Suppose that a sample of 12 pairs of data produced a test statistic of $r=0.787$. A significance level of $\alpha=$ 0.01 is to be used.
(a) Find the critical values that would be used to test for significant linear correlation. Do not type " \pm " in front of your answer (the \pm is already given in front of the answer box below). Round your answer to 3 places after the decimal point, if necessary.
\[
r= \pm
\]
(b) Is there significant linear correlation in this case?
There is not significant linear correlation.
There is significant linear correlation.

Step 1 ：We are given a sample of 12 pairs of data that produced a test statistic of \(r=0.787\). We are to use a significance level of \(\alpha=0.01\).

Step 2 ：We need to find the critical values that would be used to test for significant linear correlation. The critical value for the correlation coefficient can be calculated using the formula for the critical value of Pearson's correlation coefficient, which is given by: \[r_{critical} = \pm \sqrt{\frac{(n-2)}{n}} \times t_{critical}\] where: \(n\) is the number of pairs of data and \(t_{critical}\) is the critical value of the t-distribution at the given significance level and degrees of freedom \((n-2)\).

Step 3 ：We can find the critical value of the t-distribution using a t-table or a statistical function. For this problem, we have \(n = 12\), \(\alpha = 0.01\), \(r = 0.787\), and degrees of freedom \(df = 10\).

Step 4 ：Using these values, we find that \(t_{critical} = 3.169272667175838\) and \(r_{critical} = 2.8931368844946155\).

Step 5 ：The critical value for the correlation coefficient is approximately \(\pm 2.893\). This is the value that the test statistic \(r=0.787\) will be compared to in order to determine if there is a significant linear correlation.

Step 6 ：We compare the absolute value of the test statistic \(r=0.787\) with the absolute value of the critical value \(2.893\). If the absolute value of the test statistic is greater than the absolute value of the critical value, then there is a significant linear correlation.

Step 7 ：The absolute value of the test statistic \(0.787\) is not greater than the absolute value of the critical value \(2.893\). Therefore, there is not a significant linear correlation.

Step 8 ：Final Answer: \[\boxed{(a) \pm 2.893}\] \[\boxed{(b) \text{There is not significant linear correlation.}}\]