypothesis te…

06:48

Consider the following hyp…

04:10

02:12

If a sample of size 18 has…

07:02

(Contimuation of Exercise …

05:39

For Exercises 5 through $2…

in Problem 29. We will use hypothesis testing procedures to test if there's a significant relationship between production volume and cost were given a significance. Level off Alfa is equal to 0.5 We will include her and know the table. And to draw our conclusion and state it, we will refer back to Question 21 that gave us the data for production volume and cost and with this data will make volume I X value and cost my y value or variable. From this data, I will develop what is called the estimated regression equation, and that's simply equal to the predicted value or the dependent variable is equal to the Y intercept, plus two slope times, a predictor value or the independent variable. In order to develop his estimated regression equation, we need to use the formula for slope and why Intercept? And this formula for Slope wants me to find the product with some of the product of X minus X bar Y minus y bar, all divided by the sum of X minus X bar quantity squared. I will then use this slope value to determine the Y intercept, which is equal to Why bar minus the slope Times X bar. Now what is expire and why Bar X Bar simply refers to the mean of all the values of X and y bar refers to the mean of all the values of why, in order to find the values or the X bar, we need to take this some of all the values of X and divided by the number the total number of values similarly real do the same for why bar, which is equal to the sum of all the Y values divided by the total number of values which is n If I look in my table here, I'll see that I have one too. 3456 data points. And so my n is equal to six. The sum of all the X values is equal to 3450 I'll divide that by six and I'll get 575. The sum of all my Y values is 33,700. I'll divide that by six and I'll get 5616.67 now that I have expert and why bar my slope quantities that I need to find our X minus X bar. Why minus y bar. I need to find the product of these two quantities expanded Expert. Why minus y bar and I also need to find the quantity X minus X bar. That quantity squared my third column here, X minus X bar is in asking me to take each individual value of X and to subtract the mean of X. So I'll take 400 I'll subtract 575. And that value would be negative 175. I would then need to perform the same operation for all of the other values of X to move forward. Now that I have all those values, I need to perform a similar operation, but with the Y values and why bar this time? So I'll take 4000 and I'll subtract 5616.67 And with that I'll get negative 1616.67 then perform the same operation for the remaining values in the White Column. Now that I have all those values, I simply need to take the product of those two quantities and fill out this column. So first I'll multiply negative 175 times negative 1616.67 And with that I'll get 282,000 917 0.25 and I'll perform the same operation for the reigning values in both columns. And finally I need to find the square of the quantity X minus X bar. So the first value Alabi Square will be negative 175. And that would give me 30,000 625. We need to perform the same operation for the remaining values. And now I confined this some of my last two columns in order to determine slope. The sum of my first column here is equal to 712,500 and this some off. My last column here is equal to 93,750. If I take these two values and plugged them into the formula for slope, I will be able then to determine what my slope is and thus do the same for the y intercept. When I divide these two numbers. I simply get 7.6 on my slope is 7.6 and I'm gonna take that slow. Plug it into the equation for why intercept and why? Barware reminded is 5616.67 minus 7.6, which is my slope Times X bar, which was 5 75 This would then give us 1246.67 Now that I have the slope and the Y intercept, I can derive my estimated regression equation, which is simply equal to 1246. Why Y intercept? Plus my slope 7.6 times X is a very important equation that we will use to move forward. Now we'll perform or analysis of variation and include those values in the you know of a table. And this unova table will introduce to us to sum of squares as well as mean squares. And these were important values that will help us to test for significance a little later on. Now we need to determine the sum of squares the sum of squares due to air, the total sum of squares and the sum of squares due to regression. Now there's an important relationship between these three values, and that is the total sum of squares is equal to the sum of squares due to error, plus the sum of squares due to regression. Now, if we find two of these values, we can simply perform a little algebra to find the missing value. If I observe the quantity, the total sum of squares you'll see it's asking me to find the some off the square of why minus y bar referring back to my table. I already have y minus y bar. So all I need to do is to square this quantity and I'll be able to find my total sum of squares. So the first operation I will perform is to square negative 1616.67 And with that I'll get two million 613,621 0.889 If I square all the other values in that column, I'll be able to find my total sum of squares, and I'll do that just now. Now I simply need to take this sum of all the values in this column and I need to and that some would be five million, 648,333 0.33 for and now I have my total sum of squares. I'll then record the total sum of squares into my nova table and go back to my formula to find the other to the sum of squares due to error or this almost squares due to regression knife. I observed these two quantities. I'll see that it's It includes the predicted value in their quantity, and I can choose to solve for either one. And I solved them off screen already, and it's much easier to solve the sum of squares due to regression. The numbers are much nicer. So first we need to find the predicted values, and then we'll be able to subtract the mean of all the Y values and square that quantity and will be able to find R sum of squares due to regression. Plug those numbers into this expression here and then solve for that missing value, which would be R sum of squares due to error. So going back to my table I need to find all my predicted costs or my white hat values. So in other words, I will take my estimated regression equation and plug our volume 400 in. So we'll have 1246.67 plus 7.6 times 400. And this would yield my predicted cost of 4286.67 I'll record my values in my table 4000 to 86.67 and repeat the process for the remaining values in The X Column here. Now that I have all these values, I simply need to evaluate the quantity why hat minus y bar And I need to square that quantity. Why had minus white bar quantity squared? So the first operational perform would be 4286.67 minus y bar and that mean was 