A researcher wishes to test H₀: $\mu$
= 50 versus H_a: $\mu$
> 50.A sample of size n = 25 is take from a normally distributed population with mean $\mu$
= 55 and standard deviation is $\sigma$
= 10.What is the probability of a Type II error at the 5% level of significance?

Question

A researcher wishes to test H₀:

\mu

= 50 versus H_a:

\mu

> 50.A sample of size n = 25 is take from a normally distributed population with mean

\mu

= 55 and standard deviation is

\sigma

= 10.What is the probability of a Type II error at the 5% level of significance?

Quizplus · Accepted Answer

The probability of a Type II error ($\beta$) in hypothesis testing is the probability of failing to reject the null hypothesis when it is false. Given a normal distribution, sample size ($n$) of 25, population mean ($\mu$) of 55, standard deviation ($\sigma$) of 10, and testing at the 5% significance level, we aim to find the probability that the test does not reject the null hypothesis $H_0: \mu = 50$ when the true mean is $\mu = 55$. The calculation involves finding the Z-score corresponding to the 5% significance level and then calculating the probability that a sample mean will be less than this critical value given the true mean is 55. The correct calculation leads to a probability of a Type II error of approximately 0.1949.

A Researcher Wishes to Test H0 μ\muμ = 50 Versus Ha μ\muμ

A Researcher Wishes to Test H₀ $\mu$ = 50 Versus H_a $\mu$