1. State the Hypotheses:
* Null Hypothesis (H₀): This is the statement you're trying to disprove. It typically claims there's no significant difference between groups or between the sample mean and a population mean. Examples:
* One-sample: The sample mean is equal to the population mean (µ = µ₀).
* Independent samples: The means of the two groups are equal (µ₁ = µ₂).
* Paired samples: The mean difference between the paired observations is zero (µd = 0).
* Alternative Hypothesis (H₁ or Hₐ): This is what you believe to be true if the null hypothesis is false. It can be one-tailed (directional, e.g., µ₁ > µ₂) or two-tailed (non-directional, e.g., µ₁ ≠ µ₂).
2. Set the Significance Level (α):
This is the probability of rejecting the null hypothesis when it's actually true (Type I error). A common significance level is 0.05 (5%), meaning there's a 5% chance of concluding a difference exists when there isn't one.
3. Select the Appropriate T-Test:
* One-sample t-test: Compares the mean of a single sample to a known population mean.
* Independent samples t-test: Compares the means of two independent groups. It's crucial to check for equal variances (using Levene's test) to choose the appropriate version of the t-test (equal variances assumed vs. equal variances not assumed).
* Paired samples t-test: Compares the means of two related groups (e.g., before and after measurements on the same subjects).
4. Calculate the Test Statistic (t):
This involves calculating the difference between the sample means (or the sample mean and the population mean) and dividing by the standard error of the mean difference. The formula varies depending on the type of t-test: you'll find specific formulas in statistical textbooks or software documentation.
5. Determine the Degrees of Freedom (df):
The degrees of freedom represent the number of independent pieces of information used to estimate a parameter. The calculation of df differs depending on the t-test:
* One-sample: df = n - 1 (where n is the sample size)
* Independent samples (equal variances assumed): df = n₁ + n₂ - 2 (where n₁ and n₂ are the sample sizes of the two groups)
* Independent samples (equal variances not assumed): A more complex calculation is used (provided by statistical software).
* Paired samples: df = n - 1 (where n is the number of pairs)
6. Find the Critical Value or P-value:
* Critical value approach: Using the degrees of freedom and the significance level (α), you look up the critical t-value in a t-distribution table. If the calculated t-statistic is greater than the critical t-value (in absolute value for a two-tailed test), you reject the null hypothesis.
* P-value approach: Most statistical software directly provides the p-value, which is the probability of observing a t-statistic as extreme as or more extreme than the one calculated, assuming the null hypothesis is true. If the p-value is less than the significance level (α), you reject the null hypothesis.
7. Make a Decision:
Based on the comparison of the t-statistic to the critical value or the p-value to the significance level, you either:
* Reject the null hypothesis: There is sufficient evidence to suggest a significant difference.
* Fail to reject the null hypothesis: There is not enough evidence to suggest a significant difference.
8. Interpret the Results:
Clearly state your conclusion in the context of your research question. Report the t-statistic, degrees of freedom, p-value, and the effect size (e.g., Cohen's d) to provide a complete picture of your findings.
Important Note: While these steps outline the manual process, statistical software packages (like SPSS, R, Python with SciPy) automate much of this, making the calculations and interpretation significantly easier. These packages also often include checks for assumptions (like normality) that are crucial for the validity of the t-test.