# Statistics Dictionary

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### One-Sample t-Test

A one-sample t-test is used to test whether a population mean is significantly different from some hypothesized value.

Here is how to use the test.

• Define hypotheses. The table below shows three sets of null and alternative hypotheses. Each makes a statement about how the true population mean μ is related to some hypothesized value M. (In the table, the symbol ≠ means " not equal to ".)

Set Null hypothesis Alternative hypothesis Number of tails
1 μ = M μ ≠ M 2
2 μ > M μ < M 1
3 μ < M μ > M 1
• Specify significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.

• Find degrees of freedom. The degrees of freedom (DF) is:

DF = n - 1

where n is the number of observations in the sample.

• Compute test statistic. The test statistic is a t statistic (t) defined by the following equation.

t = (x - M ) / [ s /sqrt(n) ]

where x is the observed sample mean, M is the hypothesized population mean (from the null hypothesis), and s is the standard deviation of the sample.

• Compute P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t Distribution Calculator to assess the probability associated with the t statistic, having the degrees of freedom computed above.

• Evaluate null hypothesis. The evaluation involves comparing the P-value to the significance level, and rejecting the null hypothesis when the P-value is less than the significance level.

The one-sample t-test can be used when the population variances are equal or unequal, and with large or small samples.