If the pointsare widely scattered around the line, then it is more plausible thata line with zero or negative slope could fit the data almost as wellas our estimated line with slope of 0.23. If all of the data points in the samplelie very close to a line with slope 0.23, then we may feel quiteconfident that 0.23 is a good estimate of the slope. An estimate of 0.05 would give us less confidence.The second factor is how clearly the data in the sample are tellingus that the slope is 0.23. Other things beingequal, an estimate of 0.5 would make us more confident that B 2 is positive than the estimate we of 0.23 that weactually obtained.
Let's focus on this final question: How sure canwe be that the actual slope ( B 2) is not zero ornegative? How convincing is the evidence for a positive slope fromour sample? Two factors enter into our assessment of this question.The first is how "positive" our slope estimate is. How confident can we be, based on oureconometric regression, that the true value of the slope is notlarger than 0.30? Or smaller than 0.15? Or zero or negative? But we know that this is just an estimateand that the true value of the slope ( B 2) isprobably not exactly 0.23. That means that 0.23 is ourbest single guess at the amount of an additional dollar of incomethat will be spent on food. Suppose that we have run a linear regression offood expenditures on income and estimated the slope of the regressionline ( b 2) to be 0.23. The human resources department at a large company wants to develop a model to predict an employee’s job satisfaction from the number of hours of unpaid work per week the employee does, the employee’s age, and the employee’s income.Testing Hypotheses about Regression Coefficients REDSPOTS Testing Hypotheses about RegressionCoefficients In other words, the regression coefficient \beta_1 is not zero, and so there is a relationship between the dependent variable “job satisfaction” and the independent variable “hours of unpaid work per week.” This means that the independent variable “hours of unpaid work per week” is useful in predicting the dependent variable. This suggests that the assumption that the null hypothesis is true is most likely incorrect, and so the conclusion of the test is to reject the null hypothesis in favour of the alternative hypothesis.
So the p-value=0.0082.īecause p-value=0.0082 \lt 0.05=\alpha, we reject the null hypothesis in favour of the alternative hypothesis. The p-value for the test on the hours of unpaid work per week regression coefficient is in the bottom part of the table under the P-value column of the Hours of Unpaid Work per Week row. The regression summary table generated by Excel is shown below: SUMMARY OUTPUT Previously, we learned that the population model for the multiple regression equation is Conduct and interpret a hypothesis test on individual regression coefficients.