find a p-value and determine significance level using a t-test.

Suppose you want to determine if there is a significant difference in the average test scores between two groups of students: Group A and Group B.

Hypotheses:

• H0​: There is no difference in mean test scores between Group A and Group B.
• Ha​: There is a difference in mean test scores between Group A and Group B.

Test: Independent samples t-test

Data:

• Group A: Test scores of 20 students (mean = 75, standard deviation = 10)
• Group B: Test scores of 25 students (mean = 80, standard deviation = 12)

Calculate Test Statistic: The independent samples t-test formula is:

Where:

Substituting the values:

Calculate p-value: Using a t-table or statistical software, we find that the p-value corresponding to t=−1.08t=−1.08 with
df=n1+n2−2=20+25−2=43
df=n1​+n2​−2=20+25−2=43 degrees of freedom is approximately 0.287 (assuming a two-tailed test).

Significance Level: Let’s choose a significance level, α, of 0.05 (5%).

Comparison: Since p=0.287p=0.287 is greater than α=0.05α=0.05, we fail to reject the null hypothesis. This means that we do not have enough evidence to conclude that there is a significant difference in mean test scores between Group A and Group B at the 5% significance level.

In conclusion, based on the data and the chosen significance level, we do not find sufficient evidence to suggest that there is a significant difference in mean test scores between the two groups.