# How to do a two-sided hypothesis test for a sample mean (in Julia)

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Say we have a population whose mean $\mu$ is known. We take a sample $x_1,\ldots,x_n$ and compute its mean, $\bar x$. We then ask whether this sample is significantly different from the population at large, that is, is $\mu=\bar x$?

## Solution

This is a two-sided test with the null hypothesis $H_0:\mu=\bar x$. 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 itâ€™s actually true).

 1 2 3 4 5 6 7 8 9 10 # Replace these first three lines with the values from your situation. alpha = 0.05 pop_mean = 10 sample = [ 9, 12, 14, 8, 13 ] # The following code runs the test for your chosen alpha: using HypothesisTests p_value = pvalue( OneSampleTTest( sample, pop_mean ) ) reject_H0 = p_value < alpha alpha, p_value, reject_H0
 1 (0.05, 0.35845634462296455, false)

In this case, the $p$-value was larger than $\alpha$, so the sample does not give us enough information to reject the null hypothesis. We would continue to assume that the sample is like the population, $\mu=\bar x$.

When you are using the most common value for $\alpha$, which is $0.05$ for the $95\%$ confidence interval, you can simply print out the test itself and get a detailed printout with all the information you need, thus saving a few lines of code.

 1 OneSampleTTest( sample, pop_mean )
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 One sample t-test ----------------- Population details: parameter of interest: Mean value under h_0: 10 point estimate: 11.2 95% confidence interval: (7.986, 14.41) Test summary: outcome with 95% confidence: fail to reject h_0 two-sided p-value: 0.3585 Details: number of observations: 5 t-statistic: 1.0366421106976316 degrees of freedom: 4 empirical standard error: 1.1575836902790224

Content last modified on 24 July 2023.

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Contributed by Nathan Carter (ncarter@bentley.edu)