1. Comparing Two Population Parameters
Equal Variance t-test for Means, Section
Unequal Variance t-test Means, Section 11.3

2. Chapter Objectives
Select and use the appropriate hypothesis test in comparing
Means of two independent samples
Continuous variables
Means of two dependent samples
Proportions of two independent samples
Discrete variables - count
Variances of two independent samples
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3. Differences between Means, Independent Samples
The two samples are independent
Selection of one sample is in no way affected by the selection of the other sample
We are interested in whether men and women with a college education and working full-time have the same annual earnings
Random variable is annual earnings; two populations
Draw a sample from each population; calculate the respective sample means
Observe
Why do we observe a difference between the sample means?
The samples are drawn from populations with different means, i.e., 1  2
The samples are drawn from populations with the same population means, but because of sampling error, we observe different sample means, 1 = 2
The test that we will develop and the sampling distribution assume that the two samples are independent. That is, the selection of one sample is in no way affected by the selection of the other sample.
We are interested in whether men and women with a college education and working full-time have the same annual earnings. The random variable is annual earnings. The two populations are not related to each other. Each is a random sample. We draw a sample from each population and we calculate the respective sample means and we observe:
The problem here is, why do we observe a difference between the sample means?
The samples are drawn from populations with different means, i.e., 12.
The samples are drawn from populations with the same population means, but because of sampling error, we observe different sample means.
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4. Sampling Distribution of
To answer this question, need to know how is distributed
Difference between sample means is a statistic
Need to know the characteristics of the sampling distribution of the difference between the sample means
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5. Sampling Distribution of
Let the random variable, X, follow a normal distribution in the populations or the sample sizes, n1 and n2, are both greater than 30
Imagine drawing all possible samples from each of the two populations
Form all possible pairs of differences in sample means from population 1 and population 2
The distribution of the differences between these pairs of sample means is the sampling distribution
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6. Characteristics of Sampling Distribution
Distribution is normal
X follows normal distribution in the populations or CLT
Mean of the distribution
Unbiased estimator
Standard error is
Sampling Distribution of the Difference in Sample Means, normally distributed
1 - 2 Z
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7. Assumptions about Population Variances
Two different scenarios arise in the comparison of the population means of independent samples
Variances of the underlying populations are either known to be or assumed to be equal
Use Equal (or pooled) variance t-test
Variances are not assumed to be equal
Use Unequal (or separate) variance t-test
Two different scenarios arise in the comparison of independent samples. In the first, the variances of the underlying populations are either known to be or assumed to be equal. This leads to the pooled (or equal) variance t-test. In the second scenario, the variances are not assumed to be equal. In this case, there is considerable discussion among statisticians about the preferred approach. We will use a separate (or unequal) variance t-test.
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8. Equal Variance t-Test for Differences in Means
Two-sided test
H0:1 - 2 = 0
There is no difference in the population means
H1:1 - 2  0
There is a significant difference in the population means
One-sided tests
H0:1 - 2 ≤ 0 H1:1 - 2 > 0
H0:1 - 2 ≥ 0 H1:1 - 2 < 0
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9. Under the Null Hypothesis
What do we expect to observe under the null hypothesis?
The center of the sampling distribution is zero
If we observe a large absolute difference between the sample means, then it is unlikely that the samples came from populations with equal means
Sampling Distribution of the Difference in Sample Means, under the null
normal
What does the sampling distribution of the difference in the sample means look like under the null hypothesis?
We are given the two sample means.
Where does the observed lie?
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10. Equal Variance t-Test for Differences in Means
We need to convert into a standardized variable
If we knew the population variances, the test statistic would be
Realistically, we do not know the population variances
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11. Equal Variance t-Test for Differences in Means
Pool both samples together
Use all information to produce a more reliable estimate of the unknown population variance
Weighted average of the sample variances
Weights are the respective degrees of freedom
Substitute for unknown population variances
Pooled variance estimate of unknown population variances
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12. Equal Variance t-Test for Differences in Means
Note: the degrees of freedom are n1 + n2 - 2
Compare the value of this t-test statistic with a critical t value
DR: if (Crit. Value t ≤ t-test statistic ≤ Crit. Value t) do not reject
Or use p-value approach
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13. Problem - Negative Income Tax
Would this welfare program cause the recipients to stop working?
