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

• A. Reject $H_0$ and conclude that there is a positive correlation between $x$ and $y$ and that $\beta >0$.
• B. 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 $y$.
• C. Fail to 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 $y$.
• D. 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 $y$.

Answer: (C)

Verified step by step guidance

1. 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 \( H_a: \beta > 0 \).
2. 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: \[\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i, \quad \bar{y} = \frac{1}{n} \sum_{i=1}^n y_i\] \[S_{xx} = \sum (x_i - \bar{x})^2, \quad S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y})\]
3. 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 \( \beta \).
4. 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 \( \hat{y}_i = b_0 + b_1 x_i \).
5. 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 \( H_0 \).