Question

A biologist hypothesizes that the mean life expectancy of a Pearroach is 17 days. Researchers believe the true mean is less than 17 days and they plan a hypothesis test at the 10% significance level on a random sample of 80 of these Pearroaches. If the alternative hypothesis is correct, for which of the following values of μ is the power of the test greatest?

μ = 9
μ = 10
μ = 17
μ = 20
μ = 21

Answer 1

@Narad

Answer 2

@Narad

Answer 3

It's A, μ = 9.

This is for understanding purposes:

First set the hypotheses.
Let μ=the true population mean life expectancy of the Pearroach.
\[H _{o} : μ=17\]
\[H _{a}: μ<17\]
They are saying that α=0.1
They also said that n=80.

Their statistical test says that the alternative hypothesis is correct. This should mean that the P-value is LESS than α, or less than 0.1. Remember, the power of the test is 1-β, or the probability that a Type II error (which is approving the null when it's false) does not occur. This would mean that we have a p-value that is so low, we are farther away from failing to reject the null. This is why we need a low p-value. A low p-value corresponds with a low z-score. We are using z-score here because it's a 1 proportion z-test that they're completing. If you use the table, a p-value of less than α=0.1 would need a z-score of less than -1.29. Now let's move to the standardized test statistic, where we will figure out our answer:

\[\frac{x̄-μ }{ SE _{x̄}}\] = Z-score
We can ignore SEx̄, as it is arbitrary for our purpose of understanding. μ is already given as 17.
\[-1.29 ≥\frac{x̄-17 }{ SE _{x̄}}\]\]

Since we only consider the positive SEx̄ for hypothesis, we know that the simplified numerator must have a negative value. This is only possible if x̄ is LESS than 17.

The answer that is the least and is the farthest from 17 is option A, 9.

In simpler words,
the lower and farther away the statistic is from the parameter, the greater the power of the test would be.

Answer 4

Correction: It is not a 1 proportion z-test, but a z-test/t-test. I chose z-scores because it's simpler to work with, but you can use a t-test as well. Although the population standard deviation is not given, we don't need to use a t-score because we are only using these terms for our own understanding. Also, t-score and z-score work correlatively.

Answer 5

let me just check it

Answer 6

THANX it is rigght