Performing a one-sample T-test for means: A practical walkthrough.

📚 Quick Study Guide

• 🔢 Purpose: To determine if the mean of a single sample is significantly different from a known population mean.
• 📊 Assumptions: The data are continuous, follow a normal distribution, and are randomly sampled.
• 📝 Null Hypothesis ($H_0$): The sample mean is equal to the hypothesized population mean ($\mu = \mu_0$).
• 🧪 Alternative Hypothesis ($H_1$): The sample mean is not equal to, greater than, or less than the hypothesized population mean ($\mu \neq \mu_0$, $\mu > \mu_0$, or $\mu < \mu_0$).
• ➗ Test Statistic: Calculated as $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$, where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothesized population mean, $s$ is the sample standard deviation, and $n$ is the sample size.
• 📈 Degrees of Freedom: $df = n - 1$.
• ✅ Decision Rule: If the absolute value of the calculated t-statistic is greater than the critical t-value (or if the p-value is less than the significance level $\alpha$), reject the null hypothesis.

Practice Quiz

1. Which of the following is a primary assumption of the one-sample T-test?
   1. A. The population standard deviation is known.
   2. B. The data are normally distributed.
   3. C. The sample size is greater than 30.
   4. D. The data are discrete.
2. What does the null hypothesis typically state in a one-sample T-test?
   1. A. The sample mean is greater than the population mean.
   2. B. The sample mean is equal to the population mean.
   3. C. The sample mean is less than the population mean.
   4. D. The sample mean is not equal to the population mean.
3. What is the formula for the T-statistic in a one-sample T-test?
   1. A. $t = \frac{\bar{x} + \mu_0}{s/\sqrt{n}}$
   2. B. $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$
   3. C. $t = \frac{\bar{x} - \mu_0}{s*\sqrt{n}}$
   4. D. $t = \frac{\bar{x} - \mu_0}{s/n}$
4. What are the degrees of freedom for a one-sample T-test with a sample size of 25?
   1. A. 26
   2. B. 25
   3. C. 24
   4. D. 23
5. If the calculated T-statistic is 2.5 and the critical T-value is 2.0, what decision should be made regarding the null hypothesis?
   1. A. Accept the null hypothesis.
   2. B. Reject the null hypothesis.
   3. C. Fail to reject the null hypothesis.
   4. D. Increase the sample size.
6. What does a small p-value (e.g., p < 0.05) indicate in a one-sample T-test?
   1. A. Strong evidence in favor of the null hypothesis.
   2. B. Weak evidence against the null hypothesis.
   3. C. Strong evidence against the null hypothesis.
   4. D. The sample mean is equal to the population mean.
7. In a one-sample T-test, 's' represents which of the following?
   1. A. Population standard deviation
   2. B. Sample mean
   3. C. Sample standard deviation
   4. D. Population mean