**Instructions:** This calculator conducts a t-test for one population mean (\(\sigma\)), with unknown population standard deviation (\(\sigma\)), for which reason the sample standard deviation (s) is used instead. Please select the null and alternative hypotheses, type the hypothesized mean, the significance level, the sample mean, the sample standard deviation, and the sample size, and the results of the t-test will be displayed for you:

#### How to Conduct a T-test for One Population Mean?

More about the *t-test for one mean* so you can better interpret the results obtained by this solver: A t-test for one mean is a hypothesis test that attempts to make a claim about the population mean (\(\sigma\)). This t-test, unlike the z-test, does not need to know the population standard deviation \(\sigma\).

The test has two complementary hypotheses, the null and the alternative hypothesis. The null hypothesis is a statement about the population mean, under the assumption of no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis. The main properties of a one sample t-test for one population mean are:

- For a t-test for one mean, the sampling distribution used for the t-test statistic (which is the distribution of the test statistic under the assumption that the null hypothesis is true) corresponds to the t-distribution, with n-1 degrees of freedom (instead of being the standard normal distribution, as in the case of a z-test for one mean)
- Depending on our knowledge about the "no effect" situation, the t-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
- The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true
- In a hypothesis tests there are two types of errors. Type I error occurs when we reject a true null hypothesis, and the Type II error occurs when we fail to reject a false null hypothesis

The formula for a t-statistic is

\[t = \frac{\bar X - \mu_0}{s/\sqrt{n}}\]The null hypothesis is rejected when the t-statistic lies on the rejection region, which is determined by the significance level (\(\alpha\)) the type of tail (two-tailed, left-tailed or right-tailed) and the number of degrees of freedom \(df = n - 1\)

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