For the data in Exercise 15 of Section 10.2 "The Linear...

For the data in Exercise 15 of Section 10.2 "The Linear Correlation Coefficient" a. The business owner intends to spend $\$ 2,500$ on advertising next year. Give an estimate of next year's revenue based on this fact. b. Construct the $90 \%$ prediction interval for next year's revenue, based on the intent to spend $\$ 2,500$ on advertising.

For the data in Exercise 15 of Section 10.2 "The Linear Correlation Coefficient"
a. The business owner intends to spend $\$ 2,500$ on advertising next year. Give an estimate of next year's revenue based on this fact.
b. Construct the $90 \%$ prediction interval for next year's revenue, based on the intent to spend $\$ 2,500$ on advertising.

Key Concepts

Prediction Interval

A prediction interval provides a range within which a single future observation is expected to fall with a specified level of confidence, such as 90%. Unlike a confidence interval for the mean, a prediction interval accounts for both the uncertainty in the estimate of the regression line and the natural variability of individual data points around that line, making it wider and more applicable for predicting individual outcomes.

Simple Linear Regression

Simple linear regression is a statistical method used to model the relationship between two variables by fitting a straight line to observed data. It involves estimating the slope and intercept that best describe how changes in the independent variable (in this context, advertising spending) are associated with changes in the dependent variable (revenue). This model is the basis for making predictions about future outcomes based on the line of best fit.

Least Squares Estimation

Least squares estimation is the technique used in simple linear regression to find the regression line that minimizes the sum of the squared differences between observed values and the values predicted by the model. This method ensures that the line fits the data as closely as possible by accounting for the variation in the observations.

Transcript

00:01 Here's a solution to number 15.

00:02 So we're given a data set where the x values are the amount spent on ads, and this is in thousands of dollars.

00:11 So 1 .4 means $1 ,400.

00:13 And then this is the projected revenue that year.

00:16 And so to get the confidence interval, you'll need these things.

00:19 So i went ahead and did them.

00:19 So the x bar is the average of all the xs.

00:22 This is the average amount spent on advertising, which was 1975.

00:28 And then i went ahead and took each of these x values and i subtracted the x bar and then i squared the result and then i added those together to get the ssx and then i went ahead and took the given x subtracted the mean of the x and i squared that so the given x was 2 .5 meaning how much revenue are they going to make with $2 ,500 in spending on ads.

00:53 Okay, so all this stuff is for the confidence interval but before we get there we're asked to find the point estimate i got to change this up real quick so the y input goes from b2 to b9 and the x input goes from a 2 to a 9 okay and that gives me the regression line so you probably did this in a previous problem but part a is for the point estimate to find the point estimate for whenever x equals 2 .5 so to do that you need to get the regression line the regression line is 119 .5 plus 42 .02x.

01:30 And again, we're finding the point estimate for whenever x equals 2 .5.

01:34 So we take that 2 .5 and we plug it in for the x.

01:40 Okay.

01:41 And whenever you do that, you should get the point estimate of 224 .562, which in thousands of dollars actually means that the projected revenue is $224 ,562.

01:58 So that's the point estimate.

02:02 Okay, so now we're going to find the prediction interval.

02:05 Sorry, i think i misspoke.

02:05 I said confidence in it.

02:07 It's actually the 90 % prediction interval, which is a little bit different.

02:11 But the stuff that we need is essentially the same...

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