Final answer:
The question involves calculating the power of a hypothesis test for a single population mean. Given the parameters (population standard deviation, sample size, and significance level), a Z-test is used to find the Z-value corresponding to an alternative hypothesis mean, which can then be used to determine the probability of a Type II error (β), indicating the proportion of time the procedure will fail to recognize a true effect.
Step-by-step explanation:
The question pertains to the field of hypothesis testing in statistics, which is a method used to decide if a sample data provides enough evidence to reject a null hypothesis (H0) at a certain level of significance (α). In this scenario, we are considering the null hypothesis H0: μ = 240 and the alternative hypothesis Ha: μ < 240. We are given a significance level of α = 0.01, a sample size of 25, a population mean under the alternative hypothesis of μ = 220, and a known population standard deviation (σ) of 40. We will calculate power of the test, which is the probability that we correctly reject the null hypothesis when the alternative hypothesis is true (in this case, when μ has indeed dropped to 220).
Since we have a normal distribution, and we are given the population standard deviation, we can use the Z-test for a single mean to compute this power. The Z-value is found by taking the hypothesized mean under the alternative hypothesis, subtracting the null hypothesis mean, and dividing by the standard error of the mean, which is the standard deviation divided by the square root of the sample size (σ/√n).
To find the Z-value for the alternative hypothesis:
- Z = (220 - 240) / (40/√25) = -20 / 8 = -2.5
We then look up this Z-value in the standard normal distribution table to find the power, which is the proportion of the time we correctly reject the null hypothesis when Ha is true. However, since the typical tables give the area to the right of a Z-value, we will need to take one minus the value we find in the table to obtain the power for a left-tailed test.
The concept of failing to reject the null hypothesis when it is actually false relates to Type II error (β). The power of the test is 1 - β. Therefore, to answer the student's question directly: the procedure will fail to recognize the drop from 240 to 220 (commit a Type II error) with a likelihood of β, given the test statistics calculated above.