In a TI-84 correlation test (LinRegTTest), what does it mean for the correlation between $x$ and $y$ to be statistically significant at $\alpha$ (e.g., $\alpha =0.05$)?

• A. The test produces a p-value greater than $\alpha$, so you fail to reject $H_0:\rho =0$ and conclude there is evidence of a nonzero linear correlation.
• B. The test produces a p-value less than $\alpha$, so you reject $H_0:\rho =0$ and conclude there is evidence of a nonzero linear correlation.
• C. Statistical significance means the correlation is strong, specifically $\left|r\right|\ge 0.8$, in any dataset.
• D. The correlation is statistically significant whenever the sample correlation satisfies $r>0$, regardless of the sample size.

Answer: (B)

Verified step by step guidance

1. Understand the hypotheses involved in the correlation test: The null hypothesis \(H_0\) states that the population correlation coefficient \(\rho = 0\), meaning no linear relationship between \(x\) and \(y\). The alternative hypothesis \(H_a\) states that \(\rho \neq 0\), indicating some linear correlation exists.
2. Recognize that the test calculates a p-value, which measures the probability of observing the sample data (or something more extreme) assuming the null hypothesis is true.
3. Compare the p-value to the significance level \(\alpha\) (e.g., 0.05). If the p-value is less than \(\alpha\), it means the observed correlation is unlikely to have occurred by random chance under \(H_0\).
4. When the p-value \(< \alpha\), reject the null hypothesis \(H_0: \rho = 0\) and conclude there is statistically significant evidence of a nonzero linear correlation between \(x\) and \(y\).
5. Note that statistical significance does not necessarily imply a strong correlation (e.g., \(|r| \geq 0.8\)); it means the correlation is unlikely to be zero given the data and sample size.