Confidence Intervals for Proportions: Concepts, Formulas, and Applications

Chapter 13 (Part II): Confidence Intervals for Proportions

Introduction to Confidence Intervals for Proportions

Confidence intervals for proportions are a fundamental concept in inferential statistics, allowing us to estimate the true proportion of a population based on sample data. This chapter covers the properties of the sampling distribution of sample proportions, the construction of confidence intervals, and interpretation of results.

Properties of the Sampling Distribution of Sample Proportion p̂

• Definition: The sample proportion (p̂) is calculated from a random sample of n observations selected from a population with proportion p.

• Mean of p̂: $\mu_{\hat{p}} = p$

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

• Normal Approximation: The sampling distribution of p̂ is approximately normal if:

  • $np > 10$

  • $n(1-p) > 10$

Constructing a Confidence Interval for Population Proportion p

To estimate the unknown population proportion p, we use the sample proportion p̂ and construct a confidence interval.

• General Formula: $\hat{p} - z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ to $\hat{p} + z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• z-value: The z-value depends on the desired level of confidence and is obtained from standard normal tables.

  • For 90% confidence: $z = 1.645$

  • For 95% confidence: $z = 1.96$

  • For 99% confidence: $z = 2.576$

Conditions for Valid Confidence Intervals

• Random Sampling: The sample must be randomly selected from the population.

• Sample Size: The sample must be large enough so that:

  • $n\hat{p} > 10$

  • $n(1-\hat{p}) > 10$

Margin of Error and Standard Error

• Margin of Error (ME): The amount added and subtracted from p̂ in the confidence interval. $ME = z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Standard Error (SE): The estimated standard deviation of p̂. $SE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Alternative Formula: $\text{Estimate} \pm z \cdot SE(\hat{p})$

Interpreting Confidence Intervals

• Meaning of Confidence Level: For example, a 95% confidence interval means that if we were to take many samples and construct intervals in this way, about 95% of those intervals would contain the true population proportion.

• Effect of Confidence Level: Increasing the confidence level (e.g., from 90% to 99%) makes the interval wider, reflecting greater uncertainty.

• Effect of Sample Size: Increasing the sample size makes the interval narrower, reflecting more precise estimation.

Examples and Applications

• Crime Rate Example:

  • A commission wants to estimate the percentage of crimes involving firearms in a high-crime area. Out of 600 randomly selected crime files, 380 involved firearms.

  • To construct a 99% confidence interval for the proportion, use: $\hat{p} = \frac{380}{600} = 0.633$ $n = 600$ $z = 2.576$ (for 99% confidence)

  • Plug into the formula: $0.633 \pm 2.576 \sqrt{\frac{0.633 \times 0.367}{600}}$

• Shoplifting Example:

  • In a store, 6 out of 120 customers are detected as shoplifters. $\hat{p} = \frac{6}{120} = 0.05$ $n = 120$

  • For a 90% confidence interval ($z = 1.645$): $0.05 \pm 1.645 \sqrt{\frac{0.05 \times 0.95}{120}}$

• Coin Toss Example:

  • A quarter is spun 250 times, resulting in 140 heads ($\hat{p} = 0.56$). The 90% confidence interval is (0.51, 0.61).

  • Interpretation: We are 90% confident that the true proportion of heads for this quarter is between 51% and 61%.

Comparison of Confidence Levels and Sample Sizes

Factor	Effect on Interval Width
Higher Confidence Level	Wider interval
Larger Sample Size	Narrower interval

Common Misinterpretations

• The confidence interval does not mean that the population proportion is guaranteed to be within the interval for every sample.

• It does not mean that 51% to 61% of all quarters are unfair (in the coin example).

• It means that, in repeated sampling, 90% of constructed intervals will contain the true proportion.

Summary Table: Confidence Interval for Proportion

Symbol	Meaning	Formula
p̂	Sample proportion	$\frac{x}{n}$
p	Population proportion	Unknown
SE(p̂)	Standard error	$\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
ME	Margin of error	$z \cdot SE(\hat{p})$
CI	Confidence interval	$\hat{p} \pm z \cdot SE(\hat{p})$

Additional info: The notes infer standard textbook explanations and examples for clarity and completeness, including the interpretation of confidence intervals and the effects of sample size and confidence level.