How to do a one-sided hypothesis test for two sample means

Description

If we have two samples, $x_{1}, \dots, x_{n}$ and $x_{1}^{'}, \dots, x_{n}^{'}$ , and we compute the mean of each one, we might want to ask whether one mean is less than the other. Or more precisely, is their difference significantly less than zero?

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Using SciPy, in Python

View this solution alone.

If we call the mean of the first sample ${\bar{x}}_{1}$ and the mean of the second sample ${\bar{x}}_{2}$ , then this is a two-sided test with the null hypothesis $H_{0} : {\bar{x}}_{1} - {\bar{x}}_{2} \geq 0$ . We choose a value $0 \leq α \leq 1$ as the probability of a Type I error (false positive, finding we should reject $H_{0}$ when it’s actually true). Let’s use $α = 0.10$ as an example.

from scipy import stats

# Replace these first three lines with the values from your situation.
sample1 = [ 6, 9, 7, 10, 10, 9 ]
sample2 = [ 12, 14, 10, 17, 9 ]

# Run a one-sample t-test and print out alpha, the p value,
# and whether the comparison says to reject the null hypothesis.
stats.ttest_ind( sample1, sample2, equal_var=False, alternative="less" )

Ttest_indResult(statistic=-2.4616581720814326, pvalue=0.02548641870923849)

The output says that the $p$ -value is about $0.0255$ , which is less than $α = 0.10$ . Therefore the samples give us enough evidence to reject the null hypothesis at the $α = 0.10$ level. That is, the data suggest that ${\bar{x}}_{1} < {\bar{x}}_{2}$ .

The equal_var parameter tells SciPy not to assume that the two samples have equal variances. If in your case they do, you can omit that parameter, and it will revert to its default value of True.

Content last modified on 24 July 2023.

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Solution, in R

View this solution alone.

If we call the mean of the first sample ${\bar{x}}_{1}$ and the mean of the second sample ${\bar{x}}_{2}$ , then this is a left-tailed test with the null hypothesis $H_{0} : {\bar{x}}_{1} - {\bar{x}}_{2} \geq 0$ . We choose a value $0 \leq α \leq 1$ as the probability of a Type I error (false positive, finding we should reject $H_{0}$ when it’s actually true).

# Replace these first three lines with the values from your situation.
alpha <- 0.10
sample1 <- c( 6, 9, 7, 10, 10, 9 )
sample2 <- c( 12, 14, 10, 17, 9 )

# Run a one-sample t-test and print out alpha, the p value,
# and whether the comparison says to reject the null hypothesis.
t.test( sample1, sample2, conf.level=1-alpha, alternative = "less" )

	Welch Two Sample t-test

data:  sample1 and sample2
t = -2.4617, df = 5.7201, p-value = 0.02549
alternative hypothesis: true difference in means is less than 0
90 percent confidence interval:
      -Inf -1.605229
sample estimates:
mean of x mean of y 
      8.5      12.4 

Although we can deduce the answer to our question from the above output, by comparing the $p$ -value with $α$ manually, we can also ask R to do it.

# Is there enough evidence to reject the null hypothesis?
result <- t.test( sample1, sample2, conf.level=1-alpha, alternative = "less" )
result$p.value < alpha

[1] TRUE

In this case, the samples give us enough evidence to reject the null hypothesis at the $α = 0.10$ level. The data suggest that ${\bar{x}}_{1} < {\bar{x}}_{2}$ .

Here we did not assume that the two samples had equal variance. If in your case they do, you can pass the parameter var.equal=TRUE to t.test.

Content last modified on 24 July 2023.

See a problem? Tell us or edit the source.

Topics that include this task

Bentley University MA214

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