One-sample T-test for means: Application in various research scenarios.

📚 Quick Study Guide

• 🤔 Definition: The one-sample t-test is used to determine whether the mean of a single sample is different from a known or hypothesized population mean.
• ✅ Assumptions:
  • 📈 The data are continuous.
  • 📊 The data follow a normal distribution.
  • 🌱 The data 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 (two-tailed), greater than (right-tailed), or less than (left-tailed) the hypothesized population mean ($\mu \neq \mu_0$, $\mu > \mu_0$, $\mu < \mu_0$).
• ➗ Test Statistic: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$, 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: $df = n - 1$
• 📈 P-value: 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.
• 💡 Decision Rule:
  • If p-value $\leq \alpha$ (significance level), reject $H_0$.
  • If p-value $> \alpha$, fail to reject $H_0$.

✍️ Practice Quiz

1. Question 1: A researcher wants to test if the average height of students in a university is different from 170 cm. They collect a sample of 30 students. What is the null hypothesis?
   1. A. The average height is greater than 170 cm.
   2. B. The average height is less than 170 cm.
   3. C. The average height is equal to 170 cm.
   4. D. The average height is not equal to 170 cm.
2. Question 2: In a one-sample t-test, what does the p-value represent?
   1. A. The probability of the null hypothesis being true.
   2. B. The probability of the alternative hypothesis being true.
   3. C. The probability of observing a test statistic as extreme as, or more extreme than, the one calculated if the null hypothesis is true.
   4. D. The significance level of the test.
3. Question 3: A company claims that its batteries last 40 hours on average. A consumer group tests 20 batteries and finds a sample mean of 38 hours with a standard deviation of 4 hours. What is the test statistic?
   1. A. -2.24
   2. B. 2.24
   3. C. -11.2
   4. D. 11.2
4. Question 4: For the previous question, how many degrees of freedom are there?
   1. A. 19
   2. B. 20
   3. C. 21
   4. D. 39
5. Question 5: A researcher sets the significance level at 0.05. They perform a one-sample t-test and obtain a p-value of 0.03. What is the correct decision?
   1. A. Fail to reject the null hypothesis.
   2. B. Reject the null hypothesis.
   3. C. Increase the sample size.
   4. D. Decrease the significance level.
6. Question 6: Which of the following is NOT an assumption of the one-sample t-test?
   1. A. The data are continuous.
   2. B. The data follow a normal distribution.
   3. C. The data are independent.
   4. D. The data are randomly sampled.
7. Question 7: A doctor wants to know if a new drug affects patients' resting heart rate. Before the study, the average resting heart rate is known to be 72 bpm. The doctor measures the heart rate of 25 patients after they take the drug. This scenario is best suited for which test?
   1. A. Two-sample t-test
   2. B. Paired t-test
   3. C. One-sample t-test
   4. D. ANOVA