Which of the following statements regarding hypothesis test is FALSE?a. There is a trade-off between the probability of Type I and the probability of Type II errors.b.The smaller the significance level, the smaller the probability of a Type I error.c.If the null hypothesis is true, the distribution of the test statistic is standard normal.d.The higher the probability of a type I error, the higher the likelihood that we will retain a false null hypothesis.
Question
Which of the following statements regarding hypothesis test is FALSE?a. There is a trade-off between the probability of Type I and the probability of Type II errors.b.The smaller the significance level, the smaller the probability of a Type I error.c.If the null hypothesis is true, the distribution of the test statistic is standard normal.d.The higher the probability of a type I error, the higher the likelihood that we will retain a false null hypothesis.
Solution
Let's evaluate each statement:
a. There is a trade-off between the probability of Type I and the probability of Type II errors.
This statement is true. The probability of making a Type I error (rejecting a true null hypothesis) and the probability of making a Type II error (failing to reject a false null hypothesis) are inversely related. As one increases, the other decreases.
b. The smaller the significance level, the smaller the probability of a Type I error.
This statement is true. The significance level of a test is the probability of rejecting the null hypothesis when it is true, i.e., the probability of a Type I error. Therefore, a smaller significance level means a smaller probability of a Type I error.
c. If the null hypothesis is true, the distribution of the test statistic is standard normal.
This statement is false. The distribution of the test statistic depends on the sample size and the type of test being conducted. For large sample sizes, the Central Limit Theorem states that the distribution of the sample mean is approximately normal, but it does not have to be a standard normal distribution. For small sample sizes or non-parametric tests, the distribution may not be normal at all.
d. The higher the probability of a type I error, the higher the likelihood that we will retain a false null hypothesis.
This statement is false. A Type I error occurs when we reject a true null hypothesis, not when we retain a false null hypothesis. The latter is a Type II error.
So, the false statements are c and d.
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