# How to do a two-sided hypothesis test for two sample means (in Python, using SciPy)

## Task

If we have two samples, $x_1,\ldots,x_n$ and $x’_1,\ldots,x’_m$, and we compute the mean of each one, we might want to ask whether the two means seem approximately equal. Or more precisely, is their difference statistically significant at a given level?

Related tasks:

- How to compute a confidence interval for a population mean
- How to do a two-sided hypothesis test for a sample mean
- How to do a one-way analysis of variance (ANOVA)
- How to do a one-sided hypothesis test for two sample means
- How to do a hypothesis test for a mean difference (matched pairs)
- How to do a hypothesis test for a population proportion

## Solution

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 $H_0:\bar x_1=\bar x_2$. We choose a value $0\leq\alpha\leq1$ as the probability of a Type I error (false positive, finding we should reject $H_0$ when it’s actually true). Let’s use $\alpha=0.10$ as an example.

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from scipy import stats
# Replace these first three lines with the values from your situation.
alpha = 0.10
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 )

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Ttest_indResult(statistic=-2.4616581720814326, pvalue=0.05097283741847698)

The output says that the $p$-value is about $0.05097$, which is less than $\alpha=0.10$. In this case, the samples give us enough evidence to reject the null hypothesis at the $\alpha=0.10$ level. That is, the data suggest that $\bar x_1\neq\bar x_2$.

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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Contributed by Nathan Carter (ncarter@bentley.edu)