Connection between Hypothesis Testing and Confidence Intervals

Chapter 7. Hypothesis Testing: Introduction to Hypothesis Testing (p-value Approach)

Connection between Hypothesis Testing and Confidence Intervals

Recall that a population mean #\mu# can be estimated using a #C\%# confidence interval #(CI)#.

Reinterpreting the Confidence Level C

The confidence level #C# of a confidence interval can be reinterpreted as #(1 - \alpha)\cdot 100# for some number #\alpha#.

For example, if #C = 95#, then #\alpha = 0.05#, or if #C=99#, then #\alpha = 0.01#.

Connecting Hypothesis Testing and Confidence Intervals

Thus a #C\%\,CI# for #\mu# can be reinterpreted as a #(1-\alpha)\cdot 100\%\,CI# for #\mu#.

This enables us to establish a direct connection between a two-sided hypothesis test for #\mu# and a #(1-\alpha)\cdot 100\%# confidence interval for #\mu#:

If #\mu_0# falls inside the #(1 - \alpha)\cdot 100\%\,CI#, then #H_0: \mu=\mu_0# should not be rejected at the #\alpha# level of significance.
If #\mu_0# falls outside of the #(1 - \alpha)\cdot 100\%\,CI#, then #H_0: \mu=\mu_0# should be rejected at the #\alpha# level of significance.

A #99\%# confidence interval for a population mean #\mu#, computed based on a simple random sample from the population, is #(0.454,\,\, 1.632)#.

Suppose you use the same sample to test #H_0: \mu = 0# against #H_a: \mu \neq 0# at the #\alpha = 0.01# level of significance.

What would be the conclusion?

Reject #H_0#.

Since the #99\%# confidence interval #(0.454,\,\,1.632)# does not contain the value #\mu_0 = 0#, we would reject #H_0: \mu = 0# at the #\alpha = 0.01# level of significance.

New example