7. Hypothesis Testing: Introduction to Hypothesis Testing
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 #92\%# confidence interval for a population mean #\mu#, computed based on a simple random sample from the population, is #(6.327,\,\, 7.673)#.
Suppose you use the same sample to test #H_0: \mu = 8# against #H_a: \mu \neq 8# at the #\alpha = 0.08# level of significance.
What would be the conclusion?
Suppose you use the same sample to test #H_0: \mu = 8# against #H_a: \mu \neq 8# at the #\alpha = 0.08# level of significance.
What would be the conclusion?
Reject #H_0#.
Since the #92\%# confidence interval #(6.327,\,\,7.673)# does not contain the value #\mu_0 = 8#, we would reject #H_0: \mu = 8# at the #\alpha = 0.08# level of significance.
Since the #92\%# confidence interval #(6.327,\,\,7.673)# does not contain the value #\mu_0 = 8#, we would reject #H_0: \mu = 8# at the #\alpha = 0.08# level of significance.
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