A researcher wishes to test H₀: $\mu$
= 50 versus H_a: $\mu$
> 50.A sample of size n = 16 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 = 16 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 is the probability of failing to reject the null hypothesis when it is false. In this case, the null hypothesis is that the mean is 50, and the alternative hypothesis is that the mean is greater than 50. The sample size is 16, the population mean is 55, and the population standard deviation is 10. The level of significance is 5%.The probability of a Type II error can be calculated using the formula:$$\beta = P(Type II error) = P(Z < z_{\alpha} - \frac{\mu_0 - \mu_1}{\sigma/\sqrt{n}})$$where:* Z is the standard normal distribution* z_{\alpha} is the critical value of the standard normal distribution at the level of significance $\alpha$* $\mu_0$ is the null hypothesis mean* $\mu_1$ is the alternative hypothesis mean* $\sigma$ is the population standard deviation* n is the sample sizePlugging in the values, we get:$$\beta = P(Z < 1.645 - \frac{50 - 55}{10/\sqrt{16}}) = P(Z < 0.645) = 0.7422$$Therefore, the probability of a Type II error is 0.7422, or 74.22%.

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

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