Sampling Distribution of x̄ Exam Questions: Prepare for Your Statistics Assessment

📚 Quick Study Guide

• 📊 Definition: A sampling distribution of $\bar{x}$ is the probability distribution of the sample mean calculated from all possible samples of a specific size ($n$) taken from a population.
• 🔢 Mean of the Sampling Distribution: The mean of the sampling distribution of $\bar{x}$ (denoted as $\mu_{\bar{x}}$) is equal to the population mean ($\mu$). $\mu_{\bar{x}} = \mu$
• 📏 Standard Deviation of the Sampling Distribution (Standard Error): The standard deviation of the sampling distribution of $\bar{x}$ (denoted as $\sigma_{\bar{x}}$) is equal to the population standard deviation ($\sigma$) divided by the square root of the sample size ($n$). $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$
• ⭐ Central Limit Theorem (CLT): If the population is normally distributed, or if the sample size ($n$) is sufficiently large ($n \geq 30$), then the sampling distribution of $\bar{x}$ will be approximately normal, regardless of the shape of the population distribution.
• 📝 Z-score for Sample Mean: To calculate probabilities associated with the sample mean, we use the Z-score formula: $Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$
• 💡 Finite Population Correction Factor: If the sample size $n$ is more than 5% of the population size $N$, use the finite population correction factor when calculating the standard error: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \cdot \sqrt{\frac{N-n}{N-1}}$

🧪 Practice Quiz

1. Question 1: A population has a mean of 50 and a standard deviation of 10. What is the mean of the sampling distribution of the sample mean for samples of size 25?
1. A. 2
2. B. 10
3. C. 50
4. D. 25
2. Question 2: A population has a standard deviation of 15. What is the standard error of the mean for samples of size 9?
1. A. 1.67
2. B. 5
3. C. 15
4. D. 45
3. Question 3: According to the Central Limit Theorem, the sampling distribution of the sample mean is approximately normal if:
1. A. The population is normal.
2. B. The sample size is large enough (n ≥ 30).
3. C. Both A and B.
4. D. Neither A nor B.
4. Question 4: A random sample of size 100 is taken from a population with a mean of 75 and a standard deviation of 12. What is the standard error of the mean?
1. A. 0.12
2. B. 1.2
3. C. 7.5
4. D. 12
5. Question 5: A population consists of 500 individuals, and a sample of 50 individuals is taken. If the population standard deviation is 20, what is the standard error of the mean, considering the finite population correction factor?
1. A. 2.67
2. B. 2.77
3. C. 2.83
4. D. 2.94
6. Question 6: The sampling distribution of the sample mean is:
1. A. The distribution of all possible sample means.
2. B. The distribution of the population.
3. C. The distribution of a single sample.
4. D. Always normal.
7. Question 7: 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.