1. Inferences On Two Samples

2. Overview
We continue with confidence intervals and hypothesis testing for more advanced models
Models comparing two means
When the two means are dependent
When the two means are independent
Models comparing two proportions

3. Inference about Two Means: Dependent/paired Samples

4. Learning Objectives
Distinguish between independent and dependent sampling
Test hypotheses made regarding matched-pairs data
Construct and interpret confidence intervals about the population mean difference of matched-pairs data

5. Two populations
So far, we have covered a variety of models dealing with one population
The mean parameter for one population
The proportion parameter for one population
However, there are many real-world applications that need techniques to compare two populations

6. Examples Examples of situations with two populations
We want to test whether a certain treatment helps or not … the measurements are the “before” measurement and the “after” measurement
We want to test the effectiveness of Drug A versus Drug B … we give 40 patients Drug A and 40 patients Drug B … the measurements are the Drug A and Drug B responses

7. Dependent Sample
In certain cases, the two samples are very closely tied to each other
A dependent sample is one when each individual in the first sample is directly matched to one individual in the second
Examples
Before and after measurements (a specific person’s before and the same person’s after)
Experiments on identical twins (twins matched with each other

8. Independent Sample
On the other extreme, the two samples can be completely independent of each other
An independent sample is when individuals selected for one sample have no relationship to the individuals selected for the other
Examples
Fifty samples from one factory compared to fifty samples from another
Two hundred patients divided at random into two groups of one hundred

9. Paired Samples The dependent samples are often called matched-pairs
Matched-pairs is an appropriate term because each observation in sample 1 is matched to exactly one in sample 2
The person before  the person after
One twin  the other twin
An experiment done on a person’s left eye  the same experiment done on that person’s right eye

10. Test hypotheses made regarding matched-pairs sample

11. Analysis of Paired Samples
The method to analyze matched-pairs is to combine the pair into one measurement
“Before” and “After” measurements – subtract the before from the after to get a single “change” measurement
“Twin 1” and “Twin 2” measurements – subtract the 1 from the 2 to get a single “difference between twins” measurement
“Left eye” and “Right eye” measurements – subtract the left from the right to get a single “difference between eyes” measurement

12. Compute Difference d Specifically, for the before and after example,
d1 = person 1’s after – person 1’s before
d2 = person 2’s after – person 1’s before
d3 = person 3’s after – person 1’s before
This creates a new random variable d
We would like to reformulate our problem into a problem involving d (just one variable)

13. Test for the True Difference μd
How do our hypotheses translate?
The two means are equal -> the mean difference is zero -> μd = 0
The two means are unequal -> the mean difference is non-zero -> μd ≠ 0
Thus our hypothesis test is
H0: μd = 0
H1: μd ≠ 0
The standard deviation σd is unknown
We know how to do this!

14. Test for the True Difference
To solve
H0: μd = 0
H1: μd ≠ 0
The standard deviation σd is unknown
This is exactly the test of one population mean with the standard deviation being unknown
This is exactly the subject covered in Unit 8

15. Assumptions
In order for this test statistic to be used, the data must meet certain conditions
The sample is obtained using simple random sampling
The sample data are matched pairs
The differences are normally distributed, or the sample size (the number of pairs, n) is at least 30
These are the usual conditions we need to make our Student’s t calculations

16. Example
An example … whether our treatment helps or not … helps meaning a higher measurement
The “Before” and “After” results
Before
After
Difference
7.2
8.6
1.4
6.6
7.7
1.1
6.5
6.2
– 0.3
5.5
5.9
0.4
1.8

17. Example (continued) Hypotheses Calculations H0: μd = 0 … no difference
H1: μd > 0 … helps
(We’re only interested in if our treatment makes things better or not)
α = 0.01
Calculations
n = 5 (i.e. 5 pairs)
= .88 (mean of the paired-difference)
sd = .83

18. Example (continued) Calculations The test statistic is
sd = 0.83
The test statistic is
This has a Student’s t-distribution with 4 degrees of freedom

