In the context of the widget cost regression model, what does the $p$-value associated with the $F$ statistic indicate?

• A. It indicates the proportion of variance in widget cost explained by the model.
• B. It measures the probability that the residuals are normally distributed.
• C. It tests whether at least one predictor variable in the model has a statistically significant relationship with the response variable.
• D. It gives the likelihood that the slope of the regression line is exactly $0$.

Answer: (C)

Verified step by step guidance

1. Understand that the F statistic in a regression model is used to test the overall significance of the model, specifically whether the model explains a significant amount of variation in the response variable compared to a model with no predictors.
2. Recognize that the p-value associated with the F statistic represents the probability of observing an F value as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true.
3. Recall that the null hypothesis for the F test in regression states that all regression coefficients (except the intercept) are equal to zero, meaning none of the predictor variables have a linear relationship with the response variable.
4. Interpret the p-value: a small p-value (typically less than 0.05) indicates strong evidence against the null hypothesis, suggesting that at least one predictor variable is significantly related to the response variable.
5. Note that this p-value does not measure the proportion of variance explained, the normality of residuals, or the likelihood that a single slope coefficient is zero; it specifically tests the overall significance of the regression model.