1. Statistics Are Fun! Two-Sample Tests of Hypothesis
Chapter 11
Two-Sample Tests of Hypothesis
Statistics Is Fun!

2. Statistics Are Fun!
Goals
Conduct a test of hypothesis about the difference between two independent population means when both samples have population standard deviation that are known (z)
Conduct a test of hypothesis about the difference between two independent population means when both samples have population standard deviation that are not known (t)
Conduct a test of hypothesis about the difference between two population proportions (z)
Statistics Is Fun!

3. Compare The Means From Two Populations
Statistics Are Fun!
Compare The Means From Two Populations
Is there a difference in the mean number of defects produced on the day and the night shifts at Furniture Manufacturing Inc.?
Comparing two means from two different populations
Is there a difference in the proportion of males from urban areas and males from rural areas who suffer from high blood pressure?
Comparing two proportions from two different populations
Statistics Is Fun!

4. Two Populations Two Independent Populations
Statistics Are Fun!
Two Populations
Two Independent Populations
E-Trade index funds
Merrill Lynch index funds
Two random samples, two sample means
Mean rate of return for E-Trade index funds (10.4%)
Mean rate of return for Merrill Lynch index funds(11%)
Are the means different?
If they are different, is the difference due to chance (sampling error) or is it really a difference?
If they are the same, the difference between the two sample means should equal zero
“No difference”
Statistics Is Fun!

5. Two Populations Two Independent Populations
Statistics Are Fun!
Two Populations
Two Independent Populations
Plumbers in central Florida
Electricians in central Florida
Two samples, two sample means
Mean hourly wage rate for plumbers ($30)
Mean hourly wage rate for electricians ($29)
Are the means different?
If they are different, is the difference due to chance (sampling error) or is it really a difference?
If they are the same, the difference between the two sample means should equal zero
“No difference”
Statistics Is Fun!

6. Distribution Of Differences In The Sample Means
Statistics Are Fun!
Distribution Of Differences In The Sample Means
Statistics Is Fun!

7. Theory Of Two Sample Tests:
Statistics Are Fun!
Theory Of Two Sample Tests:
Take several pairs of samples
Compute the mean of each
Determine the difference between the sample means
Study the distribution of the differences in the sample means
If the mean of the distribution of differences is zero:
This implies that there is no difference between the two populations
If the mean of the distribution of differences is not equal to zero:
We conclude that the two populations do not have the same population parameter (example: mean or proportion)
Statistics Is Fun!

8. Theory Of Two Sample Tests:
Statistics Are Fun!
Theory Of Two Sample Tests:
If the sample means from the two populations are equal:
The mean of the distribution of differences should be zero
If the sample means from the two populations are not equal:
The mean of the distribution of differences should be either:
Greater than zero or
Less than zero
Statistics Is Fun!

9. Normal Distributions Remember from chapter 9:
Statistics Are Fun!
Normal Distributions
Remember from chapter 9:
Distribution of sample means tend to approximate the normal distribution when n ≥ 30
For independent populations, it can be shown mathematically that:
Distribution of the difference between two normal distributions is also normal
The standard deviation of the distribution of the difference is the sum of the two individual standard deviations
Statistics Is Fun!

10. Statistics Are Fun!
Test Of Hypothesis About The Difference Between Two Independent Population Means (population standard deviation known)
Same five steps:
Step 1: State null and alternate hypotheses
Step 2: Select a level of significance
Step 3: Identify the test statistic (z) and draw
Step 4: Formulate a decision rule
Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses
Fail to reject null
Reject null and accept alternate
Assumptions & Formulas 
Statistics Is Fun!

11. Distribution of Differences
Statistics Are Fun!
Formulas Two Independent Pop. Means (population standard deviation known)
Assumptions:
Two populations must be independent (unrelated)
Population standard deviation known for both
Both distributions are Normally distributed
Standard Deviation
of the
Distribution of Differences
Statistics Is Fun!

