What is the formula for calculating t test. Provide some examples in details

Answer 1

The formula for calculating a t-test depends on the context, as there are different types of t-tests for different situations. The most common types of t-tests are the independent samples t-test, paired samples t-test, and one-sample t-test.

1. Independent Samples t-test:
This test is used when you want to compare the means of two independent groups.

Formula:
t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where:
x1 and x2 are the means of the two groups,
s1 and s2 are the standard deviations of the two groups,
n1 and n2 are the sample sizes of the two groups.

Example:
Let's say you want to compare the heights of men and women in a sample population. You have a sample of 30 men with a mean height of 175 cm and a standard deviation of 5 cm. You also have a sample of 25 women with a mean height of 165 cm and a standard deviation of 4 cm.

Using these values in the t-test formula:
t = (175 - 165) / sqrt((5^2 / 30) + (4^2 / 25))

2. Paired Samples t-test:
This test is used when you want to compare the means of two related or matched groups.

Formula:
t = (M - D) / (s / sqrt(n))
where:
M is the mean of the differences between the paired samples,
D is the average of those differences,
s is the standard deviation of the differences,
n is the number of pairs.

Example:
Suppose you want to compare the effectiveness of a weight loss program. You have a sample of 35 individuals and measure their weights before and after the program. You find that the average weight loss is 3 kg with a standard deviation of 2 kg.

Using these values in the t-test formula:
t = (0 - 3) / (2 / sqrt(35))

3. One-sample t-test:
This test is used when you want to compare the mean of a single group to a known or hypothesized population mean.

Formula:
t = (x - μ) / (s / sqrt(n))
where:
x is the sample mean,
μ is the population mean,
s is the standard deviation of the sample,
n is the sample size.

Example:
Assume you want to determine if the average score of a particular class is significantly different from the national average score of 75. You collect a sample of 50 students and find that their average score is 80, with a standard deviation of 6.

Using these values in the t-test formula:
t = (80 - 75) / (6 / sqrt(50))

These examples demonstrate how to calculate t-tests for different scenarios. Note that the calculation of a t-test requires specific data such as sample means, standard deviations, and sample sizes.