Two-Sample Tests for Population Means
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Two-Sample Tests for Population Means |
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Introduction
The aim of these tests is to compare two population means, 
and 
, to the value, 
, referred here to as the hypothetical difference between the population means, based on two independent random samples, or two naturally paired random samples, 
and 
, of size 
and 
, respectively, from the corresponding populations.
Methods
The following methods assume two independent random samples, or two naturally paired random samples, and , of size and , respectively, drawn from two normally distributed populations, 
and 
.
| 1. | z-Procedure for Known Population Variances |
When the population variances, 
and 
, are known, the test statistic
, (1)
where 
.
| 2. | z-Procedure for Unknown Population Variances (Assumes Large and ) |
When the population variances, and , are not known, if the samples sizes, and , are large, we can substitute and of Eq. (1) with the unbiased sample variances, 
and 
, and say that the test statistic
, 
large. (2)
Refer to the one-sample z-Procedure for Unknown Population Variance for additional information on this subject.
t-Procedures for Unknown Population Variances
When the population variances, and , are not known and the size of the independent samples, and , is small, the unbiased sample variances, and , are not accurate enough to carry the test as described by Eq. (2). In this case, a more complex statistic, namely Student's t, is used to approximate the distribution of the difference between the means of the samples.
| 3. | Population Variances Assumed Equal (Pooled Test) |
If the population variances, and , are known to be equal, the best estimate of the pooled variance of and is given by
.
If we use this expression to replace, 
, of Eq. (2), the result is a statistic with a Student's t-distribution of 
degrees of freedom:
.
| 4. | Population Variances Assumed not Equal (Nonpooled Test) |
If the population variances, and , are known not to be equal, we can compute a similar t-deviate,
,
with 
degrees of freedom, calculated from the sample as
,
rounded down to the nearest integer.
 | When no information is available on the equality of the two population variances involved, one can resort to carrying a two-sample F test for variances based on the two independent samples at hand, so that we may be able to have a base from which to select one of the two methods just described. |
This test is a little different than the ones previously discussed. It is used to compare the means of two populations when the population members are naturally paired (e.g., mother and child, husband and wife, etc.), or to compare the effects of two different versions of an event over the same sample (e.g., the scores of two different versions of an exam over a same group of students).
In contrast to the previous methods, where the test statistic is calculated from the difference of the means of the samples, this test's statistic is calculated from the mean of the differences of the paired observations, 
, and its corresponding variance, 
, as follows:
,
where 
,
.
The resulting test statistic, 
, has a Student's t-distribution with 
degrees of freedom.
Hypotheses
The null hypothesis takes the form 
, with the following alternative hypotheses:
| § |  for two-tailed tests, |
| § |  for left-tailed tests, and |
| § |  for right-tailed tests. |
Refer to Alternative Hypothesis for additional information of this subject.
Output Formats
Two-Sample z-Test for Means (Known Variances)
Two-Sample: |
<The Two-Sample Number> |
|
Sample: |
1 |
2 |
Observations: |
<Observations in Sample 1> |
<Observations in Sample 2> |
Hypothetical Difference: |
<The Hypothetical Difference> |
|
Significance Level: |
<The Significance Level> |
|
Mean: |
<The Mean of Sample 1> |
<The Mean of Sample 2> |
Population Variance: |
<The Population Variance> |
|
z: |
<z> |
|
One-Tailed P(|z|): |
<The One-Tailed P-Value of |z| > |