12. Example 1: Comparing Two Populations
Statistics Are Fun!
Example 1: Comparing Two Populations
Two cities, Bradford and Kane are separated only by the Conewango River
The local paper recently reported that the mean household income in Bradford is $38,000 from a sample of 40 households. The population standard deviation (past data) is $6,000.
The same article reported the mean income in Kane is $35,000 from a sample of 35 households. The population standard deviation (past data) is $7,000.
At the .01 significance level can we conclude the mean income in Bradford is more?
Statistics Is Fun!

13. Example 1: Comparing Two Populations
Statistics Are Fun!
Example 1: Comparing Two Populations
We wish to know whether the distribution of the differences in sample means has a mean of 0
The samples are from independent populations
Both population standard deviations are known
Statistics Is Fun!

14. Example 1: Comparing Two Populations
Statistics Are Fun!
Example 1: Comparing Two Populations
Step 1: State null and alternate hypotheses
H0: µB ≤ µK , or µB = µK
H1: µB > µK
Step 2: Select a level of significance
 = .01 (one-tail test to the right)
Step 3: Identify the test statistic and draw
Both pop. SD known, we can use z as the test statistic
Critical value  .01 yields .49 area,  z = 2.33
Statistics Is Fun!

15. Example 1: Comparing Two Populations
Statistics Are Fun!
Example 1: Comparing Two Populations
Step 4: Formulate a decision rule
If z is greater than 2.33, we reject H0 and accept H1, otherwise we fail to reject H0
Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses
Statistics Is Fun!

16. Example 1: Comparing Two Populations
Statistics Are Fun!
Example 1: Comparing Two Populations
Because 1.98 is not greater than 2.33, we fail to reject H0
We can not conclude that the mean income in Bradford is more than the mean income in Kane
The p-value (table method) is: P(z ≥ 1.98) = = .0239
This is more area under the curve associated with z-score of 2.33 than for alpha
We conclude from the p-value that H0 should not be rejected
However, there is some evidence that H0 is not true
Statistics Is Fun!

17. Statistics Are Fun!
Test Of Hypothesis About The Difference Between Two Independent Population Means (population standard deviation not known)
Same five steps:
Step 1: State null and alternate hypotheses
Step 2: Select a level of significance
Step 3: Identify the test statistic (t) and draw
Step 4: Formulate a decision rule
Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses
Fail to reject null
Reject null and accept alternate
Assumptions & Two Formulas 
Statistics Is Fun!

18. Assumptions necessary:
Statistics Are Fun!
Assumptions necessary:
Sample populations must follow the normal distribution
Two samples must be from independent (unrelated) populations
The variances & standard deviations of the two populations are equal
Two Formulas
Pooled variance
T test statistic
Statistics Is Fun!

19. Statistics Are Fun!
Formulas Two Independent Population Means (population standard deviation not known)
Degrees of Freedom = 2
Statistics Is Fun!

20. Statistics Are Fun!
Pooled Variance
The two sample variances are pooled to form a single estimate of the unknown population variance
A weighted mean of the two sample variances
The weights are the degrees of freedom that form each sample
Why pool?
Because if we assume the population variances are equal, the best estimate will come from a weighted mean of the two variances from the two samples
Statistics Is Fun!

21. Example 2: Comparing Two Populations
Statistics Are Fun!
Example 2: Comparing Two Populations
A recent EPA study compared the highway fuel economy of domestic and imported passenger cars
A sample of 15 domestic cars revealed a mean of 33.7 MPG with a standard deviation of 2.4 MPG
A sample of 12 imported cars revealed a mean of 35.7 mpg with a standard deviation of 3.9
At the .05 significance level can the EPA conclude that the MPG is higher on the imported cars?
Assume:
Samples are independent
Population standard deviations are equal
Distributions for samples are normal
Statistics Is Fun!