5616.67 And when I performed that operation, I got negative 1300 dirty performed the same operation for the remaining values and the now square those values. Now that I have all these values, I simply need to square each of them and take the some and I'll be able to find my sum of squares due to regression. So I'll first square negative 1330 when I do that I will obtain one million 768,900. I'll perform the same operation for all the other values. Next, I will need to take the sum of all these values and I'll end up with five million, 415,000 and this is now my sum of squares due to regression. I would then record this value in Lyon Nova table and proceed to find the sum of squares to t error using my relationship equation from before SS to use equal to SSE plus ss are we already have the total sum of squares and a total sum of squares was equal to five million 648 1000 333.334 equal to SSE, which is what we need to find plus, as s our which. But we just found 5,415,000 performing a little algebra or s s E. Where sum of squares due to error would be equal to 233,333 0.334 The record R sum of squares in Terranova table and now we'll move on to mean square and degrees of freedom. Now each sum of square is associated with a number called a degree of freedom. Now the degree of freedom that's associated with the sum of the squares due to regression is equal to the number of independent variable. And in this case, a number of independent variable would be one, because our Onley independent variable here would be our volume production. The degree of freedom that is associated with this office squares due to error is equal to end minus two. And our end we knew was six minus two. And so a degree of freedom here would be four will record our degrees of freedom in our nova table and we just simply include the some, which is four plus one five. Now we need to determine our means square due to regression in our means square due to error, the formula for means square due to regression and means square due to error. ISAS follows the means great due to regression is equal to the sum of the squares due to regression divided by the number of independent variable. Therefore are means square due to regression is equal to five million, 415,000 divided by one which was simply just give us back five million, 415,000 are means square due to error is simply equal to have some of the squares due to error, which was 233,333 50.334 divided by and minus two, which we found before and that is equal to 58,000 333 0.33 35 We Nani to record our values inner and nova table and now we can finally test for significance the test for significance. We will. It will help us to determine how appropriate is for our model for us to use our model to show the relationship between a production volume and the cost. And if we recall the estimated regression equation, my part is equal to this is basically our best estimate for our population parameters, and we used a beta symbol here. Now, in order for there to be a linear relationship between our X and Y values in this case, production volume and cost, we need to have this parameter not equal to zero. The opposite of that would be that parameter being equal to zero. If it is equal to zero, then there is no relationship. But if it's not equal to zero, then there is a relationship, because if it's equal to zero, then this would disappear and there is no relationship. So we need to have our parameter not being equal to zero. So now will state are null and alternative hypotheses or no hypothesis would be your parameter. Being equal to zero because they're no hypothesis is always associated with equality, and so are alternative. Some people may write h one hypothesis would then be our parameter not equaled. 20 Next, we need to find our test statistic and we'll be using the F test because we have all the means square values and so f is equal to our mean square due to regression divided by R means square due to error. If we go back to another table. We have those values here, and we simply need to plug them into our formula. So our means. Grady to Regression was five million, 415,000 divided by 58,333 0.335 and that would give us 92.8285 What? I'm going around it, too. Two decimal places 0.8 threes of 92.83 would be our test statistic. I like to draw a distribution little F distribution here. Picture helps us to see what's going on, and I need to shade in my critical region. And if you remember earlier when we had Alfa being equal to 0.5 Alfa or the significance level is basically the probability that the test statistic will fall into this critical region here when the no hypothesis is actually true. So we're pretty confident about 95% confident that that won't happen. Next thing I need to obtain is my critical value, and we need to use the F distribution table in order to find that value. Now, in order to do so, we need to know what the degree of freedom and a numerator off that test statistic is and a degree of freedom in the denominator of our test statistic. Now, if we look at what the F test tells us, the means square due to regression is divided by the mean square due to error. If you go back to our table, we see that the degree of freedom you to regression, which is in the numerator, is equal to one and a degree of freedom due to error is equal to four. So in our table we will simply look for one in a numerator and four in the denominator at 0.5 level of significance, which is the probability of here, and that value that the table will give us is 7.71 If we observe our test statistic, which is 92.83 it falls into the critical region. It falls to the right of 7.71 other number line and so it's in our critical region. Therefore we will reject are no hypothesis and we will conclude there is a significant relationship between production volume and cost look and finally, the last thing you need to do is to plug our F value into her. No, the table. And that was 92.83 So again, we reject the null hypothesis and there is a significant relationship between the two and that's it.

View More Answers From This Book

Find Another Textbook

Numerade Educator

03:07

The admissions officer for Clearwater College developed the following estima…

04:23

In setting sales quotas, the marketing manager makes the assumption that ord…

13:26

To the Internal Revenue Service, the reasonableness of total itemized deduct…

10:26

$\begin{array}{l}{\text { Small cars offer higher fuel efficiency, are easy …

05:31

In exercise loan estimated regression equation was developed relating the pr…

02:16

The Wall Street Journal conducted a study of basketball spending at top coll…

02:58

The personnel director for Electronics Associates developed the following es…

01:59

In exercise $8,$ data on $x=$ temperature rating $\left(\mathrm{F}^{\circ}\r…

An important application of regression analysis in accounting is in the esti…

12:53

In exercise 18 the data on grade point average and monthly salary were as fo…