Question

Give as much information as you can about the \(P\) -value of a \(t\) test in each of the following situations: a. Two-tailed test, \(\mathrm{df}=9, t=0.73\) b. Upper-tailed test, \(\mathrm{df}=10, t=0.5\) c. Lower-tailed test, \(n=20, t=-2.1\) d. Two-tailed test, \(n=40, t=1.7\)

Short answer

The P-value is a measure of how extreme the data is and could be calculated based on t-value and type of the t-test. Its value helps to determine the statistical significance of the outcomes. Exact values will vary based on the t-distribution table used.

Step-by-step solution

Determine the Type of the Test

Understand whether it's a two-tailed, upper-tailed, or lower-tailed test. An upper-tailed test checks for the probability that the parameter is greater than the sample estimate, a lower-tailed checks for the opposite, and a two-tailed checks for both.

Calculate P-value for each scenario

Use the t-table (also known as Student's T-distribution table) to find out the P-value in each situation. a. For a two-tailed test, df=9, t=0.73, the P-value is equal to the proportion of the area that is greater than 0.73 or less than -0.73 in a t-distribution with 9 degrees of freedom. b. For an upper-tailed test, df=10, t=0.5, the P-value is the proportion of the area in a t-distribution with 10 degrees of freedom that is greater than 0.5. c. For a lower-tailed test, n=20 (thus, df=19), t=-2.1, the P-value is the proportion of the area in a t-distribution with 19 degrees of freedom that is less than -2.1. d. For a two-tailed test, n=40 (thus, df=39), t=1.7, the P-value is equal to the proportion of the area that is greater than 1.7 or less than -1.7 in a t-distribution with 39 degrees of freedom.

Interpret the P-values

After calculating the P-values, interpret them in terms of statistical significance. A P-value lower than 0.05 (or any other chosen significance level) would suggest that the observed data is significantly different from what was expected under the null hypothesis.