22. Example 2: Comparing Two Populations
Statistics Are Fun!
Example 2: Comparing Two Populations
Step 1: State null and alternate hypotheses
H0: µD ≥ µI
H1: µD < µI
Step 2: Select a level of significance
 = .05 (One-tail test to the left)
Step 3: Identify the test statistic and draw
Pop. SD not known, so we use the t distribution
df = – 1 – 1 = 25
One-tail test with  = .05
Critical Value =
Statistics Is Fun!

23. Example 2: Comparing Two Populations
Statistics Are Fun!
Example 2: Comparing Two Populations
Step 4: Formulate a decision rule
If our t < , we reject H0 and accept H1, otherwise we fail to reject H0
Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses
We must make the calculations for:
Pooled Variance
t-value test statistic
Statistics Is Fun!

24. Example 2: Step 5: Compute
Statistics Are Fun!
Example 2: Step 5: Compute
Statistics Is Fun!

25. Example 2: Step 5: Conclude
Statistics Are Fun!
Example 2: Step 5: Conclude
Because in not less than our critical value of , we fail to reject H0
There is insufficient sample evidence to claim a higher MPG on the imported cars
The EPA cannot conclude that the MPG is higher on the imported cars
Statistics Is Fun!

26. Statistics Are Fun!
Proportion
The fraction, ratio, or percent indicating the part of the sample or the population having a particular trait of interest
Example:
A recent survey of Highline students indicated that 98 out of 100 surveyed thought that textbooks were too expensive
The sample proportion is
98/100
.98
98%
The sample proportion is our best estimate of our population proportion
Statistics Is Fun!

27. Two Sample Tests of Proportions
Statistics Are Fun!
Two Sample Tests of Proportions
We investigate whether two samples came from populations with an equal proportion of successes (U =  M)
Assumptions:
The two populations must be independent of each other
Experiment must pass all the binomial tests
Statistics Is Fun!

28. Statistics Are Fun!
Test Of Hypothesis About The Difference Between Two Population Proportions
Same five steps:
Step 1: State null and alternate hypotheses
Step 2: Select a level of significance
Step 3: Identify the test statistic (z) and draw
Step 4: Formulate a decision rule
Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses
Fail to reject null
Reject null and accept alternate
Two Formulas 
Statistics Is Fun!

29. Formulas (Two Sample Tests of Proportions)
Statistics Are Fun!
Formulas (Two Sample Tests of Proportions)
Statistics Is Fun!

30. Example 3: Two Sample Tests of Proportions
Statistics Are Fun!
Example 3: Two Sample Tests of Proportions
Are unmarried workers more likely to be absent from work than married workers (m < u )?
A sample of 250 married workers showed 22 missed more than 5 days last year
A sample of 300 unmarried workers showed 35 missed more than 5 days last year
 = .05
Assume all binominal tests are passed
Statistics Is Fun!

31. Example 3: Two Sample Tests of Proportions
Statistics Are Fun!
Example 3: Two Sample Tests of Proportions
Step 1: State null and alternate hypotheses
H0: m ≥ u
H1: m < u
Step 2: Select a level of significance
 = .05
Step 3: Identify the test statistic and draw
Because the binomial assumptions are met, we use the z standard normal distribution
 = .05  .45  z = (one-tail test to the left)
Step 4: Formulate a decision rule
If the test statistic is less than -1.65, we reject H0 and accept H1, otherwise, we fail to reject H0
Statistics Is Fun!

32. Example 3: Two Sample Tests of Proportions
Statistics Are Fun!
Example 3: Two Sample Tests of Proportions
Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses
Statistics Is Fun!

33. Example 3: Two Sample Tests of Proportions
Statistics Are Fun!
Example 3: Two Sample Tests of Proportions
Step 5: Conclude
Because -1.1 is not less than -1.65, we fail to reject H0
We cannot conclude that a higher proportion of unmarried workers miss more than 5 days in a year than do the married workers
The p-value (table method) is:
P(z > 1.10) = = .1457
.1457 > .05, thus: fail to reject H0
Statistics Is Fun!

