Real-world examples of the sampling distribution of p̂ for college students.

📚 Quick Study Guide

🔍 The sampling distribution of $\hat{p}$ (sample proportion) is the distribution of sample proportions from all possible samples of the same size taken from a population.

📊 The mean of the sampling distribution of $\hat{p}$ is equal to the population proportion, $p$. That is, $\mu_{\hat{p}} = p$.

📈 The standard deviation of the sampling distribution of $\hat{p}$ (also known as the standard error) is given by $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$, where $n$ is the sample size.

🧪 The sampling distribution of $\hat{p}$ is approximately normal if $np \geq 10$ and $n(1-p) \geq 10$. This condition ensures the sample size is large enough.

💡 When sampling without replacement, and the sample size $n$ is more than 10% of the population size $N$, we need to apply a finite population correction factor to the standard error.

📝 The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean (or proportion) approaches a normal distribution, regardless of the shape of the population distribution.

Practice Quiz

1. What does the mean of the sampling distribution of $\hat{p}$ represent?
1. The standard deviation of the population.
2. The mean of the sample.
3. The population proportion.
4. The sample proportion.
2. What is the formula for the standard deviation of the sampling distribution of $\hat{p}$?
1. $\sqrt{\frac{n}{p(1-p)}}$
2. $\frac{p(1-p)}{n}$
3. $\sqrt{\frac{p(1-p)}{n}}$
4. $\frac{n}{p(1-p)}$
3. Under what condition is the sampling distribution of $\hat{p}$ approximately normal?
1. $n > 30$
2. $np \geq 5$ and $n(1-p) \geq 5$
3. $np \geq 10$ and $n(1-p) \geq 10$
4. $p = 0.5$
4. In a city, 60% of residents support a new park. If we take a random sample of 100 residents, what is the mean of the sampling distribution of the sample proportion who support the park?
1. 0.06
2. 0.6
3. 60
4. 6
5. Using the same scenario as above (60% support, sample of 100), what is the standard deviation of the sampling distribution of the sample proportion?
1. 0.024
2. 0.24
3. 0.049
4. 0.49
6. If you increase the sample size, what happens to the standard deviation of the sampling distribution of $\hat{p}$?
1. It increases.
2. It stays the same.
3. It decreases.
4. It fluctuates randomly.
7. A survey finds that 35% of college students prefer online textbooks. If a random sample of 200 students is taken, is the sampling distribution of $\hat{p}$ approximately normal?
1. No, because $np < 10$.
2. Yes, because $np \geq 10$ and $n(1-p) \geq 10$.
3. No, because $n$ is too small.
4. Yes, it is always normal.