Question

The power of a test is influenced by the sample size and the choice of significance level. a. Explain how increasing the sample size affects the power (when significance level is held fixed). b. Explain how increasing the significance level affects the power (when sample size is held fixed).

Short answer

Increasing the sample size increases the power of a test because a larger sample size provides a wider base to detect a false null hypothesis. Increasing the significance level likewise increases the power of a test since it lowers the threshold for rejecting the null hypothesis. However, a larger significance level also increases the risk of making a Type I error.

Step-by-step solution

Understanding Power in Hypothesis Testing

The power of a statistical test is the probability that the test will reject the null hypothesis when the alternative hypothesis is true. It's a way of avoiding Type II error (the failure to reject a false null hypothesis). Power is important in hypothesis testing and has a direct relation with sample size and significance level.

Effect of Increasing Sample Size

Part a is about discussing the effect of increasing sample size on the power of the test, while holding the significance level constant. Increasing the sample size, while holding the significance level constant, increases the power of the test. This is because a larger sample size allows for a better realm for the detection of a false null hypothesis. If the null hypothesis is false, an increased sample size improves the chances of picking sample values that are farther from the hypothesized value, making the difference more obvious and therefore leading to rejection of the null hypothesis.

Effect of Increasing Significance Level

Part b is about the effect of increasing the significance level on the power of the test, while keeping the sample size constant. A larger significance level means that we are more willing to accept the risk of a Type I error (rejecting a true null hypothesis). This makes it easier to reject the null hypothesis and thus increases the power of the test. However, increasing the significance level also increases the probability of a Type I error, which illustrates that we always have to balance power and reliability when deciding on the significance level.