Q.26 - The Practice of Statistics for AP

Question

You manufacture and sell a liquid product whose electrical conductivity is supposed to be 5. You plan to make 6 measurements of the conductivity of each lot of product. If the product meets specifications, the mean of many measurements will be 5. You will therefore test

$\begin{array}{r}{H}_0:\mu =5\\ H_a:\mu \ne 5\end{array}$

If the true conductivity is 5.1, the liquid is not suitable for its intended use. You learn that the power of your test at the 5% significance level against the alternative $\mu$=5.1 is 0.23.

(a) Explain in simple language what "power =0.23 " means in this setting.

(b) You could get higher power against the same alternative with the same $\alpha$ by changing the number of measurements you make. Should you make more measurements or fewer to increase power?

(c) If you decide to use $\alpha$=0.10 in place of $\alpha$=0.05, with no other changes in the test, will the power increase or decrease? Justify your answer.

(d) If you shift your interest to the alternative $\mu$=5.2, with no other changes, will the power increase or decrease? Justify your answer.

Step-by-Step Solution

Answer

a. It means the probability of rejecting the null hypothesis if it's false

b. make more measurements 

c. the power increases

d. the power increases 

Step 1: Given Information

The electrical conductivity of the liquid product = $5$

$\begin{array}{r}{H}_0:\mu =5\\ H_a:\mu \ne 5\end{array}$

Significance level =$5\%$

$\mu =5.1$

0.23p =$0.23$

Step 2: Explanation Part (a)

Given,

Significance level = $5\%$

$\begin{array}{c}\mu =5.1\\ p=0.23\end{array}$

"power =$0.23$ " means the probability of rejecting the null hypothesis if it's false 

Step 3: Explanation Part (b)

Given,

Significance level =$5\%$

$\begin{array}{r}\mu =5.1\\ p=0.23\end{array}$

We could get higher power against the same alternative with the same$\alpha$by changing the number of measurements you make hence we should make more measurements as more information will be needed.

Step 4: Explanation Part (c)

As the probability of making type I error increases the probability of making a type II error decreases.

We know,

$\text{ Power }=1-\beta$(the probability of making a type II error)

Hence as the probability of making a type II error decreases the power increases.

Step 5: Explanation Part (d)

Given,

Shifting the alternative to $\mu =5.2$

As per the null hypothesis,

$H_0:\mu =5$

The alternative is greater than the null hypothesis, Hence it means the power has increased.