Sampling Distributions and Confidence Intervals for Proportions: Study Notes

Sampling Distributions and Confidence Intervals for Proportions

Sampling Distributions

The concept of a sampling distribution is fundamental in inferential statistics. It describes the probability distribution of a given statistic based on a random sample.

• Definition: The sampling distribution of a statistic is the distribution of that statistic over all possible samples of a fixed size from a population.

• Sample Proportion: For categorical data, the sample proportion $\\hat{p}$ is commonly used to estimate the population proportion $p$.

• Mean and Standard Deviation:

  • The mean of the sampling distribution of $\\hat{p}$ is $p$.

  • The standard deviation (standard error) of $\\hat{p}$ is: $\\sigma_{\\hat{p}} = \\sqrt{\\frac{p(1-p)}{n}}$

• Shape: For large enough sample sizes, the sampling distribution of $\\hat{p}$ is approximately normal (by the Central Limit Theorem), provided $np \\geq 10$ and $n(1-p) \\geq 10$.

Example: If a population proportion is $p = 0.4$ and a sample of $n = 100$ is taken, the standard error is $\\sqrt{\\frac{0.4 \\times 0.6}{100}} = 0.049$.

Confidence Intervals for Proportions

Confidence intervals provide a range of plausible values for a population proportion based on sample data.

• Definition: A confidence interval for a proportion is an interval estimate, calculated from the sample data, that is likely to contain the true population proportion.

• Formula: The general form for a confidence interval for a proportion is: $\\hat{p} \\pm z^* \\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}$ where $z^*$ is the critical value from the standard normal distribution for the desired confidence level (e.g., $z^* = 1.96$ for 95% confidence).

• Interpretation: A 95% confidence interval means that, in repeated sampling, 95% of such intervals will contain the true population proportion.

• Assumptions:

  • Random sample

  • Independence (sample size less than 10% of population)

  • Sample size large enough: $n\\hat{p} \\geq 10$ and $n(1-\\hat{p}) \\geq 10$

Example: In a sample of $n = 200$ students, $\\hat{p} = 0.3$ report liking statistics. The 95% confidence interval is: $0.3 \\pm 1.96 \\sqrt{\\frac{0.3 \\times 0.7}{200}} = 0.3 \\pm 0.064$ So, the interval is (0.236, 0.364).

Summary Table: Properties of Sampling Distributions and Confidence Intervals

Concept	Definition	Formula	Conditions
Sampling Distribution of $\\hat{p}$	Distribution of sample proportions from all possible samples	$\\mu_{\\hat{p}} = p$ $\\sigma_{\\hat{p}} = \\sqrt{\\frac{p(1-p)}{n}}$	$np \\geq 10$, $n(1-p) \\geq 10$
Confidence Interval for $p$	Range likely to contain true proportion	$\\hat{p} \\pm z^* \\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}$	Random sample, independence, large sample

Additional info:

• These notes are inferred from the presence of formulas and notation for sampling distributions and confidence intervals for proportions, which are central topics in introductory statistics.

• Some context and examples have been added to ensure completeness and clarity for exam preparation.