Final answer:
A correlation of 0.29 does not necessarily lead to a very small Student's t statistic in a regression analysis with both predictor variables. The t statistic also depends on sample size and standard error. A high correlation among independent variables can, however, increase standard errors and negatively affect the t statistic due to multicollinearity.
Step-by-step explanation:
When considering a regression analysis with n=49 and two potential independent variables, a correlation of 0.29 with the dependent variable does not necessarily imply that there will be a very small Student's t statistic in the regression analysis when including both predictor variables. Let's address each option provided in the question:
- A: A high correlation among the independent variables could result in a very small Student's t statistic as the correlation increases the coefficient standard errors. This is because multicollinearity can inflate the standard errors of the coefficients, which affects the t statistics.
- B: Correlation between the independent variable and the dependent variable is not necessarily evidence of a small Student's t statistic. The magnitude of the t statistic also depends on the sample size and the standard error of the coefficient.
- C: Correlation between the independent variable and the dependent variable is evidence of a small Student's t statistic. This is incorrect as a small correlation does not automatically lead to a small t statistic; the t statistic depends on both the correlation coefficient and the standard error of the estimated coefficient.
- D: The correlation between the independent variable and the dependent variable could result in a very small Student's t statistic as the correlation creates a high variance. This is not accurate in general; the t statistic is also influenced by sample size and how the variance is partitioned in the regression model.
Ultimately, to determine the significance of the correlation coefficient, you would need to perform a hypothesis test, often with a null hypothesis stating that there is no linear relationship (correlation is zero), and an alternative hypothesis stating that there is a significant linear relationship (correlation is not zero). The actual value of the t statistic would depend on the correlation coefficient, the sample size, and the variability of the data.