Biostatistics: Student's Test | Zoology Optional Notes for UPSC PDF Download

Comprehensive Overview of the Student's t-Test and Its Variants

Introduction to the Student's t-Test: The Student's t-test is a widely utilized statistical test for the comparison of means between two groups of sample data. It serves the crucial purpose of assessing whether the means of two datasets exhibit statistically significant differences from each other. Various types of t-tests cater to different scenarios, offering valuable insights into the comparability of data:

Types of t-Tests

1. One-Sample t-Test:

   • Purpose: The one-sample t-test is employed to compare the mean of a sample population to a known theoretical value.
   • Application: This test is valuable when you need to determine how your sample mean compares to a specified population mean. It is particularly useful when the population standard deviation is unknown or when dealing with a small sample size.
2. Unpaired Two-Sample t-Test:
   • Purpose: The unpaired two-sample t-test is designed to assess the difference in means between two independent sample groups.
   • Application: It is the go-to test when you aim to compare the means of two distinct and unrelated samples, shedding light on the degree of difference between them.
3. Paired t-Test:
   • Purpose: The paired t-test is utilized to compare means within two related groups of samples.
   • Application: This test is invaluable in scenarios where you need to assess the impact of an intervention or treatment on paired observations, evaluating the changes or differences between the two sets.

One-Sample t-Test in Detail

The one-sample t-test is a specialized t-test that scrutinizes how the mean of your sample data relates to a known reference value. For instance, it helps ascertain whether your sample mean aligns with the population mean. This test is particularly applicable when you lack knowledge about the population's standard deviation or when dealing with a small sample size.

In summary, the Student's t-test and its variants offer a versatile toolkit for comparing means in diverse settings, catering to situations with known theoretical values, independent samples, or related groups. Whether it's assessing the impact of an intervention, comparing sample means to a population mean, or evaluating differences between unrelated samples, t-tests provide valuable statistical insights.

Solved Example

A professor wants to know if her introductory statistics class has a good grasp of basic math. Six students are chosen at random from the class and given a math proficiency test. The professor wants the class to be able to score above 70 on the test. The six students get scores of 62, 92, 75, 68, 83, and 95. Can the professor have 90 percent confidence that the mean score for the class on the test would be above 70?
Sol:
Step1: null hypothesis: H₀ : μ = 70
alternative hypothesis: H_a : μ > 70
Step2: compute the sample mean and standard deviation:

Step3: Compute the t‐value:
To test the hypothesis, the computed t‐value of 1.71 will be compared to the critical value in the t‐table. if the computed t‐value is larger than the critical t‐value from the table, the null hypothesis can be rejected. A 90 percent confidence level is equivalent to an alpha level of 0.10.This is a one‐tailed test, and you do not divide the alpha level by 2. The number of degrees of freedom for the problem is 6 – 1 = 5. The value in the t‐table for t .10,5 is 1.476. Because the computed t‐value of 1.71 is larger than the critical value in the table, the null hypothesis can be rejected, and the professor has evidence that the class mean on the math test would be at least 70.