### 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 #97\%# confidence interval for a population mean #\mu#, computed based on a simple random sample from the population, is #(0.131,\,\, 0.762)#.

Suppose you use the same sample to test #H_0: \mu = 0# against #H_a: \mu \neq 0# at the #\alpha = 0.03# level of significance.

What would be the conclusion?

Suppose you use the same sample to test #H_0: \mu = 0# against #H_a: \mu \neq 0# at the #\alpha = 0.03# level of significance.

What would be the conclusion?

Reject #H_0#.

Since the #97\%# confidence interval #(0.131,\,\,0.762)# does not contain the value #\mu_0 = 0#, we would reject #H_0: \mu = 0# at the #\alpha = 0.03# level of significance.

Since the #97\%# confidence interval #(0.131,\,\,0.762)# does not contain the value #\mu_0 = 0#, we would reject #H_0: \mu = 0# at the #\alpha = 0.03# level of significance.

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