One-Sample t-Test

Use this test when you have one group of observations and want to compare their mean to a specific reference value. For example, testing whether the average reaction time in your sample differs from a published norm of 250 ms, or whether a manufacturing process produces items with a mean weight equal to the target specification.

• The dependent variable is continuous.
• Observations are independent.
• The data are approximately normally distributed, or the sample size is large enough for the Central Limit Theorem to apply.

• If Pr > |t| is less than alpha, the sample mean is significantly different from the hypothesised value.
• The confidence interval for the population mean provides a range of plausible values. If the hypothesised mean falls outside this interval, the result is significant.
• The sign and magnitude of the t-statistic indicate the direction and strength of departure from the hypothesised mean.
• Consider whether the observed difference is practically meaningful, not just statistically significant.

• The test assumes the hypothesised value is a fixed constant, not estimated from data. Using a value derived from the same sample invalidates the test.
• With very large samples, even trivially small deviations from the hypothesised mean can be statistically significant. Always examine the effect size.
• Non-normal data with small samples can produce misleading results. Consider the Wilcoxon signed-rank test as a non-parametric alternative.

1. Compute the sample mean, standard deviation, and standard error.
2. Calculate the t-statistic as (sample mean - hypothesised mean) / standard error.
3. Look up the p-value from the t-distribution with N - 1 degrees of freedom.