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 |
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