An experiment in the 1960s was done to find out
Target population consisted of 10,000 low-income families in three New Jersey cities
From these families, 400 were chosen at random for the control group
Another 225 were chosen at random for the treatment group - and put on the negative income tax. All 625 families were followed for three years
The control families averaged 7,100 hours of paid work and their standard deviation was 3,900 hours
The treatment families averaged 6,200 hours and their standard deviation was 3,400 hours
Is the difference between the sample means statistically significant at ⍺ = 0.05?
In the 1960s, a “negative income tax” was proposed, supplementing low incomes instead of taxing them
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14. Problem - Negative Income Tax
H0: 1 - 2 = 0
The NIT has no effect on hours of work
H1: 1 - 2  0
The NIT has an effect on hours of work
Assume the population variances are equal
Calculate the pooled sample variance
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15. Problem - Negative Income Tax
Ask how far the observed difference in the sample means lies from the center of the sampling distribution (0) if the null hypothesis is true
Find the critical t value at ⍺ = 0.05
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16. Problem - Negative Income Tax
Degrees of freedom
n1 +n2 - 2 = = 623 =
Compare the t-test statistic with the critical value
2.90 (test statistic) > 1.96 (critical value)
Reject null hypothesis
There is a significant difference in mean hours worked between the control and the treatment group
Consumer theory would predict that as income increases the quantity demanded increases for normal goods. If we view leisure as a good, then we would expect people to demand more leisure as their income rises. Hence, their work hours should decrease as their leisure hours increase. Given consumer theory, a one-tail test would be appropriate here.
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17. Online Homework covers section 11.1-11.2
CengageNOW fifth assignment
Chapter 11:Intro to Two Samples
CengageNow sixth assignment
Chapter 11:Tests about Two Means
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18. Unequal Variance t-Test for Differences in Means
Assume variances of the two populations are not equal
How do we estimate the standard error of the difference in sample means?
Substitute sample variances
The Separate (Unequal) Variances Test
Next consider the situation in which the variances of the two populations are not assumed to be equal. In this case a modification of the previous test must be used. Instead of using
as an estimate of the common population variance 2, we substitute the sample variances for the respective unknown population variances. The appropriate test statistic is
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19. Unequal Variance t-Test for Differences in Means
Test Statistic
Use an approximation for the degrees of freedom
Behrens Fischer solution
Unlike the case in which we have equal variances, the exact distribution of t is difficult to derive. We use an approximation for the degrees of freedom (the Behrens Fischer problem)
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20. Calculating the Degrees of Freedom
Round df down to the nearest integer
Compare the test statistic with the critical t value using estimated df degrees of freedom
DR: if (Crit. Value t ≤ t-test statistic ≤ Crit. Value t) do not reject
Round v down to the nearest integer. We are approximating the distribution of t by a t distribution with v degrees of freedom. We can then proceed to compare the test statistic with the critical t value using v degrees of freedom.
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21. Two Possible Tests for Differences in Means, Independent Samples
Variances of the underlying populations are either known to be or assumed to be equal
Use Equal (or pooled) variance t-test
Variances are not assumed to be equal
Use Unequal (or separate) variance t-test
Can test whether population variances are equal
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22. P-value Approach with t - Distribution Tables
Calculate the p-value for the test statistic, 2.90
Look at the t table, infinite degrees of freedom
Find the table entries closest to the test statistic, 2.90
2.576 cuts off  = .005
It is the value closest to our test statistic
Test statistic is more extreme
So the p-value must be < .005 in one tail of the distribution or <.01 for both tails
Compare the p-value with 0.05
.01 (p-value) <.05 (level of significance)
Therefore we reject the null hypothesis
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23. T-table

24. t - Distribution for Infinite Degrees of Freedom
0.10
0.05
0.025
0.01
1.28
1.64
1.96
2.33
0.005
2.58
Find appropriate distribution in terms of degrees of freedom
Find boundaries for your particular test statistic, cuts off of t values 2.90 > cut off less than of t values, p-value < 0.005
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25. T Distribution for Infinite Degrees of Freedom
0.10
0.05
0.025
0.01
1.28
1.64
1.96
2.33
0.005
2.58
2.20
Suppose test statistics is < 2.20 < < area in tail < = p-value for one-tail test 0.02 < p-value < 0.05 for two-tail test - multiply area in tail by two
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