Respuesta :
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:
[tex]\mu=80\\\sigma = 20\\n = 16\\\alpha=0.05[/tex]
The expected mean is:
[tex]\mu_{\bar x}=80+12=92[/tex]
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:
[tex]z_{\bar x}=\frac{\mu_{\bar x}-\mu}{\sigma_{\bar x}}=\frac{92-80}{20/\sqrt{16}}=2.4[/tex]
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) = [tex]z_{\bar x}-z[/tex]
[tex]= 2.4 - 1.96 \\= -0.44[/tex]
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:
[tex]P(Z>-0.44)=1-P(Z<-0.44)=P(Z<0.44)=0.67[/tex]
Thus, the power of the test is 0.67.
