Insights into concepts of hypothesis testing (Part 11)

Demonstration of Type I Error with an Example

This example demonstrates how Type I error can be committed.  Suppose this is population data Population = {1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5}.

It is a symmetric population with Population Mean = 3, Population Mode = 3 and Population Median = 3. These are unknown, but for the sake of ease of comprehension I have mentioned it here. Let’s draw a sample of size 3.

Sample = {4, 5, 5}

Due to variations in sampling we can get such an extreme sample with sample mean = 4.66.  We test the hypothesis that this sample comes from a population with mean 3.

Null Hypothesis: Population Mean is equal to 3

Alternative Hypothesis: Population Mean is not equal to 3

Under the assumption that population standard deviation is known and is 1.1239 the Z test statistics

is 2.55 and is more than 1.96. So here the null hypothesis is rejected at 5% level of significance and we conclude that the sample doesn’t come from this population. The p value is 0.0053. If the sample mean is greater than 4.27  we have to reject the null hypothesis and conclude that the sample mean is not 3.

Sample mean more than 4.27 than we reject the null hypothesis