Question

A researcher is evaluating the influence of a treatment using a sample selected from a normally distributed population with a mean of

Answer 1

Answer:

The power of the test is 0.67.

Step-by-step explanation:

The complete question is:

A researcher is evaluating the influence of a treatment using a sample selected from a normally  distributed population with a mean of μ = 80 and a standard deviation σ = 20. The researcher  expects a 12-point treatment effect and plans to use a two-tailed hypothesis test with α = 0.05. Compute the power of the test if the researcher uses a sample of n = 16 individuals .

Solution:

The information provided are:

$\mu=80\\\sigma = 20\\n = 16\\\alpha=0.05$

The expected mean is:

$\mu_{\bar x}=80+12=92$

The critical z-score at α = 0.05 for a two-tailed test is:

z = 1.96

*Use a z-table.

Compute the test statistic value as follows:

$z_{\bar x}=\frac{\mu_{\bar x}-\mu}{\sigma_{\bar x}}=\frac{92-80}{20/\sqrt{16}}=2.4$

The power of statistical test is well-defined as the probability that we reject a false null hypothesis.

Power = Area to the right of the critical z under the assumption that H₀ is false.

Location of  critical z (in H₀ is false distribution) = $z_{\bar x}-z$

                                                                             $= 2.4 - 1.96 \\= -0.44$

This is negative because the critical z score is to the left of the mean  of the H₀ in false distribution.

Area above z = -0 .44.

Compute the value of P (Z > -0.44) as follows:

$P(Z>-0.44)=1-P(Z$

Thus, the power of the test is 0.67.