Question

A vending machine is designed to discharge at least 275mL of drink per cup on average. A number of customers complain that this is not the case and they are getting less than this amount. In response to the complaints, a random sample of 30 cups is taken in order to conduct an appropriate hypothesis test for the population mean, using a 5% level of significance. (Note: Assume that the population standard deviation is 14mL.)

Calculate the probability of Type II error if the population mean amount dispensed is actually 271mL per cup (please round your answer to three decimal places). Hint: Calculate the power of the test first.

Answer 1

Hypotheses are:

$H_{0}:\mu\geq 275$

$H_{a}:\mu< 275$

And we have

$n=30,\sigma=14$

Since test is left tailed so critical value of z for which we will reject the null hypothesis is -1.645. So critical value of sample mean for which we will reject the null hypothesis is

$z=\frac{\bar{x}-\mu}{\sigma/\sqrt{n}}$

$-1.645=\frac{\bar{x}-275}{14/\sqrt{30}}$

$\bar{x}=270.80$

The z-score for $\bar{x}=270.80$ and is

μ = 271

270.80 - 271 =-0.08 14/V30

The type II error is

3 = P(z > _0.08) = 1-P(z <-0.08) = 1-0.4681 = 0.5319

Answer: 0.532