19. Example (continued)
Use the Student’s t-distribution with 4 degrees of freedom
The right-tailed α = 0.01 critical value is 3.75 (i.e. t0.01;4 d.f. = 3.75)
2.36 is less than 3.75 (the classical method)
Thus we do not reject the null hypothesis
There is insufficient evidence to conclude that our method significantly improves the situation
We could also have used the P-Value method. P value is (note: tcdf(2.36, E99, 4) = 0.039)

20. Example (continued)
Matched-pairs tests have the same various versions of hypothesis tests
Two-tailed tests
Left-tailed tests (the alternatively hypothesis that the first mean is less than the second)
Right-tailed tests (the alternatively hypothesis that the first mean is greater than the second)
Each can be solved using the Student’s t

21. Classical and P-value Approaches
Each of the types of tests can be solved using either the classical or the P-value approach

22. Summary of the Method A summary of the method
For each matched pair, subtract the first observation from the second
This results in one data item per subject with the data items independent of each other
Test that the mean of these differences is equal to 0
Conclusions
Do not reject that μd = 0
Reject that μd = Reject that the two populations have the same mean

23. Construct and interpret confidence intervals about the population mean difference of matched-pairs data

24. Confidence Interval for the Paired Difference
We’ve turned the matched-pairs problem in one for a single variable’s mean / unknown standard deviation
We just did hypothesis tests
We can use the techniques taught in Unit 7 (again, single variable’s mean / unknown standard deviation) to construct confidence intervals
The idea – the processes (but maybe not the specific calculations) are very similar for all the different models

25. Confidence Interval for the Paired Difference
Confidence intervals are of the form
Point estimate ± margin of error
This is precisely an application of our results for a population mean / unknown standard deviation
The point estimate d
and the margin of error
for a two-tailed test

26. Confidence Interval for the Paired Difference
Thus a (1 – α) • 100% confidence interval for the difference of two means, in the matched-pair case, is
where tα/2 is the critical value of the Student’s t-distribution with n – 1 degrees of freedom

27. Example
Salt-free diets are often prescribed for people with high blood pressure. The following data was obtained from an experiment designed to estimate the reduction in diastolic blood pressure as a result of following a salt-free diet for two weeks. Assume diastolic readings to be normally distributed.
Find a 99% confidence interval for the mean reduction

28. Example (continued)
1. Population Parameter of Interest The mean reduction (difference) in diastolic blood pressure
2. The Confidence Interval Criteria
a. Assumptions: Both sample populations are assumed normal
b. Test statistic: t with df = = 7
c. Confidence level: 1 - a = 0.99
3. Sample evidence
Sample information:

29. Example 4. The Confidence Interval
a. Confidence coefficients: Two-tailed situation, a/2 = t(df, a/2) = t(7, 0.005) = 3.50
b. Maximum error:
c. Confidence limits:
5. The Results
to is the 99% confidence interval estimate for the amount of reduction of diastolic blood pressure, md..

30. Summary
Two sets of data are dependent, or matched-pairs, when each observation in one is matched directly with one observation in the other
In this case, the differences of observation values should be used
The hypothesis test and confidence interval for the difference is a “mean with unknown standard deviation” problem, one which we already know how to solve

31. Inference about Two Means: Independent Samples

32. Learning Objectives
Test hypotheses regarding the difference of two independent means
Construct and interpret confidence intervals regarding the difference of two independent means

33. Independent Samples
Two samples are independent if the values in one have no relation to the values in the other
Examples of not independent
Data from male students versus data from business majors (an overlap in populations)
The mean amount of rain, per day, reported in two weather stations in neighboring towns (likely to rain in both places)

34. Independent Samples
A typical example of an independent samples test is to test whether a new drug, Drug N, lowers cholesterol levels more than the current drug, Drug C
A group of 100 patients could be chosen
The group could be divided into two groups of 50 using a random method
If we use a random method (such as a simple random sample of 50 out of the 100 patients), then the two groups would be independent

35. Test of Two Independent Samples
The test of two independent samples is very similar, in process, to the test of a single population mean
The only major difference is that a different test statistic is used
We will discuss the new test statistic through an analogy with the hypothesis test of one mean

