12. Regression

Inferences for Slope

12. Regression

Inferences for Slope

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Multiple Choice

Using the sample data below, run a hypothesis test on $β$ to see if there is evidence that there is a positive correlation between $x$ and $y$ with $alpha equals 0.01$ .

Reject $H sub 0$ and conclude that there is a positive correlation between $x$ and $y$ and that $beta is greater than 0$ .

Fail to reject $H_0$ since there is enough evidence to suggest $\beta>0$ , but not enough evidence to suggest positive linear correlation between $x$ and $y$ .

Fail to reject $H sub 0$ since there is not enough evidence to suggest $beta is greater than 0$ and not enough evidence to suggest positive linear correlation between $x$ and $y$ .

Reject $H_0$ since there is not enough evidence to suggest $\beta>0$ and not enough evidence to suggest positive linear correlation between $x$ and $y$ .

Verified step by step guidance

Step 1: State the hypotheses for the test. The null hypothesis is that the slope $ \beta = 0 $ (no linear relationship), and the alternative hypothesis is that $ \beta > 0 $ (positive linear relationship). Formally, $ H_0: \beta = 0 $ and $ H_a: \beta > 0 $.

Step 2: Calculate the sample statistics needed for the test: the means of $ x $ and $ y $, the sample variances, and the sample covariance. Use the formulas: $$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i, \quad \bar{y} = \frac{1}{n} \sum_{i=1}^n y_i$$ $$S_{xx} = \sum (x_i - \bar{x})^2, \quad S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y})$$

Step 3: Calculate the estimated slope $ b_1 $ of the regression line using the formula: $$b_1 = \frac{S_{xy}}{S_{xx}}$$ This $ b_1 $ is the point estimate for $ \beta $.

Step 4: Compute the standard error of the slope estimate $ SE_{b_1} $. First, calculate the residual sum of squares and then use: $$SE_{b_1} = \sqrt{\frac{SSE}{(n-2) S_{xx}}}$$ where $ SSE = \sum (y_i - \hat{y}_i)^2 $ and $ \hat{y}_i = b_0 + b_1 x_i $.

Step 5: Calculate the test statistic $ t $ for the slope: $$t = \frac{b_1 - 0}{SE_{b_1}}$$ Compare this $ t $-value to the critical value from the $ t $-distribution with $ n-2 $ degrees of freedom at significance level $ \alpha = 0.01 $. If $ t $ is greater than the critical value, reject $ H_0 $; otherwise, fail to reject $ H_0 $.