Sampling Distribution Models & Confidence Intervals for Proportions

Sampling Distribution Models & Confidence Intervals for Proportions

Sampling Distribution Model for a Proportion

Sampling distributions describe the behavior of sample statistics over repeated sampling from a population. For proportions, the sampling distribution helps us understand how sample proportions vary and how to estimate population proportions.

• Parameter vs. Statistic: The true proportion in the population is called the parameter (p), while the proportion in a sample is the statistic (p̂).

• Randomness: Each sample may yield a different p̂ due to random variation.

• Center: The mean of the sampling distribution of p̂ is equal to the population proportion p.

• Spread: The standard deviation of the sampling distribution of p̂ is $\sqrt{\frac{p(1-p)}{n}}$.

• Shape: For large enough samples, the sampling distribution of p̂ is approximately normal.

• Example: If 57% of US adults believe in climate change, and a sample of 1000 is taken, the sample proportions will vary around 0.57.

Assumptions and Conditions for Normality

The normal model can be used for the sampling distribution of p̂ if certain conditions are met:

• Independence: Sample observations must be independent.

• Randomization: The sample must be randomly selected.

• Sample Size: The sample size n must be large enough so that both $np \geq 10$ and $n(1-p) \geq 10$.

• Example: In a sample of 200 students, if 31% have a BMI over 30, check if $n \times p$ and $n \times (1-p)$ are both at least 10.

Mean and Standard Deviation of Sampling Distribution

For a sample proportion p̂:

• Mean: $\mu_{p̂} = p$

• Standard Deviation: $\sigma_{p̂} = \sqrt{\frac{p(1-p)}{n}}$

• Example: For p = 0.17 and n = 99, $\sigma_{p̂} = \sqrt{\frac{0.17 \times 0.83}{99}} = 0.038$

Confidence Interval for a Proportion

A confidence interval estimates the range in which the true population proportion is likely to fall.

• Standard Error: $SE_{p̂} = \sqrt{\frac{p̂(1-p̂)}{n}}$

• Confidence Interval Formula: $p̂ \pm z^* \times SE_{p̂}$, where $z^*$ is the critical value for the desired confidence level (e.g., 1.96 for 95%).

• Example: If 36.1% of Facebook users update their status daily in a sample of 456, $SE_{p̂} = \sqrt{\frac{0.361 \times 0.639}{456}} = 0.023$

Interpreting Confidence Intervals

Confidence intervals provide a range of plausible values for the population proportion.

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

• Margin of Error: The margin of error is $z^* \times SE_{p̂}$.

• Certainty vs. Precision: Higher confidence levels increase certainty but widen the interval (less precision).

• Critical Values: For 95% confidence, $z^* = 1.96$; for 99%, $z^* = 2.58$.

One-Proportion z-Interval

When conditions are met, the one-proportion z-interval is used to estimate the population proportion.

• Formula: $p̂ \pm z^* \sqrt{\frac{p̂(1-p̂)}{n}}$

• Example: In a survey, 49% of 1015 adults think the death penalty is applied fairly. The 95% confidence interval is $0.49 \pm 1.96 \times \sqrt{\frac{0.49 \times 0.51}{1015}} = (0.46, 0.52)$

Technology Corner: Calculating Confidence Intervals

Statistical calculators and software can compute confidence intervals for proportions using built-in functions.

• Enter sample size and number of successes.

• Select the desired confidence level.

• Interpret the output as the confidence interval for the population proportion.

Common Mistakes and What Can Go Wrong

• Confusing the sampling distribution with the sample's distribution.

• Using the normal model when conditions are not met.

• Misinterpreting the confidence interval as the probability the parameter is in the interval.

• Failing to check for independence and randomization.

Summary Table: Sampling Distribution for Proportions

Statistic	Mean	Standard Deviation	Shape	Conditions
Sample Proportion (p̂)	$p$	$\sqrt{\frac{p(1-p)}{n}}$	Approximately Normal	Random, Independent, $np \geq 10$, $n(1-p) \geq 10$

Summary Table: Confidence Interval for a Proportion

Confidence Level	Critical Value ($z^*$)	Formula
90%	1.645	$p̂ \pm 1.645 \times SE_{p̂}$
95%	1.96	$p̂ \pm 1.96 \times SE_{p̂}$
99%	2.58	$p̂ \pm 2.58 \times SE_{p̂}$

Key Terms

• Parameter: A numerical summary of a population (e.g., p).

• Statistic: A numerical summary of a sample (e.g., p̂).

• Sampling Distribution: The distribution of a statistic over repeated samples.

• Standard Error: The estimated standard deviation of a sampling distribution.

• Confidence Interval: A range of values likely to contain the population parameter.

• Margin of Error: The maximum expected difference between the true parameter and a sample estimate.

• Critical Value: The z-score corresponding to the desired confidence level.

Additional info:

• These notes cover Chapter 16 topics: Sampling Distributions & Confidence Intervals for Proportion, including assumptions, calculations, interpretation, and common pitfalls.

• Examples and tables are inferred and expanded for clarity and completeness.