1. Testing Hypotheses about a Population Proportion
Lecture 30
Sections 9.3
Mon, Nov 15, 2004

2. Variations on the Method
Two variations on the method of testing a hypothesis.
p-value approach
Report the p-value and compare it to  or let the reader decide for himself.
Classical Approach
State .
Compute the critical value from  and compare the test statistic to it.

3. Let’s Do It! Let’s do it! 9.2, p. 529 – Improved Process?
Let’s do it! 9.3, p. 530 – ESP.
Let’s do it! 9.4, p. 531 – Working Part Time.
Article in the Journal of the American Medical Association.

4. JAMA Example
Results  Compared with those without the metabolic syndrome (n = 1616), elders with the metabolic syndrome (n = 1016) were more likely to have cognitive impairment (26% vs 21%, multivariate adjusted relative risk [RR], 1.20; 95% confidence interval [CI], ). There was a statistically significant interaction with inflammation and the metabolic syndrome (P = .03) on cognitive impairment. After stratifying for inflammation, those with the metabolic syndrome and high inflammation (n = 348) had an increased likelihood of cognitive impairment compared with those without the metabolic syndrome (multivariate adjusted RR, 1.66; 95% CI, ). Those with the metabolic syndrome and low inflammation (n = 668) did not exhibit an increased likelihood of impairment (multivariate adjusted RR, 1.08; 95% CI, ). Stratified multivariate random-effects models demonstrated that participants with the metabolic syndrome and high inflammation had greater 4-year decline on 3MS (P = .04) compared with those without the metabolic syndrome, whereas those with the metabolic syndrome and low inflammation did not (P = .44).

5. JAMA Example
Results  Compared with those without the metabolic syndrome (n = 1616), elders with the metabolic syndrome (n = 1016) were more likely to have cognitive impairment (26% vs 21%, multivariate adjusted relative risk [RR], 1.20; 95% confidence interval [CI], ). There was a statistically significant interaction with inflammation and the metabolic syndrome (P = .03) on cognitive impairment. After stratifying for inflammation, those with the metabolic syndrome and high inflammation (n = 348) had an increased likelihood of cognitive impairment compared with those without the metabolic syndrome (multivariate adjusted RR, 1.66; 95% CI, ). Those with the metabolic syndrome and low inflammation (n = 668) did not exhibit an increased likelihood of impairment (multivariate adjusted RR, 1.08; 95% CI, ). Stratified multivariate random-effects models demonstrated that participants with the metabolic syndrome and high inflammation had greater 4-year decline on 3MS (P = .04) compared with those without the metabolic syndrome, whereas those with the metabolic syndrome and low inflammation did not (P = .44).

6. JAMA Example
Results  Compared with those without the metabolic syndrome (n = 1616), elders with the metabolic syndrome (n = 1016) were more likely to have cognitive impairment (26% vs 21%, multivariate adjusted relative risk [RR], 1.20; 95% confidence interval [CI], ). There was a statistically significant interaction with inflammation and the metabolic syndrome (P = .03) on cognitive impairment. After stratifying for inflammation, those with the metabolic syndrome and high inflammation (n = 348) had an increased likelihood of cognitive impairment compared with those without the metabolic syndrome (multivariate adjusted RR, 1.66; 95% CI, ). Those with the metabolic syndrome and low inflammation (n = 668) did not exhibit an increased likelihood of impairment (multivariate adjusted RR, 1.08; 95% CI, ). Stratified multivariate random-effects models demonstrated that participants with the metabolic syndrome and high inflammation had greater 4-year decline on 3MS (P = .04) compared with those without the metabolic syndrome, whereas those with the metabolic syndrome and low inflammation did not (P = .44).

7. JAMA Example
Results  Compared with those without the metabolic syndrome (n = 1616), elders with the metabolic syndrome (n = 1016) were more likely to have cognitive impairment (26% vs 21%, multivariate adjusted relative risk [RR], 1.20; 95% confidence interval [CI], ). There was a statistically significant interaction with inflammation and the metabolic syndrome (P = .03) on cognitive impairment. After stratifying for inflammation, those with the metabolic syndrome and high inflammation (n = 348) had an increased likelihood of cognitive impairment compared with those without the metabolic syndrome (multivariate adjusted RR, 1.66; 95% CI, ). Those with the metabolic syndrome and low inflammation (n = 668) did not exhibit an increased likelihood of impairment (multivariate adjusted RR, 1.08; 95% CI, ). Stratified multivariate random-effects models demonstrated that participants with the metabolic syndrome and high inflammation had greater 4-year decline on 3MS (P = .04) compared with those without the metabolic syndrome, whereas those with the metabolic syndrome and low inflammation did not (P = .44).

8. JAMA Example
Results  Compared with those without the metabolic syndrome (n = 1616), elders with the metabolic syndrome (n = 1016) were more likely to have cognitive impairment (26% vs 21%, multivariate adjusted relative risk [RR], 1.20; 95% confidence interval [CI], ). There was a statistically significant interaction with inflammation and the metabolic syndrome (P = .03) on cognitive impairment. After stratifying for inflammation, those with the metabolic syndrome and high inflammation (n = 348) had an increased likelihood of cognitive impairment compared with those without the metabolic syndrome (multivariate adjusted RR, 1.66; 95% CI, ). Those with the metabolic syndrome and low inflammation (n = 668) did not exhibit an increased likelihood of impairment (multivariate adjusted RR, 1.08; 95% CI, ). Stratified multivariate random-effects models demonstrated that participants with the metabolic syndrome and high inflammation had greater 4-year decline on 3MS (P = .04) compared with those without the metabolic syndrome, whereas those with the metabolic syndrome and low inflammation did not (P = .44).

