Real-World Examples of One-Sample T-Test Applications

📚 Quick Study Guide

• 🔢 One-Sample T-Test: Used to determine if the mean of a population is equal to a specified value.
• 🧪 Null Hypothesis ($H_0$): The population mean is equal to the specified value.
• 📈 Alternative Hypothesis ($H_1$): The population mean is not equal to the specified value (two-tailed), greater than (right-tailed), or less than (left-tailed) the specified value.
• 📐 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 one computed if the null hypothesis is true.
• 💡 Decision Rule: If the p-value is less than the significance level ($\alpha$), reject the null hypothesis.

Practice Quiz

1. Which of the following is a real-world application of the one-sample t-test?
   1. Measuring the effectiveness of a new drug compared to a placebo.
   2. Comparing the average height of students in two different schools.
   3. Determining if the average weight of apples from an orchard is significantly different from a historical average.
   4. Analyzing the correlation between two variables.
2. A researcher wants to test if the average IQ score of students at a particular school is significantly different from the national average of 100. What is the null hypothesis?
   1. The average IQ score is greater than 100.
   2. The average IQ score is less than 100.
   3. The average IQ score is equal to 100.
   4. The average IQ score is not equal to 100.
3. A food company claims that their cereal boxes contain 500g of cereal. A consumer group weighs a sample of boxes and wants to test if the actual average weight is less than 500g. What type of test should they use?
   1. Two-sample t-test
   2. One-sample t-test (left-tailed)
   3. One-sample t-test (right-tailed)
   4. Chi-square test
4. In a one-sample t-test, what does the p-value represent?
   1. The probability of the null hypothesis being true.
   2. The probability of the alternative hypothesis being true.
   3. The probability of observing a test statistic as extreme as, or more extreme than, the one computed if the null hypothesis is true.
   4. The significance level of the test.
5. A factory produces light bulbs and claims their average lifespan is 1000 hours. An engineer tests 25 bulbs and finds a sample mean of 950 hours with a standard deviation of 100 hours. What are the degrees of freedom for the one-sample t-test?
   1. 24
   2. 25
   3. 26
   4. 49
6. A biologist measures the pH of 10 soil samples from a particular region and wants to determine if the average pH is significantly different from 7.0. If the calculated t-statistic is 2.5 and the p-value is 0.02, and the significance level is 0.05, what is the correct conclusion?
   1. Fail to reject the null hypothesis.
   2. Reject the null hypothesis.
   3. Accept the null hypothesis.
   4. Increase the sample size.
7. A researcher is testing if a new teaching method improves test scores. They compare the average score of students taught with the new method to the historical average score. What is a potential confounding variable that should be considered?
   1. The sample size.
   2. The standard deviation of the scores.
   3. The students' prior knowledge and abilities.
   4. The p-value of the test.