suppose that you conduct three different t tests to analyze the results of an experiment with three independent samples each individual t test
The correct answer and explanation is:
Correct Answer:
Conducting three individual t-tests on three independent samples increases the risk of Type I error, and is not appropriate if you want to compare all three groups together. Instead, one-way ANOVA should be used.
Explanation (≈300 words):
When analyzing results from an experiment involving three independent samples, it might seem logical to conduct three separate t-tests — for example, comparing:
- Group A vs. Group B
- Group A vs. Group C
- Group B vs. Group C
While each t-test is valid for comparing two means, conducting multiple t-tests on the same dataset increases the probability of making a Type I error — falsely rejecting a true null hypothesis.
Each individual t-test typically has a 5% risk (α = 0.05) of Type I error. When you conduct three tests, the family-wise error rate (FWER) increases. The FWER is calculated using:
$$
\text{FWER} = 1 – (1 – \alpha)^k
$$
Where:
- $\alpha = 0.05$
- $k = 3$ (number of comparisons)
So,
$$
\text{FWER} = 1 – (1 – 0.05)^3 = 1 – 0.95^3 ≈ 0.1426
$$
This means there’s a 14.26% chance of incorrectly finding a statistically significant result just by chance — much higher than the intended 5%.
To appropriately analyze differences among three or more independent groups, you should use a one-way ANOVA (Analysis of Variance). ANOVA tests whether there is a statistically significant difference among the means of all three groups simultaneously, while maintaining the overall Type I error at 5%.
If ANOVA reveals a significant result, post hoc tests (like Tukey’s HSD) can then determine which specific groups differ, while still controlling for multiple comparisons.
In conclusion, using multiple t-tests for more than two groups is statistically flawed. ANOVA is the correct approach for comparing three independent samples in a single experiment.