Confidence Intervals for Population Proportions (Chapter 6.3) – Study Notes

Confidence Intervals

Introduction to Confidence Intervals for Population Proportions

Confidence intervals are a fundamental concept in inferential statistics, allowing us to estimate population parameters based on sample data. In this section, we focus on constructing and interpreting confidence intervals for population proportions, which are especially useful when dealing with categorical data (e.g., success/failure outcomes).

Confidence Intervals for Population Proportions

Population Proportion and Point Estimate

• Population Proportion (p): The probability of success in a single trial of a binomial experiment. It represents the true proportion of the population with a certain characteristic.

• Sample Proportion (\( \hat{p} \)): The point estimate for the population proportion, calculated as the proportion of successes in the sample.

The sample proportion is given by:

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

where x is the number of successes in the sample and n is the sample size.

The point estimate for the population proportion of failures is:

$\hat{q} = 1 - \hat{p}$

• Notation: \( \hat{p} \) is read as "p-hat" and \( \hat{q} \) as "q-hat".

Example: Finding a Point Estimate for p

• Suppose in a survey of 1100 U.S. adults, 902 said they plan on traveling this summer.

• Here, x = 902 and n = 1100.

• The sample proportion is:

$\hat{p} = \frac{902}{1100} \approx 0.820$

This means the point estimate for the population proportion of U.S. adults who plan on traveling this summer is approximately 82%.

Constructing a Confidence Interval for a Population Proportion

A c-confidence interval for the population proportion p is an interval constructed from sample data so that, with probability c, the interval contains the true population proportion. The process assumes the estimation is repeated many times.

Steps to Construct a Confidence Interval for p

1. Calculate the sample proportion \( \hat{p} \).

2. Verify that the sampling distribution of \( \hat{p} \) can be approximated by a normal distribution (i.e., both \( n\hat{p} \geq 5 \) and \( n\hat{q} \geq 5 \)).

3. Find the critical value \( z_c \) corresponding to the desired confidence level.

4. Compute the margin of error (E):

$E = z_c \sqrt{\frac{\hat{p}\hat{q}}{n}}$

5. Construct the confidence interval:

$\left( \hat{p} - E,\ \hat{p} + E \right)$

Example: Constructing a 95% Confidence Interval for p

• Using the previous example (\( \hat{p} = 0.820 \), n = 1100), construct a 95% confidence interval.

• Assume \( z_c = 1.96 \) for 95% confidence.

• Calculate \( \hat{q} = 1 - 0.820 = 0.180 \).

• Compute the margin of error:

$E = 1.96 \sqrt{\frac{0.820 \times 0.180}{1100}} \approx 0.023$

• Confidence interval:

$(0.820 - 0.023,\ 0.820 + 0.023) = (0.797,\ 0.843)$

Interpretation: With 95% confidence, the interval from 79.7% to 84.3% contains the true population proportion of U.S. adults who plan on traveling this summer.

Example: Constructing a 99% Confidence Interval for p

• Suppose a survey of 800 U.S. adults finds that 384 prefer to get their local news from news websites or social media.

• \( \hat{p} = \frac{384}{800} = 0.48 \)

• \( \hat{q} = 1 - 0.48 = 0.52 \)

• For 99% confidence, \( z_c = 2.576 \).

• Margin of error:

$E = 2.576 \sqrt{\frac{0.48 \times 0.52}{800}} \approx 0.045$

• Confidence interval:

$(0.48 - 0.045,\ 0.48 + 0.045) = (0.435,\ 0.525)$

Interpretation: With 99% confidence, the interval from 43.5% to 52.5% contains the true population proportion of U.S. adults who prefer to get their local news from news websites or social media.

Determining Minimum Sample Size for Estimating p

To estimate a population proportion with a specified margin of error (E) and confidence level (c), the minimum sample size required is:

$n = \hat{p}\hat{q} \left( \frac{z_c}{E} \right)^2$

• If no preliminary estimate for \( \hat{p} \) is available, use \( \hat{p} = 0.5 \) (maximizes the product \( \hat{p}\hat{q} \)).

• If n is not a whole number, always round up to the next whole number.

Example: Minimum Sample Size with No Preliminary Estimate

• Estimate, with 95% confidence, the proportion of registered voters who will vote for a candidate, with a margin of error of 3% (E = 0.03).

• No preliminary estimate: \( \hat{p} = 0.5 \), \( \hat{q} = 0.5 \), \( z_c = 1.96 \).

$n = 0.5 \times 0.5 \left( \frac{1.96}{0.03} \right)^2 = 0.25 \times \left(65.33\right)^2 \approx 0.25 \times 4269.44 = 1067.36$

• Round up: n = 1068

Example: Minimum Sample Size with Preliminary Estimate

• Suppose a preliminary estimate gives \( \hat{p} = 0.4 \), \( \hat{q} = 0.6 \).

$n = 0.4 \times 0.6 \left( \frac{1.96}{0.03} \right)^2 = 0.24 \times 4269.44 = 1024.67$

• Round up: n = 1025

With no preliminary estimate, a larger sample size is required. Using a preliminary estimate can reduce the required sample size.

Summary Table: Confidence Interval Construction for Proportions

Step	Description	Formula
1	Calculate sample proportion	$\hat{p} = \frac{x}{n}$
2	Check normality	$n\hat{p} \geq 5$, $n\hat{q} \geq 5$
3	Find critical value	$z_c$ (from z-table)
4	Compute margin of error	$E = z_c \sqrt{\frac{\hat{p}\hat{q}}{n}}$
5	Construct interval	$(\hat{p} - E,\ \hat{p} + E)$

Additional info: The above notes are based on standard procedures for constructing confidence intervals for population proportions, as outlined in introductory statistics textbooks. The examples and formulas are representative of typical exam and homework problems in college-level statistics courses.