36. Test hypotheses regarding the difference of two independent means

37. Test Statistic for a Single Mean
For the test of one mean, we have the variables
The hypothesized mean (μ)
The sample size (n)
The sample mean (x)
The sample standard deviation (s)
We expect that x would be close to μ

38. Test statistic for the Difference of Two Means
In the test of two means, we have two values for each variable – one for each of the two samples
The two hypothesized means μ1 and μ2
The two sample sizes n1 and n2
The two sample means x1 and x2
The two sample standard deviations s1 and s2
We expect that x1 – x2 would be close to μ1 – μ2

39. Standard Error of the Test Statistic for a Single Mean
For the test of one mean, to measure the deviation from the null hypothesis, it is logical to take
x – μ
which has a standard deviation/standard error of approximately

40. Standard Error of the Test Statistic for the Difference of Two Means
For the test of two means, to measure the deviation from the null hypothesis, it is logical to take
(x1 – x2) – (μ1 – μ2)
which has a standard deviation/standard error of approximately

41. t -Test Statistic for a Single Mean
For the test of one mean, under certain appropriate conditions, the difference
x – μ
is Student’s t with mean 0, and the test statistic
has Student’s t-distribution with n – 1 degrees of freedom

42. t - Test Statistic for the Difference of Two Means
Thus for the test of two means, under certain appropriate conditions, the difference
(x1 – x2) – (μ1 – μ2)
is approximately Student’s t with mean 0, and the test statistic
has an approximate Student’s t-distribution

43. Distribution of the t-statistic
This is Welch’s approximation, that
has approximately a Student’s t-distribution
The degrees of freedom is the smaller of
n1 – 1 and n2 – 1
Note: Some computer or calculator calculates the degrees of freedom for this
t test statistic with a somewhat complicated formula. But, we’ll use the smaller
of n1 – 1 and n2 – 1 as the degrees of freedom.

44. A Special Case
For the particular case where be believe that the two population means are equal, or μ1 = μ2, and the two sample sizes are equal, or n1 = n2, then the test statistic becomes
with n – 1 degrees of freedom, where n = n1 = n2

45. General Test Procedure
Now for the overall structure of the test
Set up the hypotheses
Select the level of significance α
Compute the test statistic
Compare the test statistic with the appropriate critical values
Reach a do not reject or reject the null hypothesis conclusion

46. Assumptions
In order for this method to be used, the data must meet certain conditions
Both samples are obtained using simple random sampling
The samples are independent
The populations are normally distributed, or the sample sizes are large (both n1 and n2 are at least 30)
These are the usual conditions we need to make our Student’s t calculations

47. State Hypotheses & level of significance
State our two-tailed, left-tailed, or right-tailed hypotheses
State our level of significance α, often 0.10, 0.05, or 0.01

48. Compute the Test Statistic
and the degrees of freedom, the smaller of n1 – 1 and n2 – 1
Compute the critical values (for the two-tailed, left-tailed, or right-tailed test

49. Make a Statistical Decision
Each of the types of tests can be solved using either the classical or the P-value approach
Based on either of these methods, do not reject or reject the null hypothesis

50. Example We have two independent samples We test versus
The first sample of n = 40 items has a sample mean of 7.8 and a sample standard deviation of 3.3
The second sample of n = 50 items has a sample mean of 11.6 and a sample standard deviation of 2.6
We believe that the mean of the second population is exactly 4.0 larger than the mean of the first population
We use a level of significance α = .05
We test versus

51. Example (continued) The test statistic is
This has a Student’s t-distribution with 39 degrees of freedom
The two-tailed critical value is -2.02, so we do not reject the null hypothesis (notice: invT(.025,39) = or use a t-table)
Or, compute the p-value which is greater than 0.05 level of significance. (Notice that: 2*tcdf(-E99,-1.72,39) = 0.093)
We do not have sufficient evidence to state that the deviation from 4.0 is significant

52. Construct and interpret confidence intervals regarding the difference of two independent means

53. Confidence Interval of m1-m2
Confidence intervals are of the form
Point estimate ± margin of error
We can compare our confidence interval with the test statistic from our hypothesis test
The point estimate is x1 – x2
We use the denominator of the test statistic as the standard error
We use critical values from the Student’s t

54. Confidence Interval of m1-m2
Thus (1- a)100% confidence interval is
Point estimate ± margin of error
Standard error
Point estimate
where ta/2 has the degrees of freedom that is the smaller of n1-1 and n2-1 .

