Estimating a Population Proportion and Confidence Intervals

7.1 Estimating a Population Proportion

Introduction

Estimating a population proportion is a fundamental concept in inferential statistics. It involves using sample data to make inferences about the proportion of a certain characteristic in a population. This section covers point estimates, confidence intervals, margin of error, and sample size determination for population proportions.

Point Estimate

A point estimate is a single value used to estimate a population parameter. For population proportions, the sample proportion, denoted as p̂, is the best point estimate of the true population proportion p.

• Definition: The sample proportion p̂ is calculated as the number of successes divided by the total sample size.

• Example: If 329 out of 362 drive-thru orders at McDonald's are accurate, then $\hat{p} = \frac{329}{362} = 0.90912$.

Confidence Interval

A confidence interval provides a range of values that is likely to contain the true population proportion. It is more informative than a point estimate because it accounts for sampling variability.

• Definition: A confidence interval for a population proportion is an interval estimate, often denoted by CL.

• Confidence Level: The probability that the interval contains the true population parameter if the sampling process is repeated many times. Common levels are 90%, 95%, and 99%.

Confidence Level	α
90%	0.10
95%	0.05
99%	0.01

• Example: For the McDonald's data, a 95% confidence interval for the proportion of accurate orders is $0.0615 < p < 0.121$.

Interpreting Confidence Intervals

It is essential to interpret confidence intervals correctly. The correct interpretation is: "We are 95% confident that the interval contains the true value of the population proportion." This means that if we were to take many samples and construct confidence intervals for each, about 95% of those intervals would contain the true population proportion.

• Common Mistake: Saying "There is a 95% chance that the true value is in the interval" is incorrect. The interval either contains the true value or it does not; the confidence level refers to the long-run proportion of intervals that contain the true value.

• Visual Example: A diagram (Figure 7-1) shows confidence intervals from 20 samples, where most intervals contain the true proportion, but some do not.

Critical Values

Critical values are used to determine the endpoints of confidence intervals. They separate significant values from non-significant ones in the standard normal distribution.

• Definition: The critical value $z_{\alpha/2}$ is the z-score with area $\alpha/2$ to the right in the standard normal distribution.

• Example: For a 90% confidence level, $z_{\alpha/2} = 1.645$.

Confidence Level	$z_{\alpha/2}$
90%	1.645
95%	1.96
99%	2.575

Margin of Error

The margin of error is the maximum likely difference between the sample proportion and the population proportion. It is denoted by E and calculated as:

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

• Notation: E = margin of error, $z_{\alpha/2}$ = critical value, $\hat{p}$ = sample proportion, n = sample size.

• Example: For $\hat{p} = 0.0912$, $n = 362$, and $z_{\alpha/2} = 1.96$, $E = 1.96 \sqrt{\frac{0.0912 \times 0.9088}{362}} = 0.0297$.

Constructing a Confidence Interval

To construct a confidence interval for a population proportion:

1. Verify requirements: Sample must be simple random, outcomes must be independent, and probabilities must be constant.

2. Compute $\hat{p}$ and $z_{\alpha/2}$.

3. Calculate margin of error E.

4. Construct the interval: $\hat{p} - E < p < \hat{p} + E$.

• Rounding: Round to three significant digits.

• Example: For the McDonald's data, $0.0615 < p < 0.121$.

Analyzing Polls and Surveys

Polls are commonly used to estimate population proportions. However, the reliability of polls depends on the sampling method and sample size. Always ensure the sample is random and representative.

• The sample should be a simple random sample.

• The confidence level must be stated.

• The quality of results depends on sampling method and sample size.

Determining Sample Size

To estimate a population proportion with a specified margin of error, the required sample size can be calculated using:

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

If $\hat{p}$ is unknown, use $n = \frac{0.25z_{\alpha/2}^2}{E^2}$

• Rounding: Round up to the next whole number.

• Example: To estimate the percentage of California residents that are left-handed with a margin of error of 3% and 95% confidence, $n = (0.10 \times 0.90 \times 1.96^2) / 0.03^2 = 384$.

Calculator Instructions

Many calculators have built-in functions for confidence intervals. For example, on the TI-83/84 Plus:

• Press STAT, select TESTS, then 1-PropZInt.

• Enter the number of successes, sample size, and confidence level.

• Select Calculate to obtain the interval.

Summary Table: Key Formulas

Concept	Formula
Sample Proportion	$\hat{p} = \frac{x}{n}$
Margin of Error	$E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
Confidence Interval	$\hat{p} - E < p < \hat{p} + E$
Sample Size	$n = \frac{\hat{p}(1-\hat{p})z_{\alpha/2}^2}{E^2}$

Additional info:

• The confidence interval described is the Wald confidence interval. Other intervals, such as the Wilson or Agresti-Coull, may perform better for small samples or extreme proportions.

• When margin of error is given as a percentage, convert to decimal before using in formulas.

• Sample size does not depend on population size for large populations.