1. The problem: Perform a one-sample t-test to determine if the mean of a sample differs significantly from a known population mean. 2. Formula: The t-test statistic is calculated as $$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$$ where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $s$ is the sample standard deviation, and $n$ is the sample size. 3. Important rules: The t-test assumes the sample is drawn from a normally distributed population or the sample size is large enough for the Central Limit Theorem to apply. 4. Intermediate work example: Suppose a sample of size $n=10$ has $\bar{x} = 5.2$, $s = 1.1$, and the population mean $\mu = 5.0$. Calculate the t-statistic: $$t = \frac{5.2 - 5.0}{1.1 / \sqrt{10}} = \frac{0.2}{1.1 / 3.162} = \frac{0.2}{0.348} \approx 0.575$$ 5. Interpretation: Compare the calculated t-value to the critical t-value from the t-distribution table with $n-1=9$ degrees of freedom at the chosen significance level (e.g., 0.05). If $|t|$ is greater, reject the null hypothesis; otherwise, do not reject it. This process helps determine if the sample mean significantly differs from the population mean.