You have been asked to determine the effect of per capita...

You have been asked to determine the effect of per capita disposable income on retail sales using cross-section data by state. The data are contained in the data file Economic Activity. Estimate the appropriate regression equation and determine the $95 \%$ confidence interval for the expected change in retail sales that would result from a $$\$ 1,000$$ increase in per capita disposable income.

Key Concepts

Interpreting Regression Coefficients in Context

In the realm of regression analysis, the coefficient associated with the independent variable quantifies its effect on the dependent variable. When assessing the impact of a specific change (e.g., a $1,000 increase), the estimated coefficient is scaled accordingly to understand the expected change in the dependent variable within the constructed confidence interval.

Regression Equation

The regression equation represents the estimated relationship between the dependent and independent variables, typically expressed in the form Y = ?0 + ?1X + ?. Here, ?1 is the slope coefficient indicating the expected change in the dependent variable for a one-unit change in the independent variable, while ?0 is the intercept.

Confidence Interval

A confidence interval provides a range of values that is likely to contain an unknown population parameter. In regression analysis, a 95% confidence interval for a coefficient means that there is a 95% probability the interval captures the true effect size of the independent variable on the dependent variable.

Cross-Sectional Data

This refers to data collected at a single point in time across different observational units (such as states, individuals, or firms). In cross-sectional analysis, differences between units are examined to understand relationships or patterns, with the assumption that the observations are independent at the point of measurement.

Simple Linear Regression

This is a statistical method used to model the relationship between one dependent variable and one independent variable. It estimates how changes in the independent variable are associated with changes in the dependent variable by fitting a linear equation to the observed data.

Transcript

00:01 For this question, we visit back to the regression analysis for question 40.

00:06 For part a, we are asked to find a 95 % confidence interval for the slope of the regression line.

00:14 So 95 % confidence interval looks like this.

00:19 It's the slope plus or minus a critical value, which pertains to the degrees of freedom, and the confidence level times the standard error on the slope.

00:36 So we have the slope right here and we have the standard error here.

00:43 And from the question we have degrees of freedom which is n minus 2.

00:51 So we can write this out.

00:55 0 .0925 plus or minus.

01:02 Let's use excel to find the critical value.

01:06 So we use the function equals t.

01:09 I and v .2t and enter alpha of 0 .05 because we're looking for a 95 % percent.

01:16 Confidence interval and the degrees of freedom is 77 so we get 1 .99 and then the standard error on the slope and this gives us a 95 % confidence interval ranging from 0 .0776 to to 0 .1074 and this means that we are 95 % confident that the true slope of the linear relationship between sales and profits is between 0 .076 and 0 .1074.

02:22 And next for part b, we are told that an individual company brought in $9 billion worth of sales in one year, and we're asked to create a 95 % prediction interval for that company's profits.

02:36 And so this is an individual company, and so generally speaking, confidence interval on the estimated y parameter is of this form.

02:50 Is of this form, which is our predicted value plus or minus the critical value times the standard error on the predicted value.

03:28 So we can solve for all of these parts of this confidence interval equation.

03:35 So based on the regression analysis output, the predicted value, we can just call it the estimate for profits, is equal to the y intercept minus 176 .6 .6 plus plus the slope intercept, the slope coefficient, 0 .0925, and times the x value that we're given, that's x submue, which is, it's a $9 billion dollar sale.

04:28 It's $9 billion worth of sales, which is the same as $9 ,000 million, because the units of x are millions of dollars.

04:40 And this comes out to 655 .86.

04:51 And i probably should have written this out, but this is just to put it in a generic format.

04:57 We're looking at this type of equation here...

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