More sales and profits Consider again the relationship...

More sales and profits Consider again the relationship between the sales and profits of Fortune 500 companies that you analyzed in Exercise 48 . a) Find a $95 \%$ confidence interval for the slope of the regression line. Interpret your interval in context. b) Last year, the drug manufacturer Eli Lilly, Inc., reported gross sales of $\$ 23$ billion (that's $\$ 23,000$ million). Create a $95 \%$ prediction interval for the company's profits, and interpret your interval in context.

More sales and profits Consider again the relationship between the sales and profits of Fortune 500 companies that you analyzed in Exercise 48 .
a) Find a $95 \%$ confidence interval for the slope of the regression line. Interpret your interval in context.
b) Last year, the drug manufacturer Eli Lilly, Inc., reported gross sales of $\$ 23$ billion (that's $\$ 23,000$ million). Create a $95 \%$ prediction interval for the company's profits, and interpret your interval in context.

Key Concepts

Simple Linear Regression

Simple linear regression is a statistical method used to model the relationship between two quantitative variables by fitting a linear equation to observed data. It estimates the parameters, typically the slope and intercept, that define the best-fitting line representing the relationship between an independent variable and a dependent variable.

Confidence Interval for Slope

A confidence interval for the slope in simple linear regression provides a range of values within which the true slope of the relationship between the independent and dependent variables is likely to fall, given a specified level of confidence (such as 95%). This interval helps assess the significance of the relationship by indicating whether non-zero slopes (i.e., meaningful associations) are plausible based on the data.

Prediction Interval

A prediction interval in regression analysis predicts the range within which a future individual observation is expected to fall for a given predictor value, again at a specified confidence level (like 95%). It accounts for both the uncertainty in estimating the mean response and the natural variability of individual outcomes, resulting in a wider interval compared to a confidence interval for the mean response.

Transcript

00:01 Consider again the relationship between the sales and profits of fortune 500 companies that you're analyzed in the exercise 48.

00:07 We want to first find a 95 % confidence for the slope of the regression line and interpret the interval in context.

00:15 So if we're going to have a confidence of 95 % then if we look at the sample size and the given row count of the output, we see the sample size is 79, the y intercept is given in the row intercept and the column coefficient.

00:33 And we see that's negative 176 .64.

00:38 And the slope is given in the row sales and in the column coefficient of the output, which is 0 .092498.

00:49 So the equation is negative 176 .44 plus 0 .92498 times x.

01:07 And the standard deviation of the slope is given in the row of sales, and in the column se and that is, well, standard error i should say.

01:22 .0075.

01:24 Our standard error of the estimate is given as s sub e, which is 466 .2, and our mean is 4178 .29.

01:38 So for part a, officially in it now.

01:43 The critical t value can be found in your t distribution table in the appendix.

01:47 Go to degrees of freedom of n minus 2, which would be 79 minus 2 or 77, we'll go to 75 instead.

02:00 That's the nearest smaller one.

02:02 And in the column of 0 .05, you'll get a t value of 1 .992, which we're going to use from the b value, 0 .092498, plus and minus 1 .992 times the standard error to build our confidence interval.

02:28 So we'll start with subtracting to get 0 .07558 and on the upper end, 0 .107 -438.

03:00 In this context, then, we would say that we're 95 % confident...

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