The Advertising and Sales Regression
A marketing analyst collects data on monthly advertising expenditure (in thousands of dollars) and monthly sales (in thousands of dollars) for 30 randomly selected months from a retail chain. A least-squares regression line is fitted, yielding the equation: Predicted Sales = 50 + 2.5 * Advertising. The correlation coefficient is r = 0.85. The residual plot shows a random scatter with no obvious pattern, and a histogram of residuals is roughly symmetric. The analyst wants to test if there is a significant linear relationship between advertising and sales at the 5% level. However, a colleague warns that extrapolation beyond the observed advertising range could be misleading.
1. State the null and alternative hypotheses for testing the slope of the regression line, and explain what the slope means in context.
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Hypotheses: H0: β = 0 (no linear relationship between advertising and sales) versus Ha: β ≠ 0 (there is a linear relationship). The slope of 2.5 means that for each additional $1,000 spent on advertising, predicted sales increase by $2,500, on average. The slope measures the change in the predicted response for a one-unit increase in the predictor.
2. The standard error of the slope is 0.6. Compute the t-statistic for the slope and make a conclusion at the 5% significance level. Assume the conditions for inference are met.
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The t-statistic is t = (2.5 - 0) / 0.6 ≈ 4.17. With df = n - 2 = 28, the critical t-value for a two-tailed test at α = 0.05 is about 2.048. Since 4.17 > 2.048, we reject H0. There is convincing evidence of a significant positive linear relationship between advertising expenditure and sales. Alternatively, the p-value is less than 0.05.