9. JAMA Example
Results  Compared with those without the metabolic syndrome (n = 1616), elders with the metabolic syndrome (n = 1016) were more likely to have cognitive impairment (26% vs 21%, multivariate adjusted relative risk [RR], 1.20; 95% confidence interval [CI], ). There was a statistically significant interaction with inflammation and the metabolic syndrome (P = .03) on cognitive impairment. After stratifying for inflammation, those with the metabolic syndrome and high inflammation (n = 348) had an increased likelihood of cognitive impairment compared with those without the metabolic syndrome (multivariate adjusted RR, 1.66; 95% CI, ). Those with the metabolic syndrome and low inflammation (n = 668) did not exhibit an increased likelihood of impairment (multivariate adjusted RR, 1.08; 95% CI, ). Stratified multivariate random-effects models demonstrated that participants with the metabolic syndrome and high inflammation had greater 4-year decline on 3MS (P = .04) compared with those without the metabolic syndrome, whereas those with the metabolic syndrome and low inflammation did not (P = .44).

10. JAMA Example
Results  Compared with those without the metabolic syndrome (n = 1616), elders with the metabolic syndrome (n = 1016) were more likely to have cognitive impairment (26% vs 21%, multivariate adjusted relative risk [RR], 1.20; 95% confidence interval [CI], ). There was a statistically significant interaction with inflammation and the metabolic syndrome (P = .03) on cognitive impairment. After stratifying for inflammation, those with the metabolic syndrome and high inflammation (n = 348) had an increased likelihood of cognitive impairment compared with those without the metabolic syndrome (multivariate adjusted RR, 1.66; 95% CI, ). Those with the metabolic syndrome and low inflammation (n = 668) did not exhibit an increased likelihood of impairment (multivariate adjusted RR, 1.08; 95% CI, ). Stratified multivariate random-effects models demonstrated that participants with the metabolic syndrome and high inflammation had greater 4-year decline on 3MS (P = .04) compared with those without the metabolic syndrome, whereas those with the metabolic syndrome and low inflammation did not (P = .44).

11. JAMA Example
Results  Compared with those without the metabolic syndrome (n = 1616), elders with the metabolic syndrome (n = 1016) were more likely to have cognitive impairment (26% vs 21%, multivariate adjusted relative risk [RR], 1.20; 95% confidence interval [CI], ). There was a statistically significant interaction with inflammation and the metabolic syndrome (P = .03) on cognitive impairment. After stratifying for inflammation, those with the metabolic syndrome and high inflammation (n = 348) had an increased likelihood of cognitive impairment compared with those without the metabolic syndrome (multivariate adjusted RR, 1.66; 95% CI, ). Those with the metabolic syndrome and low inflammation (n = 668) did not exhibit an increased likelihood of impairment (multivariate adjusted RR, 1.08; 95% CI, ). Stratified multivariate random-effects models demonstrated that participants with the metabolic syndrome and high inflammation had greater 4-year decline on 3MS (P = .04) compared with those without the metabolic syndrome, whereas those with the metabolic syndrome and low inflammation did not (P = .44).

12. The Classical Approach
1. State the hypotheses.
2. State the level of significance.
3. Find the critical value of the test statistic and the rejection region.
4. State the decision rule.
5. Compute the value of the test statistic.
6. State the conclusion.

13. The Classical Approach
1. State the hypotheses.
2. State the level of significance.
3. Find the critical value of the test statistic and the rejection region.
4. State the decision rule.
5. Compute the value of the test statistic.
6. State the conclusion.

14. Step 3: The Critical Value and the Rejection Region
In a one-tailed test, the critical value c is the value of the test statistic that cuts off a tail of area  in the direction of extreme.
Rejection region – the interval that starts at c and goes in the direction of extreme. z p0 c

15. The Critical Value and the Rejection Region
In a one-tailed test, the critical value c is the value of the test statistic that cuts off a tail of area  in the direction of extreme.
Rejection region – the interval that starts at c and goes in the direction of extreme. 
Rejection region z p0 c

16. The Critical Value and the Rejection Region
In a two-tailed test, the critical value c is the value that cuts off an upper tail of area /2.
The rejection region is the two intervals that start at c and go in the directions of extreme. z –c p0 c

17. The Critical Value and the Rejection Region
In a two-tailed test, the critical value c is the value that cuts off an upper tail of area /2.
The rejection region is the two intervals that start at c and go in the directions of extreme.
/2
/2
Rejection region
Rejection region z –c p0 c

18. Step 4: The Decision Rule
Decision rule – A statement that exactly which values of the test statistic will lead to reject H0.
One-tailed test, extreme to the right.
Reject H0 if z > c.
One-tailed test, extreme to the left.
Reject H0 if z < c.
Two-tailed test.
Reject H0 if either z > c or z < –c.

19. Example
Rework the example of the 520 male births out of 1000 births using the classical approach.