When conducting multiple t-tests, each with an individual significance level ($\alpha = 0.05$), the cumulative probability of making at least one Type I error across all tests increases. This is known as the "family-wise error rate."

For three independent tests, the family-wise error rate ($\alpha_{\text{FW}}$) can be calculated using the formula: $\alpha_{\text{FW}} = 1 - (1 - \alpha)^k$ where $k$ is the number of tests (in this case, $k = 3$) and $\alpha$ is the significance level for each test (here, $\alpha = 0.05$).

Calculating this: $\alpha_{\text{FW}} = 1 - (1 - 0.05)^3$ $\alpha_{\text{FW}} = 1 - (0.95)^3$ $\alpha_{\text{FW}} \approx 1 - 0.8574$ $\alpha_{\text{FW}} \approx 0.1426$

Therefore, the probability of making at least one Type I error across the three tests is approximately 0.1426 or 14.26%.

Would you like more details on family-wise error rate control methods or the implications of multiple testing? Here are some further questions to consider:

1. How does the family-wise error rate change with an increasing number of tests?
2. What methods can be used to control the family-wise error rate in multiple comparisons?
3. Why is controlling the family-wise error rate important in experimental research?
4. How does the Bonferroni correction work to adjust the significance level?
5. How does the probability of a Type I error impact the conclusions drawn from hypothesis testing?

Tip: When conducting multiple tests, consider using methods like the Bonferroni correction to reduce the likelihood of a Type I error.