1. Hypothesis Test II: t tests
Test for one population mean
Test for the difference of two population means

2. Hypothesis Tests for Population Parameters
Population proportion
Population mean
Difference between two population means

3. 5 Steps in Hypothesis Tests
Step 1. Determine the null (Ho) and alternative (Ha) hypotheses
Ho : the population parameter = a constant (called the null value)
Ha : upper or lower or two-tailed

4. 5 Steps in Hypothesis Tests
Step 2. Find an appropriate test statistic and pre-set the level of significance (called a level)
Test statistic =
(Sample estimate – null value)
(S.E. of sample null value)
a= 0.05 (or 0.01 or 0.10)

5. 5 Steps in Hypothesis Tests
Step 3. Check the conditions and find the p-value assuming Ho is true
The calculation of the p-value depends on the
Ha.

6. 5 Steps in Hypothesis Tests
Step 4. Decide whether or not reject Ho based on the p-value
p-value > a  fail to reject Ho
p-value < a  reject Ho (= accept Ha)

7. 5 Steps in Hypothesis Tests
Step 5. Report the result in the context of the situation
Reject Ho 
There is sufficient evidence to support (Ha).
Fail to reject Ho 
There is not sufficient evidence to support (Ha).

8. Hypothesis Test for a Population Mean
Step 1. Ho: m = mo
Three possible Ha’s:
Ha: m = mo (two-tailed) or
Ha: m < mo (lower-tailed)
Ha: m > mo (upper-tailed)

9. Hypothesis Test for a Population Mean
Step 2. test statistic

10. Hypothesis Test for a Population Mean
Step 3:
Condition: Must be a random sample and the data must be bell-shaped (or normal)
When the above conditions are met, use t-Table to find p-value (the required df = n-1)
This test is called “one-sample t-test.”

11. Examples
The average cost of 30 randomly selected used cars in a used car lot was $5,000 with sample standard deviation of $1,000.
Find the p-value for testing
Ho: m = $4,500 vs. Ha: m = $4,500.
Answer: t = 2.74 and < p-value < 0.016

12. Examples Find the p-value for testing
Ho: m = $4,500 vs. Ha: m > $4,500.
Ho: m = $4,500 vs. Ha: m < $4,500.

14. Hypothesis Test for the Difference of Two Population Means
Step 1. Ho: m1 = m2
Three possible Ha’s:
Ha: m1 = m2 (two-tailed) or
Ha: m1 < m2 (lower-tailed)
Ha: m1 > m2 (upper-tailed)

15. Hypothesis Test for the Difference between Two Population Means
Step 2. test statistic

16. Hypothesis Test for the Difference between Two Population Means
Step 3:
Must be two independent random samples; the both samples must be bell-shaped (or normal)
When the above conditions are met, use t-Table to find the p-value (the required df is the smaller one of (n1-1) and (n2-1))
This test is called “Two-sample t-test.”

17. Examples
The average waiting time for 50 students to connect to the college server from the dormitories was 5 seconds, while the average time to connect for 20 students living off-campus apartments was 10 seconds. Assume the times to connect in the dormitories and the apartments are both bell-shaped. The sample standard deviations were 2 seconds for dormitories and 8 seconds for apartments

18. Examples Find the p-value for testing Ho: m1 = m2 vs. Ha: m1 = m2
Answer: t = -2.76
1: dormitories; 2: apartments