Example of the power of the hypothesis test



Introduction

While performing hypothesis tests we may come across errors while making decisions. One might be doubtful about the reliability of the hypothesis test. To check whether the results come from the hypothesis test are reliable or not we calculate the power of the hypothesis test. Many times, it is a brainteasing task to calculate the power of the test. In this article, you will learn how to calculate the power of the hypothesis test systematically. Consider a real-life example as follows.


Example

A researcher collected a sample of 50 students on the heights of students in a college. He used a statistical hypothesis test to test if the average of the student’s height is less than 172 cm.

The researcher assumed that the population is normally distributed with the standard deviation of heights of the students is 12 cm. Calculate the power of the hypothesis test if the average height of students in the college is 168 cm.


Solution

Consider the researcher’s point of view. Therefore, we will set up the whole problem as follows.

In this context, we need to test if the average of the student’s height is less than 172 cm. We assumed that the population is normally distributed with the standard deviation is 12 cm. Therefore, we use a one-sample Z test to test the claim.

To know why we chose the Z test in this context, please refer to the article on one sample mean test.


Step 1: Fix the null and alternative hypotheses

The null and alternative hypotheses for the one-sample Z test are as follows.

H0: µ = 172

Vs

H1: µ < 172

Since the alternative hypothesis contains < sign, the tail of the test is left-tailed.


Step 2: The formula for calculating the test statistic

We use the following formula to find the value of the test statistic.


Where M is the sample mean, µ0 is the value of mean in the hypotheses, σ is the population standard deviation, n is the sample size.

Since the test is left-tailed the critical value Zc at α=0.05 is -1.6449. Let us find Mc as follows.



Step 3: Definition of the power of the hypothesis test

The power of the hypothesis test is the probability of correctly rejecting the false null hypothesis. For this, we will find the probability of committing a Type II error. Accepting the false null hypothesis is the Type II error. The probability of Type II error is denoted by ß. We can write this as follows.

ß = P (Z > Zc | µ < µ0) = P (M > Mc | µ < µ0)


Step 4: Calculations

In the given example, we have µ0=172, σ=12, n=50. Moreover, the actual population mean (µ) is 168 cm. This gives,


Therefore, we get the probability of committing Type II error is as follows.


Note that the variable Z follows the standard normal distribution. We need to find the standard normal probability. Here we use the excel formula to find the probability. Refer to the article on excel for normal distribution probabilities.

This gives, P(Z < 0.7121) = 0.2382.

ß = 1-0.7618 = 0.2382.

From this, we get the power of the hypothesis test is 1-ß = 1-0.2382 = 0.7618.

 

Conclusion

In the context of the given problem, we can interpret the power of the hypothesis as follows.

When the actual population mean of the students’ height in the college is 168 cm then the power is the probability of making the correct decision.