The researcher wanted to test the hypothesis that Ho: μ=3, knowing the population, which was normal, had a variance of 1.00.
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How powerful would this test be against the alternative hypothesis that
Ha: μ=4, given that n=15, α = .05
The probability of making a Type II error is equal to beta. A Type II error is failure to reject the null when it is false. The power of the test is 1-beta and is the correct decision of rejecting the null when it is false. The alpha level directly affects the power of a test. The higher the level, the more powerful the test. Sample size also affects power. I'm not sure if your question is asking to calculate the actual power of the test or just determine if the test itself is powerful. If I'm interpreting the question correctly, then the test has power at alpha .05.
I hope this will be some help to you.
To determine the power of the test against the alternative hypothesis, we need to calculate the probability of rejecting the null hypothesis when it is false, i.e., finding evidence in favor of the alternative hypothesis.
In this case, our null hypothesis (Ho) is μ=3, and the alternative hypothesis (Ha) is μ=4. We also know that the population is normal and has a variance of 1.00.
To calculate the power of the test, we will need to use the concept of the standard error (SE) and the critical value (Zα) associated with the significance level (α).
The standard error (SE) of the mean is calculated using the formula:
SE = σ / sqrt(n)
where σ is the population standard deviation and n is the sample size.
In this case, since we know the population variance (1.00), we can use this as the population standard deviation (σ).
SE = 1.00 / sqrt(15)
Next, we need to find the critical value (Zα) associated with the significance level (α = 0.05). This value is obtained by looking up the appropriate z-score from a standard normal distribution table. For a two-tailed test with α = 0.05, we divide the significance level equally between the two tails, giving us α/2 = 0.025 for each tail.
Using a standard normal distribution table, we find that the critical value (Zα/2) is approximately 1.96.
Now, to calculate the power of the test, we need to determine the test statistic (Z) based on the alternative hypothesis (Ha) and the critical value (Zα).
The test statistic (Z) is calculated using the formula:
Z = (x̄ - μ) / SE
where x̄ is the sample mean.
Since our alternative hypothesis (Ha) is μ=4, the test statistic is:
Z = (4 - 3) / SE
Finally, we can calculate the power of the test by finding the probability that the test statistic (Z) exceeds the critical value (Zα), given that the alternative hypothesis (Ha) is true. This can be done by using a standard normal distribution table or using statistical software to calculate the area under the curve (probability).
Note: The power of a test is influenced by several factors, including the sample size (n), the population standard deviation (σ), the significance level (α), and the true population mean (μ) under the alternative hypothesis.