Independent Two-Sample t-Statistic

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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.



References

2017a

  • 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

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