To determine which of the two statements regarding hypothesis testing are false, let's analyze each statement:

1. There is a trade-off between the probability of Type I and the probability of Type II errors.

   • True: This statement is correct. When the significance level (α) is decreased, the probability of a Type I error (rejecting a true null hypothesis) decreases, but the probability of a Type II error (failing to reject a false null hypothesis) increases. This creates a trade-off between the two errors.

2. The null hypothesis is typically assumed to be true until there is sufficient statistical evidence to reject it.

   • True: This statement is also correct. In hypothesis testing, the null hypothesis (H₀) is assumed true, and the objective is to determine if there is enough evidence to reject it in favor of the alternative hypothesis (H₁).

3. The smaller the significance level, the larger the probability of Type I error.

   • False: This statement is incorrect. The significance level (α) directly represents the probability of a Type I error. A smaller significance level means a lower probability of making a Type I error, not a larger one.

4. The higher the probability of Type I error, the higher the probability of Type II error.

   • False: This statement is also incorrect. There is an inverse relationship between Type I and Type II errors. Increasing the probability of a Type I error (by increasing α) decreases the probability of a Type II error (β), and vice versa.

The two false statements are:

• Statement 3: "The smaller the significance level, the larger the probability of Type I error."
• Statement 4: "The higher the probability of Type I error, the higher the probability of Type II error."

Would you like more details or have any other questions? Here are some related questions:

1. What are the implications of increasing the sample size on Type I and Type II errors?
2. How do you choose an appropriate significance level for a hypothesis test?
3. What are some strategies to reduce both Type I and Type II errors?
4. Can you explain the consequences of Type I and Type II errors in real-world scenarios?
5. How does statistical power relate to Type II error?

Tip: Lowering the significance level reduces the risk of a Type I error but increases the risk of a Type II error, highlighting the importance of balancing the two based on the context of the test.