The two-sample t-test, also known as the independent samples t-test, is a statistical hypothesis test used to compare the means of two independent groups to determine if there is a significant difference between them. This test is appropriate when the data are continuous and follow a normal distribution.
For example, you might use a two-sample t-test to determine whether the average heights of men and women are significantly different.
The null hypothesis (H0) for the two-sample t-test states that there is no significant difference between the means of the two groups (μ1 = μ2). The alternative hypothesis (H1) states that there is a significant difference between the means (μ1 ≠ μ2).
To perform the two-sample t-test, follow these steps:
- Calculate the difference between the sample means: This is the observed difference between the means of the two groups.Difference = (M1 – M2)
where M1 and M2 are the sample means of groups 1 and 2, respectively.
- Calculate the pooled variance: The pooled variance is a weighted average of the sample variances from both groups. It is used to estimate the population variance.Pooled variance = [(n1 – 1) * s1² + (n2 – 1) * s2²] / (n1 + n2 – 2)
where s1² and s2² are the sample variances of groups 1 and 2, and n1 and n2 are the sample sizes of groups 1 and 2, respectively.
- Calculate the standard error: The standard error measures the variability of the difference between the two sample means.Standard error = sqrt[(s1² / n1) + (s2² / n2)]
- Calculate the test statistic: The test statistic is a t-score, which measures how many standard errors away the observed difference between the sample means is from the expected difference (under the null hypothesis).t = (Difference – 0) / Standard error
- Determine the critical value and p-value: Compare the test statistic (t-score) to a critical value from the t-distribution based on the desired significance level (usually 0.05) and degrees of freedom (calculated as n1 + n2 – 2). Calculate the p-value, which represents the probability of observing a test statistic as extreme as or more extreme than the one calculated if the null hypothesis were true.
- Make a decision: If the test statistic is greater than the critical value or the p-value is less than the significance level, reject the null hypothesis. This would suggest that there is a significant difference between the means of the two groups.
The two-sample t-test is a statistical method used to compare the means of two independent groups. It helps determine if there is a significant difference in the means based on the collected data.