A small business tracks how advertising spending relates to weekly sales.(C)(C) Create a prediction interval for the value in part (B)(B).

( 9066.62 , 11434.18 ) \left(9066.62,11434.18\right) ( 9066.62 , 11434.18 )

A small business tracks how advertising spending relates to weekly sales. ( C ) (C) Create a prediction interval for the value in part ( B ) (B) .

Welcome back. In this problem, we have a small business tracking how advertising spending relates to weekly sales. In part a, we got the scatter plot for our data. We also found the regression line equation and got r, the correlation coefficient. And in part b, we predicted the y value, weekly sales, if the x value, advertising, was 400. In part c, we'd like to create a 90% prediction interval for the value in part b or a 90% prediction interval for the weekly sales if $400 is spent in advertising. Before we create a prediction interval, we have to check two conditions. First, we need to make sure that there is strong linear correlation between our two variables. We know that the correlation coefficient r is 0.985, which is really close to one, so there is strong linear correlation. We also need our x value, in this case, 400, to be within the x range. Looking at our data, we can see that there are values below 400, like 210, and there are values above 400, like 450, and our x data for advertising. So we can say that 400 is definitely in range. And with both conditions met, we can create our prediction interval. Now the first step would be finding the point estimate, which would be the predicted y value when x is 400. We found that in part b to be 10,250.4. So next, we need to find our margin of error, e, which we're going to do in steps because it is quite the calculation. Let's start by getting our n value. For a regression model, n is going to be the number of coordinate pairs we have, which in this case is going to be the number of weeks where we collected data, and we can see that that's going to be 15. Next, we want to find our t critical value. We're going to use the same function that we've used for confidence intervals, equals t dot I n v dot two t. Probability, the first input, is going to be the alpha value, which is one minus the prediction interval percentage. In this case, 90% or point nine. So we'll do one minus point nine to get point one, and point one is going to be probability, the first input. We'll do a comma, and next we need degrees of freedom. As a reminder, the degrees of freedom is n minus two, so we'll click on our n value of 15 and do minus two. Once that's all set, we get our t critical value of about 1.77. Next, we need the standard error. If it hasn't already been given to us, we will need to get it from the regression readout using the data analysis tool pack. So to get mine, since it's not in the problem, I'm gonna head up to the top menu, go to data, click on data analysis to the right, scroll to regression, click okay, and input in the information I need, starting with the y data. We'll click on this arrow and click and select our y data sales, making sure we don't get any empty cells. Click on this arrow button here. Then for x, do the same thing. Click and select our label advertising and our advertising data. Then click on this arrow button here and check off labels because we did get our labels when getting our data. Once that's all set, I'm not gonna check off anything else because I want my readout to be pretty simple. I'm gonna click on okay, and I'll be automatically taken to the readout. The standard error appears in the top table here. We see this row, standard error, and we can see the value is about 643.48. That's 643.48. So let's head back to the example, and our standard error is approximately six forty three point four eight. Next, we wanna find some of the pieces inside of our square root. It's pretty messy, so we don't wanna do it all in one go. The first thing we might wanna find is x bar, which is going to be the mean of our x values. So we'll type equals average, open parenthesis, and click and select our x data. Clicking enter, we end up getting x bar at four seventy two point seven approximately. Next, we wanna find the sum of our x values. We'll use the equal sum function. So equals sum, open parenthesis, click and select the x data, and we end up getting the sum of our x values at 7,090. Next, we need the sum of the square of our x values. We can use the equals sum s q function. Again, click and select our x data, and we end up getting that at a really large number. Now, technically, we could end up getting our margin of error with just these values, but we do have that messy fraction inside of the square root, and it is gonna be helpful to find the numerator and denominator separately so that we don't have a really huge calculation. So I'm gonna label this n u m for numerator, type equals, and the numerator is n. So let's click on that times, open parenthesis, x naught, which is the x value we're making the prediction interval for, in this case, 400, minus our x bar values. Let's click on that. End parenthesis, caret two to square it, and we can click enter. Now we get another really large value. And next, we wanna find our denominator, so we'll label that denom. And that's going to be equals n, so we'll click on that, times the sum of our x squareds, so we can click on that, minus open parenthesis, click on our sum x, end parenthesis, and then caret two to square it, and click enter. And that's another really big value. Now that we have our big pieces, let's find the margin of error e. Gonna start with an equal sign. Click on the t critical value, that 1.77 times, then click on the standard error times s q r t for square root, open parenthesis, one, plus, open parenthesis, one divided by, click on n, end parenthesis. Then we wanna do plus, open parenthesis, that numerator value divided by, then the denominator value, end parenthesis, and then do an end parenthesis for the square root. Once I check everything to make sure that everything looks good, we can click enter, and we get the margin of error at about 1,184. Now we wanna find the lower bound and the upper bound of our prediction interval. As a reminder, the lower bound is just going to be our point estimate minus our margin of error. Equals, we can click on the point estimate from part b minus our margin of error e, and we end up getting about 9,066.62. For the upper bound, we're going to do something very similar equals. We can click on that point estimate. Now plus margin of error. And we end up getting an upper bound of about 11,434.18. So let's create our prediction interval. Our prediction interval ends up being about 9,066.62 to about 11,434.18. So we predict that if $400 is spent on advertising, weekly sales will be somewhere between $9,066.62 and $11,434.18. Great work.

