**Instructions:** This calculator conducts a Sign Test. Please select the null and alternative hypotheses, enter the number of positives (+) and the number of negatives (-), along with the significance level, and the results of the sign test will be displayed for you (please disregard the ties):

## How to do a Sign Test?

More about the *sign test* for you to understand in a better way the results presented above: A sign test is a parametric test used to assess claims about a population median. It is typically used when the assumptions for a z-test for one mean are not met (namely, when the distribution departs significantly from normality). The test has, as every other hypothesis test, two non-overlapping hypotheses, the null and the alternative hypothesis. The null hypothesis is a statement about the population median, under the assumption of no effect, and the alternative
hypothesis is the complementary hypothesis to the null hypothesis.

- The sign test is a non-parametric test, and as such, it does not require the sample to come from a normally distributed population
- The sign test is very flexible and can be used in many contexts where it is possible to measure the outcome as "positive" or "negative" (such as being above or below the median, etc.)
- Depending on our knowledge about the "no effect" situation, the sign-test can be two-tailed, left-tailed or right-tailed
- The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true
- If the sample size is small enough, then we need to use a comparison with a critical value (which depends on the significance level provided) that is obtained from a sign test table (check the back of your book).
- If the sample size is large enough, then normal approximation can be used, and an appropriate z-test can be used.

If \(X^+\) and \(X^-\) are the number of positive and negative signs, respectively, then the test statistic is computed as \(X = \min\{X^+, X^-\}\). The null hypothesis of the sign test is rejected if \(X \le X*\), where \(X*\) is the critical value for the Sign Test, for the significance level provided and the type of tails specified. If the sample size is large enough, a formula for a z-statistic can be used, and it is

\[z = \frac{X + 0.5 - n/2 }{\sqrt{n}/2}\]If the sample size is large enough, the null hypothesis is rejected when the z-statistic lies on the rejection region, which is determined by the significance level (\(\alpha\)) and the type of tail (two-tailed, left-tailed or right-tailed).

The sign test can be used in case that the assumptions are not met for a one-sample t-test. If instead, the assumptions are met, then you can use our t-test for one mean calculator.

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