A t-test is a type of inferential statistical test used to determine if there is a significant difference between the means of two groups, which may be related in certain features. It is most commonly applied when the data sets are normally distributed, and when the sample sizes are small (generally n < 30), or when the population standard deviation is unknown. Here's a detailed look into the t-test:
History
The t-test was developed by William Sealy Gosset, who worked for the Guinness brewery in Dublin, Ireland. Under the pseudonym "Student," Gosset published the theory of the t-test in 1908 in the journal Biometrika. His work was initially intended to help brewery workers make decisions about the quality of their product. The test is often referred to as "Student's t-test" due to this pseudonym.
- Independent Samples t-test: Used when comparing the means of two separate groups to see if they come from populations with different means. This is also known as the unpaired or two-sample t-test.
- Paired Sample t-test: Applied when the same subjects are measured twice, allowing for comparison of means from the same group over time or under different conditions.
- One Sample t-test: Used to determine whether the mean of a sample is significantly different from a known or hypothesized population mean.
Assumptions
- The data should be continuous.
- The data are independent.
- The variances of the two groups are equal (homogeneity of variance assumption), although there are versions of the t-test that can handle unequal variances (Welch's t-test).
- The distribution of the data is approximately normal, especially for small sample sizes.
Procedure
The t-test calculates a t-statistic by taking the difference between the means of two groups and dividing it by a measure of the variability of the groups:
t = (X̄₁ - X̄₂) / √[(s₁²/n₁) + (s₂²/n₂)]
X̄₁, X̄₂ are the sample means of the two groups.s₁², s₂² are the sample variances.n₁, n₂ are the sample sizes.
The resulting t-value is then compared against a critical value from the t-distribution with the appropriate degrees of freedom to determine significance.
Application
The t-test is widely used in:
- Medicine for comparing treatment effects.
- Business to compare customer satisfaction scores between different groups.
- Psychology to test the effectiveness of psychological interventions.
- Education to evaluate the impact of teaching methods.
Limitations
While the t-test is robust, it has limitations:
- It assumes normality, which might not always hold true, particularly with small sample sizes.
- The power of the test decreases with unequal sample sizes.
- It can be less effective with ordinal data or when the data does not meet the assumptions.
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