Two independent sample T-test

(Reading time 10 min) This blog is about two sample t test theory, solved example and steps to perform analysis online along with interpretation in Agri Analyze platform

1. 1. Introduction

A two-sample independent t-test is a statistical method used to determine if there is a significant difference between the means of two independent groups. This test is particularly useful in comparing the average performance, outcomes or measurements of two distinct groups to understand if they significantly differ from each other, beyond what might be expected by random chance.

2. 2. When to use two sample t-test

The two-sample t-test, also known as the independent samples t-test, is used to compare the means of two independent groups to determine if there is a statistically significant difference between them.

3. 3. Assumption of two sample independent t-test

For the two-sample independent t-test to be valid, certain assumptions must be met:

· Independence of observation: The observations in each group must be independent of each other. This means that the measurement of one participant should not influence or be related to the measurement of another participant within the same group or across groups.

· Normality: The data in each group should be approximately normally distributed. This assumption is particularly important for small sample sizes (n < 30).

· Homogeneity of variance: The variances of the two groups should be equal. This assumption is known as homoscedasticity. If the variances are not equal, a modified version of the t-test, known as Two sample independent t-test with unequal variance (Welch's t-test), should be used. The Levene's test or F-test can be used to check for equality of variances.

· Scale of measurement: The variables should be continuous and have meaningful numerical values with equal intervals between them.

4. 4. Step to check assumptions:

· Normality: Use graphical methods such as histograms or Q-Q plots to visually inspect the distribution of the data. Conduct normality tests like the Shapiro-Wilk test or Kolmogorov-Smirnov test.

· Homogeneity of variance: Perform Levene's test, the F-test and Bartlett’s test to check if the variances of the two groups are significantly different.

1. Two sample independent t-test with equal variance

The two-sample t-test with equal variance assesses whether there is a significant difference between the means of two independent groups. This statistical test assumes homogeneity of variances and normal distribution within each group, making it suitable for comparing the means of two normally distributed populations.

· Checking homogeneity of variance by Bartlett’s test:

Bartlett's test evaluates the homogeneity of variances across multiple samples. By calculating a test statistic from the sample variances, it determines if there are significant differences. A p-value below a chosen significance level (e.g., 0.05) indicates unequal variances, while a higher p-value suggests equal variances.

Example of Bartlett’s test:

A study is conducted to compare the effectiveness of two different teaching methods (Method A and Method B) on student test scores. The experiment involves two groups of students, with 10 students in each group. The test scores of the students are recorded after they have been taught using their respective methods.

· Method A: 60, 65, 70, 75, 80, 85, 90, 95, 100, 105

· Method B: 55, 60, 65, 70, 75, 80, 85, 90, 95, 100

Solution:

Hypothesis:

· Null hypothesis (H₀): The variances of the test scores for both methods are equal.

· Alternative Hypothesis (H₁): The variances of the test scores for both methods are not equal.

1. 5. Determine the p-value:

The test statistic T follows a chi-squared distribution with k – 1 = 2 - 1 = 1 degree of freedom. In this case, T = 0, corresponding to a very high p-value (1). Thus, we do not reject the null hypothesis of equal variances.

2. 6. Conclusion: Based on Bartlett's test, there is no significant evidence to suggest that the variances of test scores differ between Method A and Method B.

· To perform a two-sample independent t-test with equal variances based on the provided example, we'll use the test to compare the means of the test scores for Method A and Method B.

Hypothesis:

• Null hypothesis (H₀): The means of the test scores for Method A and Method B are equal.

• Alternative Hypothesis (H₁): The means of the test scores for Method A and Method B are not equal.

1. Calculate the mean and variance for each group:

5. Find the critical value:

· For a two tailed test with = 0.05 and df = 18, the critical value is approximately 2.101.

6. Compare the t-statistic and critical value:

· Calculated t (0.7385) is less than critical value of t (2.101), so we fail to reject null hypothesis.

7. Conclusion: There is not enough evidence to reject the null hypothesis. This suggests that there is no significant difference in the mean test scores between Method A and Method B.

2. 2. Two sample independent t-test with unequal variance:

The two-sample t-test with unequal variances, also known as Welch's t-test, compares the means of two independent groups when their variances differ. It's robust when the assumption of equal variances is violated. Welch's t-test adjusts the degrees of freedom and provides a more reliable comparison under heterogeneous variances.

Example of Two sample independent t-test with unequal variance:

A new variety of cotton was evolved by a breeder. In order to compare its yielding ability with that of a ruling variety, an experiment was conducted in CRD. The kapas yield (kg/plot) was observed. The summary of the results are given below. Test whether the new variety of cotton gives higher yield than the ruling variety.