A small business tracks how advertising spending relates to weekly sales. ( C ) (C) Create a prediction interval for the value in part ( B ) (B) .

( 9066.62 , 11434.18 ) \left(9066.62,11434.18\right) ( 9066.62 , 11434.18 )

( 9016.22 , 11383.78 ) (9016.22,11383.78) ( 9016.22 , 11383.78 )

( 8755.90 , 11644.10 ) (8755.90,11644.10) ( 8755.90 , 11644.10 )

( 8806.30 , 11694.50 ) (8806.30,11694.50) ( 8806.30 , 11694.50 )

12. Regression

Prediction Intervals - Excel

Statistics for Business

12. Regression

Prediction Intervals - Excel

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Multiple Choice

A small business tracks how advertising spending relates to weekly sales.
$(C)$ Create a prediction interval for the value in part $(B)$ .

$(9016.22,11383.78)$

$(8755.90,11644.10)$

$$\left$(9066.62,11434.18$\right$)$

$(8806.30,11694.50)$

Verified step by step guidance

Step 1: Understand the problem context. We have data on weekly advertising spending (independent variable) and corresponding weekly sales (dependent variable). The goal is to create a prediction interval for a predicted sales value based on a given advertising spending amount from part (B).

Step 2: Fit a simple linear regression model to the data, where sales (Y) is predicted by advertising spending (X). The regression equation is $Y = \beta_0 + \beta_1 X + \epsilon$, where $\beta_0$ is the intercept, $\beta_1$ is the slope, and $\epsilon$ is the error term.

Step 3: Use the regression model to predict the sales value for the specific advertising spending amount given in part (B). This predicted value is $\hat{Y} = \hat{\beta}_0 + \hat{\beta}_1 X_{B}$, where $X_{B}$ is the advertising spending from part (B).

Step 4: Calculate the standard error of the prediction, which accounts for both the variability in the regression estimate and the variability of individual observations. The formula for the standard error of prediction is: \[ SE_{pred} = s \sqrt{1 + \frac{1}{n} + \frac{(X_B - \bar{X})^2}{\sum (X_i - \bar{X})^2}} \] where $s$ is the standard error of the regression, $n$ is the number of observations, $X_B$ is the advertising spending for prediction, and $\bar{X}$ is the mean of the advertising spending values.

Step 5: Construct the prediction interval using the t-distribution critical value $t_{\alpha/2, n-2}$: \[ \hat{Y} \pm t_{\alpha/2, n-2} \times SE_{pred} \] This interval gives a range where we expect the actual sales value to fall with a certain confidence level (usually 95%).

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