34. Statistics Are Fun!
Goals
Understand the difference between dependent and independent samples
Conduct a test of hypothesis about the mean difference between paired or dependent observations
Statistics Is Fun!

35. Understand The Difference Between Dependent And Independent Samples
Statistics Are Fun!
Understand The Difference Between Dependent And Independent Samples
Independent samples are samples that are not related in any way
Dependent samples are samples that are paired or related in some fashion
Statistics Is Fun!

36. Dependent Samples The samples are paired or related in some fashion:
Statistics Are Fun!
Dependent Samples
The samples are paired or related in some fashion:
1st Measurement of item, 2nd measurement of item
If you wished to buy a car you would look at the same car at two (or more) different dealerships and compare the prices (1 price, 2 price)
Before & After
If you wished to measure the effectiveness of a new diet you would weigh the dieters at the start and at the finish of the program (1 weight, 2 weight)
Statistics Is Fun!

37. Distribution Of The Differences In The Paired Values
Statistics Are Fun!
Distribution Of The Differences In The Paired Values
Follows Normal Distribution
If the paired values are dependent, be sure to use the dependent formula, because it is a more accurate statistical test than the formula 11-3 in our textbook
Formula 11-7 helps to:
Reduce variation in the sampling distribution
Two kinds of variance (variation between 1st & 2nd categories for paired values, &, variation between values) are reduced to only one (variation between 1st & 2nd categories for paired values)
Reduced variation leads to smaller standard error, which leads to larger test statistic and greater chance of rejecting H0
DF will be smaller
Statistics Is Fun!

38. Make the following calculations when the samples are dependent:
Statistics Are Fun!
Test Of Hypothesis About The Mean Difference Between Paired Or Dependent Observations
Make the following calculations when the samples are dependent:
where d is the difference
where is the mean of the differences
is the standard deviation of the differences
n is the number of pairs (differences)
Statistics Is Fun!

39. Statistics Are Fun!
EXAMPLE 4
An independent testing agency is comparing the daily rental cost for renting a compact car from Hertz and Avis
At the .05 significance level can the testing agency conclude that there is a difference in the rental charged?
A random sample of eight cities revealed the following information
Statistics Is Fun!

40. EXAMPLE 4 continued Miami Seattle 41 46 39 50 Statistics Are Fun!
Statistics Is Fun!

41. EXAMPLE 4 continued Step 1: Step 2: Step 3: Step 4:  = .05
Statistics Are Fun!
EXAMPLE 4 continued
Step 1:
Step 2:
 = .05
Step 3:
Use t because n < 30, df = 7, critical value = 2.365
Step 4:
If t < or t > 2.365, H0 is rejected and H1 accepted, otherwise, we fail to reject H0
Statistics Is Fun!

42. Example 4 continued City Hertz Avis d d2 Atlanta 42 40 2 4
Statistics Are Fun!
Example 4 continued
City Hertz Avis d d2
Atlanta
Chicago
Cleveland
Denver
Honolulu
Kansas City
Miami
Seattle
Totals
Statistics Is Fun!

43. Statistics Are Fun!
Example 4 continued
Statistics Is Fun!

44. Example 4 continued Step 5:
Statistics Are Fun!
Example 4 continued
Step 5:
Because is less than the critical value, do not reject the null hypothesis
There is no difference in the mean amount charged by Hertz and Avis
Statistics Is Fun!

45. Statistics Are Fun!
Summarize Chapter 11
Conduct a test of hypothesis about the difference between two independent population means when both samples have 30 or more observations
Conduct a test of hypothesis about the difference between two independent population means when at least one sample has less than 30 observations
Conduct a test of hypothesis about the difference between two population proportions
Understand the difference between dependent and independent samples
Conduct a test of hypothesis about the mean difference between paired or dependent observations
Statistics Is Fun!