Confidence Intervals for Proportions: Concepts, Calculations, and Applications

Confidence Intervals for Proportions

Introduction

Confidence intervals are a fundamental concept in inferential statistics, allowing us to estimate population parameters based on sample data. This study guide focuses on confidence intervals for proportions, including their calculation, interpretation, and the factors that affect their width and reliability.

Parameter Estimation and Sampling Distributions

Parameter Estimation

Parameter estimation involves using sample data to estimate an unknown population parameter. For categorical data, the parameter of interest is often the population proportion, denoted as p.

• Sample Proportion (p̂): The sample proportion is calculated as the number of individuals in the sample with the specified characteristic divided by the sample size.

$\hat{p} = \frac{x}{n}$

• x: Number of individuals in the sample with the characteristic

• n: Sample size

Example: In a university poll, 349 out of 1745 students believe extraterrestrial life exists. The point estimate is:

$\hat{p} = \frac{349}{1745} = 0.2$

Statistical Inference about Proportion

• Sample Proportion (SP): Denoted by $\hat{p}$, it is a random variable representing the proportion of successes in a sample.

• Observed Sample Proportion: The value of $\hat{p}$ computed from an observed sample is a single number.

Sampling Distribution of the Sample Proportion

Shape, Center, and Spread

• Shape: As sample size increases, the sampling distribution of $\hat{p}$ becomes approximately normal.

• Center: The mean of the sampling distribution equals the population proportion $p$.

• Spread: The standard deviation decreases as sample size increases.

Conditions for Normality

• The sample is a simple random sample.

• Sampled values are independent (sample size < 5% of population).

• $np(1-p) \geq 10$

Formulas for Sampling Distribution

• Mean: $\mu_{\hat{p}} = p$

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

Example: Sampling Distribution

Suppose $p = 0.76$ and $n = 60$:

$\sigma_{\hat{p}} = \sqrt{\frac{0.76(1-0.76)}{60}} = 0.055$

The distribution of $\hat{p}$ is approximately normal with mean $0.76$ and standard deviation $0.055$.

Probability Calculations for Sample Proportions

Using the Central Limit Theorem (CLT)

• Check $n < 5\%$ of population and $np(1-p) \geq 10$.

• Approximate the sampling distribution of $\hat{p}$ as normal.

Example: If 18.8% of children are overweight, in a sample of 90, what is the probability that at least 19% are overweight?

• Mean: $0.188$

• Standard deviation: $\sqrt{\frac{0.188(1-0.188)}{90}} = 0.0412$

If 24 out of 90 are overweight, $\hat{p} = \frac{24}{90} = 0.2667$. This is an unusual sample if the true proportion is $0.188$ (only about 3 in 100 samples would be this high or higher).

Confidence Intervals for Proportions

Definition and Level of Confidence

• Confidence Interval: An interval of numbers based on a point estimate, used to estimate an unknown parameter.

• Level of Confidence: The expected proportion of intervals that will contain the parameter if many samples are taken. Denoted $(1-\alpha) \times 100\%$.

Interpretation

• A 95% confidence level ($\alpha = 0.05$) means that 95 out of 100 intervals constructed from different samples will contain the true parameter.

• Because the sampling distribution is normal, 95% of sample proportions lie within 1.96 standard deviations of $p$.

Formula for Confidence Interval

• General form: point estimate ± margin of error

• Margin of error for 95% confidence: $1.96 \sigma_{\hat{p}}$

$\text{CI} = \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• $z_{\alpha/2}$ is the critical value for the desired confidence level.

Critical Values Table

Level of Confidence	Area in Each Tail	Critical Value $z_{\alpha/2}$
95%	0.025	1.96
99%	0.005	2.575

Example: Confidence Interval Calculation

Suppose $n = 100$, $x = 25$ (left-handed):

• $\hat{p} = \frac{25}{100} = 0.25$

• 95% CI: $0.25 \pm 1.96 \sqrt{\frac{0.25(1-0.25)}{100}} = 0.25 \pm 0.085$

• Interval: (0.165, 0.335)

Margin of Error

• The margin of error $E$ is given by:

$E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Example: Parent-Teen Cell Phone Survey

• $n = 800$, $x = 272$, $\hat{p} = 0.34$

• 95% CI: $0.34 \pm 1.96 \sqrt{\frac{0.34(1-0.34)}{800}} = 0.34 \pm 0.033$

• Interval: (0.307, 0.373)

Effect of Confidence Level on Margin of Error

• Increasing the confidence level (e.g., from 95% to 99%) increases the critical value $z_{\alpha/2}$, resulting in a larger margin of error and a wider interval.

• Example: For 99% confidence, $z_{\alpha/2} = 2.575$, margin of error increases to $0.043$.

Effect of Sample Size on Confidence Interval

• Larger sample sizes decrease the standard error, resulting in a smaller margin of error and a narrower confidence interval.

• This is a consequence of the Law of Large Numbers.

Interpretation and Cautions

Correct Interpretation

• A 95% confidence interval does not mean there is a 95% probability that the interval contains the parameter. The parameter is fixed; the interval is random.

• Whether a confidence interval contains the parameter depends on the sample statistic. Intervals from samples in the tails of the sampling distribution may not contain the parameter.

Visualisation

• Most sample proportions will result in intervals that contain the population proportion, but those in the tails will not.

Summary Table: Key Concepts

Concept	Definition/Formula	Notes
Sample Proportion	$\hat{p} = \frac{x}{n}$	Point estimate for population proportion
Standard Error	$\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	Spread of sampling distribution
Confidence Interval	$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	Interval estimate for population proportion
Margin of Error	$E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	Width of confidence interval
Critical Value	$z_{\alpha/2}$	Depends on confidence level

Conclusion

Confidence intervals for proportions are a powerful tool for statistical inference. Their accuracy and precision depend on sample size, confidence level, and adherence to the conditions for normality. Understanding their correct interpretation is essential for sound statistical reasoning.