What are the mean and standard deviation of the sampling distribution of x̄? What are the mean and standard deviation of the sampling distribution of p̂?

Verified step by step guidance

1. Identify the population mean (\( \mu \)) and population standard deviation (\( \sigma \)) for the variable \( x \). These are the parameters from which the sampling distributions are derived.
2. Recall that the mean of the sampling distribution of the sample mean \( \overline{x} \) is equal to the population mean: \( \mu_{\overline{x}} = \mu \).
3. Calculate the standard deviation of the sampling distribution of \( \overline{x} \), also called the standard error, using the formula: \( \sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}} \), where \( n \) is the sample size.
4. For the sampling distribution of the sample proportion \( \hat{p} \), identify the population proportion \( p \). The mean of \( \hat{p} \) is equal to \( p \): \( \mu_{\hat{p}} = p \).
5. Calculate the standard deviation (standard error) of \( \hat{p} \) using the formula: \( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \), where \( n \) is the sample size.

Key Concepts

Sampling Distribution of the Sample Mean (x̄)

The sampling distribution of the sample mean is the probability distribution of all possible sample means from samples of a given size. Its mean equals the population mean (μ), reflecting that sample means are unbiased estimators of the population mean.

Standard Deviation of the Sampling Distribution of x̄ (Standard Error)

The standard deviation of the sampling distribution of the sample mean, called the standard error, measures the variability of sample means around the population mean. It is calculated as the population standard deviation (σ) divided by the square root of the sample size (n), i.e., σ/√n.

Sampling Distribution of the Sample Proportion (p̂)

The sampling distribution of the sample proportion describes the distribution of sample proportions from repeated samples. Its mean equals the population proportion (p), and its standard deviation (standard error) is √[p(1-p)/n], reflecting variability in sample proportions.