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 $H_0$ and conclude that there is a positive correlation between $x$ and $y$ and that $\beta>0$ .

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 $H_0$ since there is not enough evidence to suggest $\beta>0$ 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: 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 $\beta > 0$.

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.

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.

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.

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 $y$.

4:30m

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