Question

A doctor claims that less than 30 percent of all individuals exposed to a certain amount of radiation will feel any ill effects. To test her claim, she takes a random sample of 64 individuals exposed to such radiation and carries out a hypothesis test. For the significance level α=0.05, what is the power of the test when the proportion of all individuals exposed to the radiation who feel any ill effects is 21 percent?

Answer 1

The power of the test is the probability of correctly rejecting the null hypothesis when it is false. It requires calculating the Z-value for a sample proportion of 21% and finding the corresponding area under the standard normal curve, typically using statistical software.

Step-by-step explanation:

The power of a hypothesis test is the probability that the test correctly rejects a false null hypothesis. To calculate the power for this test, one would need to perform calculations using the hypothesized proportion under the null hypothesis (30 percent) and the true proportion (21 percent). Essentially, the power is the probability that we will observe a sample statistic that leads to the rejection of the null hypothesis when the alternative hypothesis is true. A common way to calculate the power is to use software that performs these calculations using the sample size, significance level, and the true proportion.

In this scenario, given the significance level α = 0.05 and the sample size of 64, we would typically use a statistical software to determine the Z-value for the sample proportion under the alternate hypothesis (21 percent). Then we would find the area to the left of this Z-value under the standard normal distribution to find the probability of rejecting the null when it is false, which gives us the power of the test.