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 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 (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.

• Formula: where is the number of successes and is the sample size.

• Example: If 329 out of 362 drive-thru orders at McDonald's are accurate, then .

Confidence Interval for a Population Proportion

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 is a range (or interval) of values used to estimate the value of a population parameter.

• Notation: Confidence intervals are sometimes denoted by CI.

• Confidence Level: The probability (e.g., 0.90, 0.95, 0.99) that a confidence interval contains the true population parameter if the situation is repeated many times.

• Table: Confidence Level and Corresponding α

• Example: For a 95% confidence interval for the inaccurate drive-thru orders, .

Interpreting Confidence Intervals

It is essential to interpret confidence intervals correctly. The confidence level refers to the proportion of intervals that will contain the true population parameter if the process is repeated many times.

• Correct Interpretation: "We are 95% confident that the interval 0.0615 to 0.121 contains the true value of the population proportion."

• Common Mistake: "There is a 95% chance that the true value of p is between 0.0615 and 0.121." This is incorrect because the true value is fixed; the interval either contains it or not.

• Visual Example: A diagram showing confidence intervals from 20 samples, where most intervals contain the true value .

Critical Values

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

• Definition: A critical value is a value that separates sample statistics from significantly lower or higher values that are not significant.

• Notation: is the critical value (z-score) with area to the right in the standard normal distribution.

• Table: Common Critical Values

• Example: For a 90% confidence level, .

Margin of Error

The margin of error is the maximum likely difference between the sample proportion and the population proportion.

• Formula:

• Example: For , , and :

Constructing a Confidence Interval

To construct a confidence interval for a population proportion, follow these steps:

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

2. Compute the sample proportion .

3. Find the critical value for the desired confidence level.

4. Calculate the margin of error .

5. Construct the interval:

• Example: For , , the 95% confidence interval is .

Analyzing Polls and Reliability

Polls are commonly used to estimate population proportions, but their reliability depends on sampling methods and sample size.

• The sample should be a simple random sample.

• The confidence level must be specified.

• The quality of poll results depends on the sampling method and sample size, not the size of the population.

Determining Sample Size

Calculating the required sample size is important for estimating a population proportion with a specified margin of error.

• Formula (when is known):

• Formula (when is unknown):

• Example: To estimate the percentage of California residents that are left-handed with a margin of error of 3% and 95% confidence level:

Calculator Instructions

Many calculators have built-in functions to compute confidence intervals for proportions. 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 confidence interval.

Summary Table: Key Concepts

Additional info:

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

• Always round sample size up to the next whole number.

• Margin of error must be converted to decimal form before use in formulas.