Test Questions on Sampling Distributions of Means and Proportions

📚 Quick Study Guide

• 📈 Sampling Distribution of the Mean: The distribution of sample means from repeated samples of the same size taken from a population.
• 📊 Central Limit Theorem (CLT): States that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.
• 📏 Standard Error of the Mean: The standard deviation of the sampling distribution of the mean, calculated as $\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.
• 🧪 Sampling Distribution of the Proportion: The distribution of sample proportions from repeated samples of the same size taken from a population.
• 🧮 Standard Error of the Proportion: The standard deviation of the sampling distribution of the proportion, calculated as $\sqrt{\frac{p(1-p)}{n}}$, where $p$ is the population proportion and $n$ is the sample size.
• 📝 Conditions for Normality (Proportions): The sampling distribution of the proportion is approximately normal if $np \geq 10$ and $n(1-p) \geq 10$.
• 💡 Z-score for Sample Mean: $z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size.

Practice Quiz

1. A random sample of 100 light bulbs has a mean lifetime of 500 hours with a standard deviation of 50 hours. What is the standard error of the mean?
   1. A. 5 hours
   2. B. 0.5 hours
   3. C. 50 hours
   4. D. 0.05 hours
2. The Central Limit Theorem states that, under certain conditions, the sampling distribution of the sample mean is approximately:
   1. A. Uniform
   2. B. Exponential
   3. C. Normal
   4. D. Skewed
3. In a population, 40% of people prefer coffee over tea. If you take a random sample of 200 people, what is the standard error of the proportion?
   1. A. 0.0346
   2. B. 0.0012
   3. C. 0.04
   4. D. 0.4899
4. Which of the following is NOT a condition for using the normal approximation for the sampling distribution of the proportion?
   1. A. $np \geq 10$
   2. B. $n(1-p) \geq 10$
   3. C. The sample is randomly selected
   4. D. $n \geq 30$
5. A population has a mean of 75 and a standard deviation of 12. If you take a sample of size 36, what is the probability that the sample mean will be greater than 78?
   1. A. 0.0668
   2. B. 0.9332
   3. C. 0.4332
   4. D. 0.5668
6. What happens to the standard error of the mean as the sample size increases?
   1. A. It increases
   2. B. It decreases
   3. C. It stays the same
   4. D. It becomes zero
7. A researcher wants to estimate the proportion of students who support a new policy. How large a sample should they take to ensure the margin of error is no more than 0.05 with 95% confidence, assuming they have no prior estimate of the proportion?
   1. A. 385
   2. B. 196
   3. C. 97
   4. D. 769