In a test of hypotheses, the null hypothesis is that the mean is 100 and the alternate hypothesis is that the mean is greater than 100. The power of the test when the mean is 101 would be greatest for which of the following choices of sample size n and significance level α?
a) n=10, α=0.05
b) n=10, α=0.01
C) n=20, α=0.05
d) n=20, α=0.01
e) It cannot be determined from the information given
Could you explain why the answer is a?
To determine the sample size and significance level that would result in the greatest power of the test when the mean is 101, we need to understand the power of a statistical test.
The power of a hypothesis test is the probability of correctly rejecting the null hypothesis when it is false (i.e., when the alternate hypothesis is true). It is influenced by various factors, including the sample size (n) and the significance level (α).
A higher sample size generally leads to greater power because it reduces the variability in the data and allows for more precise estimation of the true mean. Similarly, a higher significance level increases the chance of rejecting the null hypothesis, resulting in greater power.
In this scenario, we want to maximize the power of the test when the mean is 101. Since the alternate hypothesis is that the mean is greater than 100, we are interested in the right-tailed test.
Comparing the options provided:
a) n=10, α=0.05: This combination has a small sample size and a relatively high significance level.
b) n=10, α=0.01: This combination has the same sample size as option (a), but a lower significance level.
c) n=20, α=0.05: This option has a larger sample size but the same significance level as option (a).
d) n=20, α=0.01: This combination has both a larger sample size and a lower significance level.
Based on the principles mentioned earlier, a larger sample size and a lower significance level are likely to result in greater power. Therefore, option (d) with n=20 and α=0.01 would have the greatest power of the test when the mean is 101.
Therefore, the correct answer is option (d) n=20, α=0.01.