# Two-sample t-test with Unequal Sample Sizes in R

The two-sample t-test with unequal sample sizes can be performed using the built-in `t.test()`

function from
in R.

In case of unequal sample sizes, you should check the **assumption of the equality of variances (homoscedasticity)**.

If the variances are not equal you should perform the Welch’s t-test (which does not assume equal variances between the two samples).

You should use **Student’s two-sample t-test** with unequal sample sizes when variances are equal

```
t.test(sample1, sample2, var.equal = TRUE)
```

`var.equal`

parameter, `t.test()`

will perform a two-sample t-test assuming non-equal
variances between two samples.You should use **Welch’s t-test** with unequal sample sizes when variances are not equal

```
# Welch t-test
t.test(sample1, sample2, var.equal = FALSE)
```

Welch’s t-test is appropriate for unequal sample sizes when variances are not equal as it adjusts for differences in sample size and variance.

The following examples demonstrate how to perform a two-sample t-test with unequal sample sizes using the built-in `t.test()`

function in R.

## Create dataset

Create datasets with unequal sizes for two samples ,

```
sample1 <- c(28, 35, 45, 65, 44, 56, 40, 42, 35, 34, 44)
sample2 <- c(15, 10, 40, 25, 26, 21)
```

The sample1 has 11 observations whereas the sample2 has 6 observations.

## Check assumption of equality of variances (homoscedasticity)

Before performing the two-sample t-test with unequal sample sizes, you should check the assumption of the equality of variances.

You can use **Levene’s test** to assess whether the variances of the two samples are equal

```
# load package
library(car)
leveneTest(c(sample1, sample2),
group = factor(rep(c("sample1", "sample2"), c(length(sample1), length(sample2)))))
# output
Levene's Test for Homogeneity of Variance (center = median)
Df F value Pr(>F)
group 1 2e-04 0.9897
15
```

As the *p* value (0.9897) from Levene’s test is greater than significance level (0.05), you would fail to reject the null hypothesis of equal
variances i.e. variances are equal between the two samples.

## Perform two-sample t-test

As the variances are equal between two unequal sizes samples, you should perform the two-sample t-test using `t.test()`

function
(assuming equal variance).

```
t.test(sample1, sample2, var.equal = TRUE)
# output
Two Sample t-test
data: sample1 and sample2
t = 3.715, df = 15, p-value = 0.002074
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
8.402563 31.021680
sample estimates:
mean of x mean of y
42.54545 22.83333
```

As the *p* value (0.002) from the two-sample t-test is less than a significance level (0.05), you
should reject the null hypothesis and conclude that there is a statistically significant difference
between the means of the two samples with unequal sizes.