University statistics t-distribution exam questions and solutions

📚 Quick Study Guide

• 🔢 Definition: The t-distribution, also known as Student's t-distribution, is a probability distribution that arises when estimating the mean of a normally distributed population when the sample size is small and the population standard deviation is unknown.
• 🧪 Degrees of Freedom (df): The shape of the t-distribution depends on the degrees of freedom, which is typically calculated as $df = n - 1$, where $n$ is the sample size.
• 📈 Comparison to Normal Distribution: The t-distribution is similar to the normal distribution but has heavier tails, meaning it is more prone to producing values that fall far from its mean. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution.
• 📝 T-Statistic Formula: 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 population mean, $s$ is the sample standard deviation, and $n$ is the sample size.
• 📊 Using T-Tables: T-tables provide critical values for different degrees of freedom and significance levels. These values are used to determine whether a null hypothesis should be rejected in a t-test.
• 💡 Assumptions: The t-test assumes that the data is normally distributed, the samples are independent, and the variances are equal (in the case of a two-sample t-test).

Practice Quiz

1. Question 1: What happens to the t-distribution as the degrees of freedom increase?
   1. A. It becomes more skewed.
   2. B. It approaches the standard normal distribution.
   3. C. It becomes flatter.
   4. D. Its variance increases.
2. Question 2: A researcher is testing the hypothesis that the mean height of students is 170cm. They collect a sample of 25 students. What are the degrees of freedom for the t-test?
   1. A. 26
   2. B. 25
   3. C. 24
   4. D. 23
3. Question 3: Which of the following is NOT an assumption of the independent samples t-test?
   1. A. The data is normally distributed.
   2. B. The samples are independent.
   3. C. The variances are equal.
   4. D. The sample sizes are equal.
4. Question 4: The t-statistic is calculated to be 2.5 with 15 degrees of freedom. Using a significance level of 0.05, what would you conclude? (Assume a one-tailed test and critical value is 1.753).
   1. A. Fail to reject the null hypothesis.
   2. B. Reject the null hypothesis.
   3. C. The test is invalid.
   4. D. More information is needed.
5. Question 5: What does a large t-statistic indicate?
   1. A. Strong evidence against the null hypothesis.
   2. B. Strong evidence in favor of the null hypothesis.
   3. C. The sample mean is close to the population mean.
   4. D. The sample size is small.
6. Question 6: In the t-statistic formula, what does 's' represent?
   1. A. Population standard deviation.
   2. B. Sample standard deviation.
   3. C. Population mean.
   4. D. Sample mean.
7. Question 7: When is it appropriate to use a t-test instead of a z-test for comparing means?
   1. A. When the population standard deviation is known.
   2. B. When the sample size is large (n > 30).
   3. C. When the population standard deviation is unknown and the sample size is small.
   4. D. When comparing proportions.