University Statistics Quiz: One-sample T-test for proportions mastery.

📚 Quick Study Guide

• 📊 Definition: A one-sample T-test for proportions is used to determine if a sample proportion significantly differs from a hypothesized population proportion.
• 🧮 Null Hypothesis ($H_0$): The sample proportion is equal to the hypothesized population proportion ($p = p_0$).
• 🧪 Alternative Hypothesis ($H_1$): The sample proportion is not equal to, greater than, or less than the hypothesized population proportion ($p \neq p_0$, $p > p_0$, or $p < p_0$).
• 📐 Test Statistic: Calculated as: $t = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$, where $\hat{p}$ is the sample proportion, $p_0$ is the hypothesized population proportion, and $n$ is the sample size.
• 📈 Degrees of Freedom: $df = n - 1$. However, since we are dealing with proportions, a z-test is typically more appropriate, but if using a t-test due to small sample size adjustments, the df calculation might be ignored. If using a t-test approximation, df = n-1.
• 🔑 Assumptions: Random sample, independence, and approximately normal sampling distribution. For proportions, $np_0 \geq 10$ and $n(1-p_0) \geq 10$ are usually checked for normality approximation.
• 💡 P-value: The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.
• ✅ Decision Rule: If the p-value is less than or equal to the significance level ($\alpha$), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Practice Quiz

1. A researcher believes that 60% of students at a university support a new policy. A random sample of 100 students reveals that 52 support the policy. What is the test statistic for a one-sample T-test (approximating a Z-test) for proportions?

   1. -1.63
   2. 0.82
   3. -1.33
   4. 1.63

2. Using the information from the previous question, what is the null hypothesis ($H_0$)?

   1. $p < 0.60$
   2. $p > 0.60$
   3. $p = 0.52$
   4. $p = 0.60$

3. What condition must be met to assume that the sampling distribution of the sample proportion is approximately normal?

   1. $n > 30$
   2. $np \geq 5$ and $n(1-p) \geq 5$
   3. $n \geq 100$
   4. $np \geq 10$ and $n(1-p) \geq 10$

4. In a one-sample T-test (approximating Z-test) for proportions, a p-value of 0.03 is obtained. If the significance level is 0.05, what is the correct decision?

   1. Fail to reject the null hypothesis.
   2. Reject the alternative hypothesis.
   3. Reject the null hypothesis.
   4. Accept the null hypothesis.

5. A survey of 200 adults found that 60% prefer coffee over tea. If the hypothesized population proportion is 50%, what is the standard error of the sample proportion?

   1. 0.035
   2. 0.00125
   3. 0.0025
   4. 0.05

6. What does a large p-value (e.g., 0.8) in a one-sample T-test (approximating Z-test) for proportions indicate?

   1. Strong evidence against the null hypothesis.
   2. Strong evidence for the null hypothesis.
   3. Insufficient evidence to reject the null hypothesis.
   4. The sample proportion is significantly different from the hypothesized proportion.

7. In a hypothesis test for a single proportion, the alternative hypothesis is $p > 0.7$. What type of test is this?

   1. Two-tailed test
   2. Left-tailed test
   3. Right-tailed test
   4. Central test