Independent Two-Sample t-Statistic

An Independent Two-Sample t-Statistic is a t-statistic used in a independent two-sample t-tests to measure the difference between sample means or population means.

• AKA: Unpaired Two-Sample t-Statistic, Pooled Variance t-Test Statistic.
• Context:
  • It is defined as [math]\displaystyle{ t = \frac{(\overline{x}_1 - \overline{x}_2) - d_0}{s_p \sqrt{1/n_1+1/n_2}} }[/math] where [math]\displaystyle{ \overline{X_1} }[/math] is the sample mean of sample drawn from population 1,[math]\displaystyle{ \overline{x_1} }[/math] is the sample mean of sample drawn from population 1, [math]\displaystyle{ d_0 }[/math] is a constant (hypothesized difference between population means), [math]\displaystyle{ s_p = \sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}} }[/math] is pooled standard deviation, [math]\displaystyle{ n_1 }[/math] and [math]\displaystyle{ n_2 }[/math] are the sample sizes, [math]\displaystyle{ s_2 }[/math] are the standard deviation for each sample. .
• Example(s)
  • An unpaired two-sample t-statistic where the population means are equal (i.e. [math]\displaystyle{ d_0=0 }[/math]): [math]\displaystyle{ t = \frac{\overline{x}_1 - \overline{x}_2}{s_p \sqrt{1/n_1+1/n_2}} }[/math]
  • An unpaired two-sample t-statistic where the sample sizes are the same (i.e. [math]\displaystyle{ n_1=n_2=n }[/math]): [math]\displaystyle{ t = \frac{(\overline{x}_1 - \overline{x}_2) - d_0}{\sqrt{(s_1^2+s_2^2)/n}} }[/math].
  • …
• Counter-Example(s):
  • One-Sample t-Statistic.
  • Matched-pair t-Statistic.
• See: Test Statistic, Comparison of Means Test, Parametric Statistical Test.

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References

2017a

• (Wikipedia, 2017) ⇒ http://en.wikipedia.org/wiki/Student%27s_t-test#Independent_two-sample_t-test
  • Given two groups (1, 2), this test is only applicable when:

• the two sample sizes (that is, the number, n, of participants of each group) are equal;
• it can be assumed that the two distributions have the same variance;

Violations of these assumptions are discussed below.

The t statistic to test whether the means are different can be calculated as follows:

[math]\displaystyle{ t = \frac{\bar {X}_1 - \bar{X}_2}{s_p \sqrt{2/n}} }[/math]

where

[math]\displaystyle{ \ s_p = \sqrt{\frac{s_{X_1}^2+s_{X_2}^2}{2}} }[/math]

Here [math]\displaystyle{ s_p }[/math] is the pooled standard deviation for n=n₁=n₂ and [math]\displaystyle{ s_{X_1}^2 }[/math] and [math]\displaystyle{ s_{X_2}^2 }[/math] are the unbiased estimators of the variances of the two samples. The denominator of t is the standard error of the difference between two means.

For significance testing, the degrees of freedom for this test is 2n − 2 where n is the number of participants in each group.

2017b

• (Stattrek, 2017) ⇒ http://stattrek.com/hypothesis-test/difference-in-means.aspx?Tutorial=AP
  • This lesson explains how to conduct a hypothesis test for the difference between two means. The test procedure, called the two-sample t-test, is appropriate when the following conditions are met:
    • The sampling method for each sample is simple random sampling.
    • The samples are independent.
    • Each population is at least 20 times larger than its respective sample.
    • The sampling distribution is approximately normal, which is generally the case if any of the following conditions apply.
      • The population distribution is normal.
      • The population data are symmetric, unimodal, without outliers, and the sample size is 15 or less.
      • The population data are slightly skewed, unimodal, without outliers, and the sample size is 16 to 40.
      • The sample size is greater than 40, without outliers.

2017c

• (QCP Glossary, 2017) ⇒ https://www.quality-control-plan.com/StatGuide/ttest_unpaired.htm
  • The two-sample unpaired t test is used to test the null hypothesis that the two population means corresponding to the two random samples are equal.

Assumptions:

• Within each sample, the values are independent, and identically normally distributed (same mean and variance).
• The two samples are independent of each other.
• For the usual two-sample t test, the two different samples are assumed to come from populations with the same variance, allowing for a pooled estimate of the variance. However, if the two sample variances are clearly different, a variant test, the Welch-Satterthwaite t test, is used to test whether the means are different.

2017D

• http://www.evanmiller.org/ab-testing/t-test.html Does the average value differ across two groups?

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