|
One-Tailed z Critical (+/-): |
<The One-Tailed Critical Point> |
|
Left-Tailed Test: |
"[Do not ]Reject Ho" |
|
Right-Tailed Test: |
"[Do not ]Reject Ho" |
|
Two-Tailed 2x P(|z|): |
<The Two-Tailed P-Value of |z|, doubled> |
|
Two-Tailed z Critical (+/-): |
<The Two-Tailed Critical Point> |
|
Two-Tailed Test: |
"[Do not ]Reject Ho" |
|
Two-Sample z-Test for Means (Unknown Variances)
Two-Sample: |
<The Two-Sample Number> |
|
Sample: |
1 |
2 |
Observations: |
<Observations in Sample 1> |
<Observations in Sample 2> |
Hypothetical Difference: |
<The Hypothetical Difference> |
|
Significance Level: |
<The Significance Level> |
|
Mean: |
<The Mean of Sample 1> |
<The Mean of Sample 2> |
Sample Variance: |
<The Variance of Sample 1> |
<The Variance of Sample 2> |
z: |
<z> |
|
One-Tailed P(|z|): |
<The One-Tailed P-Value of |z| > |
|
One-Tailed z Critical (+/-): |
<The One-Tailed Critical Point> |
|
Left-Tailed Test: |
"[Do not ]Reject Ho" |
|
Right-Tailed Test: |
"[Do not ]Reject Ho" |
|
Two-Tailed 2x P(|z|): |
<The Two-Tailed P-Value of |z|, doubled> |
|
Two-Tailed z Critical (+/-): |
<The Two-Tailed Critical Point> |
|
Two-Tailed Test: |
"[Do not ]Reject Ho" |
|
Two-Sample t-Test (Pooled) for Means
Two-Sample: |
<The Two-Sample Number> |
|
Sample: |
1 |
2 |
Observations: |
<Observations in Sample 1> |
<Observations in Sample 2> |
Hypothetical Difference: |
<The Hypothetical Difference> |
|
Significance Level: |
<The Significance Level> |
|
Mean: |
<The Mean of Sample 1> |
<The Mean of Sample 2> |
Sample Variance: |
<The Variance of Sample 1> |
<The Variance of Sample 2> |
Pooled Variance: |
<The Pooled Variance> |
|
t: |
<t> |
|
Degrees of Freedom: |
<Degrees of Freedom> |
|
One-Tailed P(|t|): |
<The One-Tailed P-Value of |t| > |
|
One-Tailed t Critical (+/-): |
<The One-Tailed Critical Point> |
|
Left-Tailed Test: |
"[Do not ]Reject Ho" |
|
Right-Tailed Test: |
"[Do not ]Reject Ho" |
|
Two-Tailed 2x P(|t|): |
<The Two-Tailed P-Value of |t|, doubled> |
|
Two-Tailed t Critical (+/-): |
<The Two-Tailed Critical Point> |
|
Two-Tailed Test: |
"[Do not ]Reject Ho" |
|
Two-Sample t-Test (Unpooled) for Means
Two-Sample: |
<The Two-Sample Number> |
|
Sample: |
1 |
2 |
Observations: |
<Observations in Sample 1> |
<Observations in Sample 2> |
Hypothetical Difference: |
<The Hypothetical Difference> |
|
Significance Level: |
<The Significance Level> |
|
Mean: |
<The Mean of Sample 1> |
<The Mean of Sample 2> |
Sample Variance: |
<The Variance of Sample 1> |
<The Variance of Sample 2> |
t: |
<t> |
|
Degrees of Freedom: |
<Degrees of Freedom> |
|
One-Tailed P(|t|): |
<The One-Tailed P-Value of |t| > |
|
One-Tailed t Critical (+/-): |
<The One-Tailed Critical Point> |
|
Left-Tailed Test: |
"[Do not ]Reject Ho" |
|
Right-Tailed Test: |
"[Do not ]Reject Ho" |
|
Two-Tailed 2x P(|t|): |
<The Two-Tailed P-Value of |t|, doubled> |
|
Two-Tailed t Critical (+/-): |
<The Two-Tailed Critical Point> |
|
Two-Tailed Test: |
"[Do not ]Reject Ho" |
|
Two-Sample t-Test (Paired) for Means
Two-Sample: |
<The Two-Sample Number> |
|
Sample: |
1 |
2 |
Observations: |
<Observations in Sample 1> |
<Observations in Sample 2> |
Hypothetical Difference: |
<The Hypothetical Difference> |
|
Significance Level: |
<The Significance Level> |
|
Mean: |
<The Mean of Sample 1> |
<The Mean of Sample 2> |
Sample Variance: |
<The Variance of Sample 1> |
<The Variance of Sample 2> |
Differences Mean: |
<The Mean of the Paired Differences> |
|
Differences Variance: |
<The Variance of the Paired Differences> |
|
Pearson's Correlation: |
<Pearson's Correlation> |
|
t: |
<t> |
|
Degrees of Freedom: |
<Degrees of Freedom> |
|
One-Tailed P(|t|): |
<The One-Tailed P-Value of |t| > |
|
One-Tailed t Critical (+/-): |
<The One-Tailed Critical Point> |
|
Left-Tailed Test: |
"[Do not ]Reject Ho" |
|
Right-Tailed Test: |
"[Do not ]Reject Ho" |
|
Two-Tailed 2x P(|t|): |
<The Two-Tailed P-Value of |t|, doubled> |
|
Two-Tailed t Critical (+/-): |
<The Two-Tailed Critical Point> |
|
Two-Tailed Test: |
"[Do not ]Reject Ho" |
|