The t-test is a statistical test used to determine whether there is a significant difference between the means of two groups, especially when the sample size is small (n < 30) and the population standard deviation is unknown.
It is widely used in inferential statistics to test hypotheses about population means based on sample data.
> Developed by: William Sealy Gosset under the pseudonym "Student", hence often called Student’s t-test.
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? Significance of t-Test
Ideal for testing hypotheses involving small sample sizes.
Helps in determining whether differences between means are statistically significant.
Useful in scientific research, medicine, education, and business studies.
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? Types of t-Tests
Type of t-Test Used When Example
One-sample t-test Comparing sample mean to known population mean Comparing average marks of students to pass marks
Two-sample (Independent) t-test Comparing means of two independent samples Comparing sales figures of two different stores
Paired t-test Comparing means from the same group at different times or under two conditions Before-and-after testing on the same group
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? Paired t-Test in Detail
Also known as the dependent sample t-test.
It is used when the same subjects are tested twice (e.g., pre-test and post-test), or when samples are matched pairs.
It helps determine whether the mean difference between paired observations is zero or significant.
Formula:
t = \frac{\bar{d}}{s_d/\sqrt{n}}
= Mean of the differences
= Standard deviation of the differences
= Number of pairs
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? Assumptions for t-Tests
Data should be approximately normally distributed.
Observations are independent.
Homogeneity of variance (for independent t-tests).
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? Test of Significance Based on t-Statistic (Small Sample)
The t-statistic is used for:
Testing the population mean when population variance is unknown.
Comparing two means when sample size is less than 30.
> Test Statistic:
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
= Sample mean
= Population mean
= Sample standard deviation
= Sample size
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⚠️ Limitations of t-Test
Assumes normality, which may not hold in small or skewed samples.
Sensitive to outliers.
Not suitable when comparing more than two groups (use ANOVA instead).
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? Conclusion
The t-test is a fundamental tool for testing hypotheses involving means, especially with small samples. Whether it's comparing two independent groups or assessing the impact of a treatment within the same group (paired t-test), this test enables evidence-based decision-making in various fields.
