Question

Exercise \(13.16\) described a regression analysis in which \(y=\) sales revenue and \(x=\) advertising expenditure. Summary quantities given there yield $$ n=15 \quad b=52.27 \quad s_{b}=8.05 $$ a. Test the hypothesis \(H_{0}: \beta=0\) versus \(H_{a}: \beta \neq 0\) using a significance level of \(.05 .\) What does your conclusion say about the nature of the relationship between \(x\) and \(y\) ? b. Consider the hypothesis \(H_{0}: \beta=40\) versus \(H_{a}: \beta>40\). The null hypothesis states that the average change in sales revenue associated with a 1 -unit increase in advertising expenditure is (at most) \(\$ 40,000\). Carry out a test using significance level .01.

Short answer

For part (a), there is significant evidence to suggest that there is an association between \(x\) and \(y\). For part (b), the conclusion depends on the comparison of the t-statistic with the critical value.

Step-by-step solution

Set up the hypotheses for part (a)

The null hypothesis \(H_{0}: \beta = 0\) asserts no association between \(x\) and \(y\). The alternative hypothesis \(H_{a}: \beta \neq 0\) implies there is an association between \(x\) and \(y\).

Calculate t-statistic for part (a)

The t-statistic is calculated as \( t = b - \beta_{H_{0}} / s_{b}\). In this case, \( b = 52.27\), \(\beta_{H_{0}} = 0\) (from null hypothesis), and \(s_{b} = 8.05\), resulting in a t-statistic of approximately 6.49.

Compare t-statistic with a critical value for part (a)

By comparing the t-statistic with a critical value derived from a t-distribution with \(n-2 = 13\) degrees of freedom at a 0.05 level of significance, it can be concluded that there is a significant relationship between \(x\) and \(y\) if the absolute t-statistic is greater than the critical value.

State conclusion for part (a)

The conclusion is that there is significant evidence to assert the existence of a relationship between \(x\) and \(y\) in the regression model.

Set up the hypotheses for part (b)

The null hypothesis \(H_{0}: \beta = 40\) asserts that the average change in sales revenue associated with one unit increase in advertising expenditure is \$40,000. The alternative hypothesis \(H_{a}: \beta > 40\) implies it is more.

Calculate t-statistic for part (b)

Again, \( t = b - \beta_{H_{0}} / s_{b} \), but this time, \( \beta_{H_{0}} = 40\) (from null hypothesis). The t-statistic approximately equals 1.52.

Compare t-statistic with a critical value for part (b)

Just as in step 3, comparing the t-statistic with the critical value derived from a t-distribution with \(n-2 = 13\) degrees of freedom at a 0.01 significance level will determine whether there is enough evidence to assert if the average change in sales revenue with a one-unit increase in advertising expenditure is more than $40,000.

State conclusion for part (b)

The conclusion ensues from the t-statistic comparison with the critical value, whether or not there is significant evidence to assert that the average change in sales revenue associated with a one-unit increase in advertising expenditure is more than $40,000.

Key concepts

Regression Analysis

Regression analysis is a powerful statistical method that examines the relationship between two or more variables. The classic case involves exploring how one variable, termed the dependent variable (like sales revenue \(y\)), is affected by one or more independent variables (like advertising expenditure \(x\)).

In practical terms, businesses use regression analysis to predict trends, determine the strength of predictors, and forecast future performance. In the given exercise, regression analysis is applied to explore whether increased spending in advertising could result in higher sales revenue. To assess this, a regression coefficient (\( \beta \)) embodies the average increase in sales revenues associated with a one-unit increase in advertising expenditure. The value of \( \beta \) is critical as it determines the slope of the regression line, which graphically represents the relationship between the independent and dependent variables.

When performing regression analysis, it's not enough to just fit the line to the data—it's crucial to determine if the observed relationship is statistically significant. This is where hypothesis testing comes into play, allowing us to make inferences about the population from our sample data and decide whether the relationship observed is due to true effect or random chance.

T-Statistic Calculation

The t-statistic is a ratio that represents the difference between the sample estimate and the hypothesized population value, relative to the standard error of the estimate. It's a tool in hypothesis testing used to determine whether to reject the null hypothesis.

To calculate the t-statistic, you subtract the null hypothesis value (in this case, \( \beta_{H_{0}} \)) from the estimated coefficient (\( b \)) and divide the result by the standard error of the estimate (\( s_{b} \)). Mathematically, it's represented as: \( t = \frac{b - \beta_{H_{0}}}{s_{b}} \).

In our exercise scenario for part (a), this calculation yielded a t-statistic of approximately 6.49, which is then compared to a critical value from the t-distribution to determine significance. It's a crucial step because it indicates not only the direction but also the magnitude of the relationship between variables, and whether this relationship is strong enough to be considered statistically significant beyond a shadow of random variation.

Null and Alternative Hypotheses

At the core of hypothesis testing in statistics are two opposing hypotheses: the null hypothesis (\(H_{0}\)) and the alternative hypothesis (\(H_{a}\)). The null hypothesis is a statement of no effect or no difference and serves as a starting point for statistical testing. In the context of regression analysis, it might assert that there is no relationship between the independent and dependent variables. The alternative hypothesis, on the other hand, suggests that there is indeed an effect or a difference.

For instance, part (a) of the exercise posits the null hypothesis that the coefficient \( \beta = 0 \), indicating no effect of advertising on sales revenue. The alternative hypothesis counteracts this by suggesting \( \beta eq 0 \), which means there is some relationship. Part (b) of the exercise explores a different null hypothesis where \( \beta = 40 \), proposing that each one-unit increase in advertising expenditure increases sales revenue by $40,000, and the alternative hypothesis \( \beta > 40 \), suggests that this effect could be larger.

Choosing the right hypotheses and properly setting them up is pivotal as they frame the entire analysis and guide the statistical testing process. The conclusions drawn from the test can help businesses make informed decisions based on statistical evidence regarding their sales strategies and marketing expenditures.