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: Define the hypotheses for the test. The null hypothesis (\(H_0\)) states that there is no positive correlation between \(x\) and \(y\), which means \(\beta \leq 0\). The alternative hypothesis (\(H_a\)) states that there is a positive correlation, so \(\beta > 0\).
2. Step 2: Calculate the sample correlation coefficient \(r\) between the paired \(x\) and \(y\) values using the formula: \[r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}\] where \(n\) is the number of pairs.
3. Step 3: Compute the test statistic \(t\) for the correlation using the formula: \[t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\] This follows a \(t\)-distribution with \(n-2\) degrees of freedom.
4. Step 4: Determine the critical value from the \(t\)-distribution for a one-tailed test at the significance level \(\alpha = 0.01\) with \(n-2\) degrees of freedom.
5. Step 5: Compare the calculated test statistic \(t\) to the critical value. If \(t\) is greater than the critical value, reject \(H_0\) and conclude there is evidence of a positive correlation. Otherwise, fail to reject \(H_0\), indicating insufficient evidence to support a positive correlation between \(x\) and \(y\).