Final answer:
A t value with a 95% two-sided confidence interval and 19 degrees of freedom is approximately 2.093. It can be obtained from a probability table or calculator. In a two-sided confidence interval for a t-distribution, there is 2.5% probability in each tail.
Step-by-step explanation:
The t value with a 95% confidence level depends on the specific distribution one is working with and the degrees of freedom associated with that distribution. For a t-distribution with 19 degrees of freedom, a 95% two-sided confidence interval has a t value of approximately 2.093. This can be obtained using a probability table or a calculator function such as invT(0.975, 19). When constructing a two-sided 95 percent confidence interval, there will be 2.5% probability in each tail of the distribution because the total area in the tails must equal 5% (100% - 95%).
The critical value, ta, is associated with both the tail areas and the degrees of freedom of the distribution. If a calculated t value is less than the critical value for a two-tailed test, as in the case of a t distribution with 29 degrees of freedom and a critical value of 2.045, it suggests that one fails to reject the null hypothesis, implying no significant correlation is found between the variables.
The p-value represents the probability of observing a test statistic at least as extreme as the one observed, given the null hypothesis is true. The p-value can help in determining whether to reject the null hypothesis. For a t-distribution with nine degrees of freedom, a p-value would be computed for the area to the right of a given t-score, such as a p-value of 0.0396 for t = 1.98.