Test Your Knowledge: Calculating Probabilities for Sample Proportions.

📚 Quick Study Guide

🔢 Sample Proportion: The sample proportion, denoted as $\hat{p}$, is the proportion of individuals in a sample who have a specific characteristic. It's calculated as $\hat{p} = \frac{x}{n}$, where $x$ is the number of individuals with the characteristic and $n$ is the sample size. 📊 Mean of the Sampling Distribution of $\hat{p}$: The mean of the sampling distribution of $\hat{p}$ is equal to the population proportion, $p$. That is, $\mu_{\hat{p}} = p$. 📉 Standard Deviation of the Sampling Distribution of $\hat{p}$: The standard deviation (also called standard error) of the sampling distribution of $\hat{p}$ is given by $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$, where $n$ is the sample size. 🧭 Conditions for Normality: The sampling distribution of $\hat{p}$ is approximately normal if $np \geq 10$ and $n(1-p) \geq 10$. 📝 Z-score for Sample Proportions: To calculate probabilities, standardize $\hat{p}$ using the z-score: $z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}$.

Practice Quiz

1. What is the formula for the standard deviation of the sampling distribution of a sample proportion?
   1. $\sigma_{\hat{p}} = \sqrt{\frac{n}{p(1-p)}}$
   2. $\sigma_{\hat{p}} = \frac{p(1-p)}{n}$
   3. $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$
   4. $\sigma_{\hat{p}} = \frac{n}{\sqrt{p(1-p)}}$
2. If a population proportion $p = 0.4$ and the sample size $n = 100$, what is the mean of the sampling distribution of the sample proportion?
   1. 0.6
   2. 0.04
   3. 0.4
   4. 0.06
3. For a sample proportion to be approximately normally distributed, which conditions must be met?
   1. $n > 30$
   2. $np \geq 5$ and $n(1-p) \geq 5$
   3. $np \geq 10$ and $n(1-p) \geq 10$
   4. $n > 50$
4. In a survey of 500 people, 300 said they prefer coffee over tea. What is the sample proportion of people who prefer coffee?
   1. 0.3
   2. 0.5
   3. 0.6
   4. 0.4
5. Given $p = 0.6$ and $n = 400$, calculate the standard error of the sample proportion.
   1. 0.024
   2. 0.0006
   3. 0.6
   4. 0.06
6. What does $\hat{p}$ represent?
   1. The population parameter
   2. The sample size
   3. The sample proportion
   4. The standard deviation of the population
7. If $\hat{p} = 0.7$, $p = 0.6$, and $n = 100$, what is the z-score?
   1. 2.04
   2. 2.15
   3. 2.25
   4. 2.5