Sampling Distributions and Confidence Intervals for Proportions

Chapter 7: Sampling Distribution of Sample Proportions, Confidence Intervals

Sampling Distribution of Sample Proportions

The sampling distribution of sample proportions describes the behavior of proportions calculated from random samples drawn from a population. This concept is fundamental for making statistical inferences about population proportions.

• Population Proportion (p): The true proportion of individuals in the population with a certain characteristic.

• Sample Proportion (\hat{p}): The proportion observed in a sample, calculated as \( \hat{p} = \frac{x}{n} \), where x is the number of successes and n is the sample size.

• Mean of Sampling Distribution: The mean of all possible sample proportions equals the population proportion, \( E[\hat{p}] = p \).

• Standard Deviation (Standard Error): The standard deviation of the sampling distribution is \( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \).

• Normal Approximation: For large samples, the sampling distribution of \( \hat{p} \) is approximately normal if both \( np \geq 10 \) and \( n(1-p) \geq 10 \).

Example: If 60% of college students in a population are female (\( p = 0.6 \)), and a sample of 100 students is taken, the probability that the sample proportion is above 0.65 can be found using the normal model.

Confidence Intervals for Proportions

Confidence intervals provide a range of plausible values for the population proportion based on sample data. The interval is constructed using the sample proportion and its standard error.

• Formula for Confidence Interval: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ where \( z^* \) is the critical value from the standard normal distribution for the desired confidence level.

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

• Effect of Confidence Level: Increasing the confidence level widens the interval, making it less precise but more likely to contain the true proportion.

• Effect of Sample Size: Larger samples yield narrower (more precise) confidence intervals.

Example: In a sample of 900 residents, 627 are right-handed. The 99% confidence interval for the proportion of right-handed residents is calculated using the formula above.

Comparing Two Proportions

When comparing two groups, we may be interested in the difference between their proportions. The sampling distribution of the difference is also approximately normal for large samples.

• Difference in Proportions: \( \hat{p}_1 - \hat{p}_2 \)

• Standard Error for Difference: $\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

• Confidence Interval for Difference: $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

Example: Two samples are taken: one of 1,000 males and one of 900 females. The confidence interval for the difference in proportions of a certain characteristic is calculated using the formula above.

Normal Model and Probability Calculations

The normal model is used to approximate probabilities for sample proportions when the sample size is sufficiently large.

• Standardization: To find probabilities, standardize the sample proportion: $z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}$

• Application: Use the standard normal table to find probabilities associated with the calculated z-score.

Example: If the sample proportion is 0.75 and the population proportion is 0.7, with a sample size of 100, calculate the probability that the sample proportion is at least 0.75.

Summary Table: Key Formulas for Proportions

Concept	Formula	Notes
Sample Proportion	$\hat{p} = \frac{x}{n}$	x = number of successes, n = sample size
Standard Error	$\sqrt{\frac{p(1-p)}{n}}$	Use p if known, \hat{p} if unknown
Confidence Interval	$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	z* depends on confidence level
Difference in Proportions	$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$	Comparing two groups

Additional info:

• When constructing confidence intervals, always check that the sample size is large enough for the normal approximation to be valid.

• Interpretation of confidence intervals should be in the context of repeated sampling, not as a probability for a single interval.