1. Hypothesis Testing and Confidence Intervals
Two Population Means
Hypothesis Testing and Confidence Intervals
With Known Standard Deviations

2. SITUATION: 2 Populations
Mean = 2 St’d Dev. = 2
Population 1
Mean = 1 St’d Dev. = 1
Salaries in Chicago
Salaries in St. Louis
Women’s Test Scores
Men’s Test Scores
Lakers Attendance
Clippers Attendance
Anaheim Sales
Irvine Sales

3. KEY ASSUMPTIONS Sampling is done from two populations.
Population 1 has mean µ1 and variance σ12.
Population 2 has mean µ2 and variance σ22.
A sample of size n1 will be taken from population 1.
A sample of size n2 will be taken from population 2.
Sampling is random and both samples are drawn independently.
Either the sample sizes will be large or the populations are assumed to be normally distribution.

4. The Problem OBJECTIVES 1 and 2 are unknown
1 and 2 may or may not be known
(In this module we assume they are known.)
OBJECTIVES
Test whether 1 > 2 (by a certain amount)
or whether 1  2
Determine a confidence interval for the difference in the means: 1 - 2

6. Key Concepts About the Random Variable .
is the difference in two sample means.
Its mean is the difference of the two individual means:
If the variables are independent (which we assumed), the variance (not the standard deviation) of the random variable of the differences = the sum (not the difference) of the two variances:
Thus its standard deviation is:
Its distribution is:
Normal if σ1 and σ2 are known
t if σ1 and σ2 are unknown

7. Hypothesis Test Statistics for Difference in Means, Known σ’s
We will be performing hypothesis tests with null hypotheses, H0, of the form:
From the general form of a test statistic, the required test statistic will be:
H0: µ1 - µ2 = v

8. Confidence Intervals for µ1 - µ2 Known σ’s
Recall the general form of a confidence interval is:
Thus when the σ’s are known this becomes:
(Point Estimate) ± zα/2(Appropriate Standard Error)

9. EXAMPLE Hypothesis Test: 1, 2 Known
Test whether starting salaries for secretaries in Chicago are at least $5 more per week than those in St. Louis.
GIVEN:
Salaries assumed to be normal
Standard Deviations known: Chicago $10; St. Louis $15
Sample Results
Sampled 100 secretaries in Chicago; 75 secretaries in St. Louis
Sample averages: Chicago - $550, St. Louis -$540

10. Hypothesis Test H0: 1 - 2 = 5 HA: 1 - 2 > 5 Use  = .05
Reject H0 (Accept HA) if z > z.05 = 1.645

11. Calculating z
Remember

12. Conclusion Since 2.5 > 1.645 The p-value:
It can be concluded that the average starting salary for secretaries in Chicago is at least $5 per week greater than the average starting salary in St. Louis.
The p-value:
The area above z= 2.5 on the normal curve = = .0062
Since is low (compared to α), it can be concluded that the average starting salary for secretaries in Chicago is at least $5 per week greater than the average starting salary in St. Louis.

13. EXAMPLE Confidence Interval: 1, 2 Known
Construct a 95% confidence for the difference in average between weekly starting salaries for secretaries in Chicago and St. Louis.
$10 ± $3.92
$6.08 ↔ $13.92

14. z-test: Two Sample for Means
Excel Approach
Suppose, as shown on the next slide the data for Chicago is given in column A (A2:A101) and the data for St. Louis is given in column B (B2:B76).
The analysis can be done using an entry from the Data Analysis Menu:
z-test: Two Sample for Means

15. z-Test: Two Sample For Means
Go to Tools
Data Analysis
Go to Tools
Data Analysis
Select
z-Test: Two Sample For Means

16. For 1-tail tests, input columns so that the test is a “>” test.
Enter
Hypothesized Difference
Enter Variances
Not
Standard Deviations
Check
Labels
Enter
Beginning Cell
For Output

17. p-value for “>” test

18. =(E4-F4)-NORMSINV(0.975)*SQRT(E5/E6+F5/F6)
Highlight formula in cell E15—press F4.
Drag to cell E16 and change “-” to “+”.

19. Estimating Sample Sizes
Usual Assumptions:
Same sample size from each pop.: n1 = n2 = n
Standard deviations, s1, s2 known
Calculate n from the “±” part of the confidence interval for known 1and 2

20. Example
How many workers would have to be surveyed in Chicago and St. Louis to estimate the true average difference in starting weekly salary to within $3?

21. Review Mean and standard deviation for X1 -X2
Assumptions for tests and confidence intervals
z-tests for differences in means when 1 and 2 are known:
By Formula
By Excel data analysis tool
Confidence intervals for differences in means when 1 and 2 are known:
Estimating Sample Sizes