Question

A researcher is concerned that the true

population mean could be as much as 4.8 greater

than the accepted population mean, but the

researchers hypothesis test fails to find a

significant difference. The power for this study

was 0.5, the researcher probably should

A) accept the outcome and move on

B) repeat the experiment with a smaller α

C) repeat the experiment with a larger α

D) repeat the experiment with a larger n

E) repeat the experiment and hope that the next sample

mean is significantly different than the hypothesized

mean.

Failing to observe a treatment affect

for Rogaine, when in reality Rogaine reduces

hair loss, would be...

A) impossible

B) an error with probability equal to α

C) an error with probability equal to β

D) an error with probability equal to 1-β

Answer 1

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According to the given information, let 'd' is the difference between sample mean and population mean. Hence we can take

Null Hypothesis: H0 : d ≤ 4.8

Alt Hypothesis:   H1 : d > 4.8.

Since, researcher failed to find significant difference in the hypothesis testing, that means the null hypothesis is rejected here.

Given that, the power (1 - β) for the study was 0.5.

• Power is the probability of rejecting the null hypothesis when, in fact, it is false.
• Power is the probability of making a correct decision (to reject the null hypothesis) when the null hypothesis is false.

That means in this case there is only 50% chance that the hypothesis testing was correct.

Power of test lies between 0 and 1. If the power is close to 1, the hypothesis test is very good at detecting a false null hypothesis. β is commonly set at 0.2, but may be set by the researchers to be smaller to make the power (1-β) close to 1.

We can express power as:

i.e. Power is increased when a researcher increases sample size, as well as when a researcher selects stronger effect sizes and significance levels (i.e. with less value of α).

Since, collecting more data to increase sample size may be time consuming, in this case researcher should go with the option (B) repeat the experiment with a smaller α.

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We need to remember

α = Type I error = Probability (Rejecting H0 when H0 is true)

β = Type II error = Probability (Rejecting H1 when H1 is true)

(1-β) = Power = Probability (Accepting H1 when H1 is true)

                         = Probability (Rejecting H0 when H0 is false)

In second query, let us take the two hypotheses as

Null Hypothesis H0 : Rogaine reduces hair loss.

Alt Hypothesis H1 : Rogaine does NOT reduce hair loss.

So, the scenario of “Failing to observe a treatment affect for Rogaine, when in reality Rogaine reduces hair loss” would be represented by the case (Rejecting H0 when H0 is true) which is at the option (B) an error with probability α = Probability (Rejecting H0 when H0 is true)

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