8.6 Relationship Between Confidence Intervals and Hypothesis Tests
Confidence intervals (CI) and hypothesis tests should give consistent results: we should not reject
A
Table 8.3: Relationship Between Confidence Interval and Hypothesis Test
Null hypothesis | |||
---|---|---|---|
Alternative | |||
Decision |
Reject
|

Here is the reason we should reject
Take the right-tailed test for example, we should reject
Example: Relationship Between Confidence Intervals and Hypothesis Tests
The ankle-brachial index (ABI) compares the blood pressure of a patient’s arm to the blood pressure of the patient’s leg. The ABI can be an indicator of different diseases, including arterial diseases. A healthy (or normal) ABI is 0.9 or greater. Researchers obtained the ABI of 100 women with peripheral arterial disease and obtained a mean ABI of 0.64 with a standard deviation of 0.15.
- At the 5% significance level, do the data provide sufficient evidence that, on average, women with peripheral arterial disease have an unhealthy ABI?
Steps:- Set up the hypotheses:
versus . - The significance level is
. - Compute the value of the test statistic:
with (not given in Table IV, use 95, the closest one smaller than 99). - Find the P-value. For a left-tailed test, the P-value is the area to the left of the observed test statistic
. since . - Decision: Since the P- value
, we should reject the null hypothesis . - Conclusion: At the 5% significance level, the data provide sufficient evidence that, on average, women with peripheral arterial disease have an unhealthy ABI.
- Set up the hypotheses:
- Obtain a confidence interval corresponding to the test in part a).
For a left-tailed test at the significance level , we should obtain a lower-tailed interval. For , not given in Table IV, use .Interpretation: We are 95% confident that women with peripheral arterial disease have an average ABI below 0.665.
- Does the interval in part b) support the conclusion in part a)?
In part a), we reject and claim that the mean ABI is below 0.9 for women with peripheral arterial disease. In part b), we are 95% confident that the mean ABI is less than 0.9 since the entire confidence interval is below 0.9. In other words, the hypothesized value 0.9 is outside the corresponding confidence interval, we should reject the null. Therefore, the results obtained in parts a) and b) are consistent.