One-Sample T-Test Definition: Explaining the T-Statistic and P-Value

📚 Quick Study Guide

• 🧪 The One-Sample T-Test determines if the mean of a single sample is significantly different from a known or hypothesized population mean.
• 🔢 The T-statistic is calculated as: $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$, where $\bar{x}$ is the sample mean, $\mu$ is the hypothesized population mean, $s$ is the sample standard deviation, and $n$ is the sample size.
• 📊 Degrees of freedom (df) are calculated as $df = n - 1$.
• 📈 The P-value represents the probability of observing a test statistic as extreme as, or more extreme than, the statistic obtained from the sample, assuming the null hypothesis is true.
• 🎯 A small P-value (typically $\leq 0.05$) suggests strong evidence against the null hypothesis.
• 💡 Assumptions include: the data are continuous, the data follow a normal distribution, and the data are randomly sampled.

Practice Quiz

1. What is the primary purpose of a One-Sample T-Test?

   • A) To compare the means of two independent samples.
   • B) To determine if the mean of a sample is significantly different from a known value.
   • C) To assess the correlation between two variables.
   • D) To calculate the variance within a sample.

2. In the formula for the T-statistic, what does $\mu$ represent?

   • A) The sample mean.
   • B) The population standard deviation.
   • C) The hypothesized population mean.
   • D) The sample size.

3. What does the P-value indicate in a One-Sample T-Test?

   • A) The probability of the null hypothesis being true.
   • B) The probability of observing a test statistic as extreme as, or more extreme than, the one calculated if the null hypothesis is true.
   • C) The effect size of the sample.
   • D) The confidence interval of the sample mean.

4. What is the typical threshold for the P-value to reject the null hypothesis?

   • A) P $\leq$ 0.10
   • B) P $\leq$ 0.01
   • C) P $\leq$ 0.05
   • D) P $\geq$ 0.05

5. What is the formula for calculating degrees of freedom (df) in a One-Sample T-Test, where 'n' is the sample size?

   • A) df = n + 1
   • B) df = n * 2
   • C) df = n - 1
   • D) df = n

6. Which of the following is an assumption of the One-Sample T-Test?

   • A) The data are discrete.
   • B) The data follow a non-normal distribution.
   • C) The data are randomly sampled.
   • D) The data are dependent.

7. If the calculated T-statistic is 2.5 and the critical T-value is 2.0, what can you conclude?

   • A) Fail to reject the null hypothesis.
   • B) Reject the null hypothesis.
   • C) The sample mean is equal to the population mean.
   • D) There is no significant difference.