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. Lower-tailed test, \(n=20, t=-5.1\) e. Two-tailed test, \(n=40, t=1.7\)

Short answer

a. Do not reject the null hypothesis. b. Do not reject the null hypothesis. c. Reject the null hypothesis. d. Strongly reject the null hypothesis. e. Do not reject the null hypothesis.

Step-by-step solution

Analyzing the first case

For a two-tailed test with degree of freedom 9, and a t-value 0.73, the P-value is obtained by looking at a t-value table. In this case, P-value is more than 0.05 (common threshold), which means we do not reject the null hypothesis.

Analyzing the second case

For an upper-tailed test with a degree of freedom 10, and a t-value -0.5, the P-value is again obtained by looking at a t-value table. Here, the P-value is again more than 0.05, so we still do not reject the null hypothesis.

Analyzing the third case

For a lower-tailed test with sample size 20 (thus, degree of freedom 19), and a t-value -2.1, by consulting a t-distribution table, we find that the P-value less than 0.05, hence we reject the null hypothesis.

Analyzing the fourth case

For a lower-tailed test with a sample size 20 and a t-value -5.1, the P-value is significantly less than 0.05, rendered almost zero. So, in this case, we strongly reject the null hypothesis.

Analyzing the last case

For a two-tailed test with a sample size 40 and a t-value 1.7, the P-value is more than 0.05. Hence, we do not reject the null hypothesis in this case as well.