1. Summary of Tests Confidence Limits
Proportions and means

2. One sample Tests

3. Z test for a proportion Test statistic Null Hypothesis Alt. Hypothesis
Critical Region
H0: p = p0
H0: p ≠ p0
z < -za/2 or z > za/2
H0: p > p0
z > za
H0: p < p0
z < -za

4. Confidence interval for a proportion
(1 – a)100% confidence limits for p

5. Z test for the mean of a Normal Population
Test statistic
Null Hypothesis
Alt. Hypothesis
Critical Region
H0: m = m 0
H0: m ≠ m 0
z < -za/2 or z > za/2
H0: m > m 0
z > za
H0: m < m 0
z < -za

6. Confidence interval for a mean of a normal population
(1 – a)100% confidence limits for m or
for large samples

7. The t test for the mean of a Normal Population
Test statistic
Null Hypothesis
Alt. Hypothesis
Critical Region
H0: m = m 0
H0: m ≠ m 0
t < -ta/2 or z > ta/2
H0: m > m 0
t > ta
H0: m < m 0
t < -ta
df = n -1

8. Confidence interval for a mean of a normal population (small samples)
(1 – a)100% confidence limits for m
df = n -1

9. Two sample Tests

10. Z test for a comparing two proportions
Test statistic
where
Null Hypothesis
Alt. Hypothesis
Critical Region
H0: p1 = p2
H0: p1 ≠ p2
z < -za/2 or z > za/2
H0: p1 > p2
z > za
H0: p1 < p2
z < -za

11. Confidence intervals for the difference in two proportions
(1 – a)100% confidence limits for p1 – p2

12. Z test for a comparing two means of Normal Populations
Test statistic
Null Hypothesis
Alt. Hypothesis
Critical Region
H0: m1 = m2
H0: m1 ≠ m2
z < -za/2 or z > za/2
H0: m1 > m2
z > za
H0: m1 < m2
z < -za

13. Confidence intervals for the difference in two means of normal populations
(1 – a)100% confidence limits for m1 – m2 or
for large samples

14. Test statistic where df = n + m - 2
The t test for a comparing two means of Normal Populations (variances assumed equal, sample sizes small)
Test statistic
where
Null Hypothesis
Alt. Hypothesis
Critical Region
H0: m1 = m2
H0: m1 ≠ m2
z < -za/2 or z > za/2
H0: m1 > m2
z > za
H0: m1 < m2
z < -za
df = n + m - 2

15. Confidence intervals for the difference in two means of normal populations (small sample sizes equal variances)
(1 – a)100% confidence limits for m1 – m2
where

16. Tests, Confidence intervals for the difference in two means of normal populations (small sample sizes, unequal variances)

17. Consider the statistic
For large sample sizes this statistic has standard normal distribution. For small sample sizes this statistic has been shown to have approximately a t distribution with

18. The approximate test for a comparing two means of Normal Populations (unequal variances)
Test statistic
Null Hypothesis
Alt. Hypothesis
Critical Region
H0: m1 = m2
H0: m1 ≠ m2
t < -ta/2 or t > ta/2
H0: m1 > m2
t > ta
H0: m1 < m2
t < -ta

19. Confidence intervals for the difference in two means of normal populations (small samples, unequal variances)
(1 – a)100% confidence limits for m1 – m2
with

20. Review Assignment
solutions

26. A third possible solution is to use the approximate t distribution
with
t0.025 = and t0.005 = for 21 df.

27. (1 – a)100% confidence limits for m1 – m2 are:
29.95 to 48.63
99% confidence limits
26.57 to 52.01