55. Example
A recent study reported the longest average workweeks for non-supervisory employees in private industry to be chef and construction
Find a 95% confidence interval for the difference in mean length of workweek between chef and construction. Assume normality for the sampled populations and that the samples were selected randomly.

56. Example
1. Parameter of interest The difference between the mean hours/week for chefs and the mean hours/week for construction workers, m1 - m2
2. The Confidence Interval Criteria
a. Assumptions: Both populations are assumed normal and the samples were random and independently selected
b. Test statistic: t with df = 11; the smaller of n1 - 1 = = 17 or n2 - 1 = = 11
c. Confidence level: 1 - a = 0.95
3. The Sample Evidence
Sample information given in the table
Point estimate for m1 - m2:

57. Example 4. The Confidence Interval a. Confidence coefficients:
t0.025, 11d.f.= 2.20
b. Margin of error:
c. Confidence limits:
4.1 – 3.77 = to = 7.87
5. The Results
0.33 to 7.87 is a 95% confidence interval for the difference in mean hours/week for chefs and construction workers. ( It also means that there is a significant difference between the mean hours/week for chefs and the mean hours/week for construction workers at 0.05 level of significance, since the interval does not contain zero.)

58. Summary
Two sets of data are independent when observations in one have no affect on observations in the other
In this case, the differences of the two means should be used in a Student’s t-test
The overall process, other than the formula for the standard error, are the general hypothesis test and confidence intervals process

59. Inference about Two Population Proportions

60. Learning Objectives
Test hypotheses regarding two population proportions
Construct and interpret confidence intervals for the difference between two population proportions

61. Test hypotheses regarding two population proportions

62. Inference about Two Proportions
This progression should not be a surprise
One mean and one proportion
Unit 7 – confidence intervals
Unit 8 – hypothesis tests
Two means
Unit 9 - hypothesis tests and confidence intervals
Now for two proportions …

63. Examples
We now compare two proportions, testing whether they are the same or not
Examples
The proportion of women (population one) who have a certain trait versus the proportion of men (population two) who have that same trait
The proportion of white sheep (population one) who have a certain characteristic versus the proportion of black sheep (population two) who have that same characteristic

64. Two Population Proportions
The test of two populations proportions is very similar, in process, to the test of one population proportion and the test of two population means
The only major difference is that a different test statistic is used
We will discuss the new test statistic through an analogy with the hypothesis test of one proportion

65. Test of One Proportion
For the test of one proportion, we had the variables of
The hypothesized population proportion (p0)
The sample size (n)
The number with the certain characteristic (x)
The sample proportion ( )
We expect that should be close to p0

66. Test of Two Proportions
In the test of two proportions, we have two values for each variable – one for each of the two samples
The two hypothesized proportions (p1 and p2)
The two sample sizes (n1 and n2)
The two numbers with the certain characteristic (x1 and x2)
The two sample proportions ( and )
We expect that should be close to p1 – p2

67. Test Statistic of One Proportion
For the test of one proportion, to measure the deviation from the null hypothesis, we took
which has a standard deviation of

68. Test Statistic of Two Proportions
For the test of two proportions, to measure the deviation from the null hypothesis, it is logical to take
which has a standard deviation of

69. Test Statistic for One Proportion
For the test of one proportion, under certain appropriate conditions, the difference
is approximately normal with mean 0, and the test statistic
has an approximate standard normal distribution

70. Test Statistic for Two Proportions
Thus for the test of two proportions, under certain appropriate conditions, the difference
is approximately normal with mean 0, and the test statistic
has an approximate standard normal distribution

