Question 1206173: i have tried answering this problem, i can figure out the critcal values & Z score but my answers don't match up to the choices given...
It is desired to test H0: μ = 55 against H1: μ < 55 using α = 0.10. The population in question is normally distributed with a standard deviation of 20. A random sample of 64 will be drawn from this population. If μ is really equal to 50, what is the probability that the hypothesis test would lead the investigator to commit a Type II error?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! two tail 90% confidence interval generates a critical z-score of plus or minus 1.644885.
the population mean is assumed to be 55
the population standard deviation is equal to 20.
the standard error is used.
standard error = standard deviation / sqrt(sample size).
sample size is 64.
standard error = 20 / sqrt(64) = 2.5.
z-score formula is z = (x-m)/s
when z = -1.644885, formula becomes -1.644885 = (x-55)/2.5
solve for x to get x = 50.888
when z = 1.644885, formula becomes 1.644885 = (x-55)/2.5
solve for x to get x = 59.112
90% confidence interval is 50.888 to 59.112
if we assume that the sample mean was 50, then z-score formula becomes z = (50 - 55)/2.5 = -2.
since the critical z-score is -1.64485, the results would be significant and the conclusion would be that 55 is not the real mean, and that the real mean was closer to 50.
a type 1 error is when you reject the null hypothesis but it is actually true.
a type 2 error is when you reject the alternate hypothesis but it is actually true.
i'm not exactly sure how to calculate the probability of a type 2 error in this problem, so i declined to answer this part of your problem.
let me know if what i provided you helped and, if not, where the deficiency lies.
here's a reference on type 1 and type 2 errors.
it may help, although it wasn't enough for me to understand fully how to calculate the probability of a type 2 error.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2996198/#:~:text=A%20type%20I%20error%20(false,actually%20false%20in%20the%20population.
theo
P.S.
i found a calculator online that calculates power and beta of a test.
it can be found at https://www.statology.org/type-ii-error-calculator/
it told me that beta = .23624.
power of the test is 1 minus beta.
here's what the results of my using that calculator looks like.
i still don't understand how it was calculated, but at least i got an answer.
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