In a two-tailed hypothesis test for a population correlation coefficient using a TI-84, you have $n=15$ paired observations and significance level $\alpha =0.05$. What is the correct way to obtain the critical value(s) of the correlation coefficient $r$ (the cutoff values for rejecting $H_0:\rho =0$) on a TI-84?

• A. Use $df=n-2$ in invT to find $t_{\alpha /2}$, then convert with $r=\frac{t_{\alpha /2}}{\sqrt{t_{\alpha /2}^2+df}}$, and use critical values $\pm r$.
• B. Use invNorm with $\frac{\alpha }{2}$ to find $z_{\alpha /2}$, then set the critical values to $\pm z_{\alpha /2}$ directly.
• C. Use the t critical value with $df=n-1$ and take the critical correlation as $r=\frac{1}{t_{\alpha /2}}$.
• D. Use the TI-84 LinReg menu to display $r$, and reject $H_0$ whenever the computed $r$ is greater than $\frac{\alpha }{2}$.

Answer: (A)

Verified step by step guidance

1. Identify the degrees of freedom (df) for the test. For testing a population correlation coefficient, df is calculated as \(df = n - 2\), where \(n\) is the number of paired observations. In this case, \(df = 15 - 2\).
2. Determine the critical t-value using the TI-84 calculator's inverse t-distribution function (invT). Since this is a two-tailed test with significance level \(\alpha = 0.05\), find the critical t-value for \(\frac{\alpha}{2} = 0.025\) and \(df = 13\).
3. Convert the critical t-value to the critical correlation coefficient \(r\) using the formula: \[r = \frac{t}{\sqrt{t^2 + df}}\] where \(t\) is the critical t-value obtained from the previous step.
4. The critical values for rejecting the null hypothesis \(H_0: \rho = 0\) are \(\pm r\). This means you reject \(H_0\) if the sample correlation coefficient is less than \(-r\) or greater than \(+r\).
5. Use the TI-84 to perform these calculations step-by-step: first use invT to find the critical t-value, then apply the formula above to find the critical \(r\) values, and finally compare your sample correlation coefficient to these critical values to decide whether to reject \(H_0\).