71. Test Statistic for Equal Proportions
For the particular case where we believe that the two population proportions are equal, or p1 = p2 (i.e.
p1 – p2 = 0). Thus
and
Here, since two population proportions are the same under the null hypothesis, we use , an estimated common proportion for both p1 and p2, which is computed by combining two samples together to calculate an estimated common sample proportion. That is,

72. General Test Procedure
Now for the overall structure of the test
Set up the hypotheses
Select the level of significance α
Compute the test statistic
Compare the test statistic with the appropriate critical values
Reach a do not reject or reject the null hypothesis conclusion

73. Assumptions
In order for this method to be used, the data must meet certain conditions
Both samples are obtained independently using simple random sampling
Each sample size is large
These are the usual conditions we need to make our test of proportions calculations

74. Hypotheses and Level of Significance
State our two-tailed, left-tailed, or right-tailed hypotheses
State our level of significance α, often 0.10, 0.05, or 0.01

75. Test Statistic and Critical Values
Compute the test statistic
which has an approximate standard normal distribution
Compute the critical values (for the two-tailed, left-tailed, or right-tailed test)

76. Make Statistical Decision
Each of the types of tests can be solved using either the classical or the P-value approach
Based on either of these two methods, do not reject the null hypothesis

77. Example We have two independent samples
55 out of a random sample of 100 students at one university are commuters
80 out of a random sample of 200 students at another university are commuters
We wish to know of these two proportions are equal
We use a level of significance α = .05
Both samples sizes are large so our method can be used

78. Example (continued) The test statistic is Notice that
The critical values for a two-tailed test using the normal distribution are ± 1.96, thus we reject the null hypothesis
Or, we calculate P-value which is less than the 0.05 level of significance. ( Notice: 2*normalcdf(2.46,E99) = 0.014)
We conclude that the two proportions are significantly different

79. Confidence Interval of p1 – p2
Thus confidence intervals are
Point estimate ± margin of error
Standard error
Point estimate
Here, for calculating the standard error, we use separate estimates of the population proportions, instead of the common estimate

80. Example
A consumer group compared the reliability of two similar microcomputers from two different manufacturers. The proportion requiring service within the first year after purchase was determined for samples from each of two manufacturers.
Find a 98% confidence interval for p1 - p2, the difference in proportions needing service

81. Example (continued)
Population Parameter of Interest : The difference between the proportion of microcomputers needing service for manufacturer 1 and the proportion of microcomputers needing service for manufacturer 2, that is, p1- p2
Point estimate:
Confidence coefficients: z(a/2) = z(0.01) = 2.33 98 .
z(0.01)

82. Example (continued) Margin of error: Confidence limits:
0.06 – = to =
Results
to is a 98% confidence interval for the difference in proportions

83. Summary We can compare proportions from two independent samples
We use a formula with the combined sample sizes and proportions for the standard error
The overall process, other than the formula for the standard error, are the general hypothesis test and confidence intervals process

84. Inferences on Two Samples
Summary

85. Summary
The process of hypothesis testing is very similar across the testing of different parameters
The major steps in hypothesis testing are
Formulate the appropriate null and alternative hypotheses
Calculate the test statistic
Determine the appropriate critical value or values
Reach the reject / do not reject conclusions

86. Tests for Means and Proportions
Similarities in hypothesis test processes
Parameter
Mean (one population)
Two Means
(Independent)
(Dependent)
Two Proportions
H0:
μ = μ0
μ1 = μ2
p1 = p2
(2-tailed) H1:
μ ≠ μ0
μ1 ≠ μ2
p1 ≠ p2
(L-tailed) H1:
μ < μ0
μ1 < μ2
p1 < p2
(R-tailed) H1:
μ > μ0
μ1 > μ2
p1 > p2
Test statistic
Difference
Critical value
Normal
Student t

87. Summary
We can test whether sample data from two different samples supports a hypothesis claim about a population mean or proportion
For two population means, there are two cases
Dependent (or matched-pair) samples
Independent samples
All of these tests follow very similar processes, differing only in their test statistics and the distributions for their critical values