Overview
This lecture explains the steps for performing a one-sample t-test, including calculation of the test statistic, determination of significance, and interpretation of the p-value.
Calculating the t-Test Statistic
- A one-population t-test compares the sample mean (x̄) to the population mean (μ) using the t-distribution.
- The formula: t = (x̄ − μ) / (s / √n), where s is sample standard deviation and n is sample size.
- This formula determines how many standard errors the sample mean is from the population mean.
- In the example: t = (11.898 − 11) / (6.043 / √324) ≈ 2.675.
- A positive t statistic indicates the sample mean is above the population mean; negative means below.
Interpreting the t-Test Statistic
- The t statistic tells us how significantly the sample mean differs from the population mean in units of standard error.
- t and z test statistics standardize data for comparison, regardless of original units.
- For a 5% significance level in a two-tailed test, split the area to 2.5% in each tail.
- Critical t values with 323 degrees of freedom are approximately ±1.967.
- If the t statistic falls outside these critical values, the result is significant.
p-Value and Statistical Significance
- The p-value is the probability, under the null hypothesis, of observing a test statistic as extreme as the one calculated.
- For t = 2.675 and df = 323, the one-tail p = 0.0039; two-tail p-value = 0.0078 (both tails).
- A p-value of 0.0078 (0.78%) is less than the standard significance level (α = 0.05 or 5%).
- If p < α, we reject the null hypothesis.
Drawing Conclusions
- A low p-value indicates it is unlikely the observed sample mean occurred by random chance if the null hypothesis is true.
- Rejecting the null hypothesis means there is significant evidence against the claim that the population mean is 11.
Key Terms & Definitions
- t-Test Statistic — A standardized value showing how many standard errors the sample mean is from the population mean.
- Standard Error (SE) — s / √n; estimates the standard deviation of the sample mean distribution.
- Critical Value — The cutoff value(s) separating significant results from non-significant ones in hypothesis testing.
- p-Value — The probability of observing a test statistic as extreme as the observed value under the null hypothesis.
- Significance Level (α) — The threshold for rejecting the null hypothesis, commonly set at 0.05 (5%).
- Null Hypothesis (H₀) — The default assumption that there is no effect or difference, e.g., population mean = 11.
Action Items / Next Steps
- Practice calculating t-test statistics and p-values for sample problems.
- Review the use of statistical software or calculators for t-distributions and critical values.
- Read about assumption checks for one-sample t-tests.