Section 9.1: Estimating a Population Proportion

Objective 1: Obtain a Point Estimate for the Population Proportion

Estimating a population proportion is a fundamental task in inferential statistics. A point estimate provides a single value as an estimate of a population parameter, such as the proportion of individuals in a population with a certain characteristic.

• Point Estimate: The value of a statistic that estimates the value of a population parameter. For a population proportion, the point estimate is the sample proportion, denoted as or , where is the number of successes and is the sample size.

• Example: A poll of 1015 Americans aged 18 and older found that 731 said they pay income tax. The point estimate for the proportion is .

Objective 2: Construct and Interpret a Confidence Interval for the Population Proportion

Because sample statistics vary from sample to sample, we use confidence intervals to express a range of plausible values for the population proportion. The confidence interval provides an interval estimate, which is more informative than a single point estimate.

• Confidence Interval: An interval of numbers based on a point estimate that is likely to contain the true value of an unknown population parameter.

• Level of Confidence: The expected proportion of intervals that will contain the parameter if a large number of samples are taken. Common levels are 90%, 95%, and 99%.

• Interpretation: A 95% confidence interval means that if we were to take many samples and construct intervals in the same way, about 95% of those intervals would contain the true population proportion.

• Formula for Confidence Interval for Population Proportion:

• = sample proportion

• = critical value from the standard normal distribution for the desired confidence level

• = sample size

• Critical Value: The value corresponds to the desired confidence level (e.g., 1.96 for 95%).

• Example: In a poll, 731 out of 1015 Americans said they pay income tax. For a 95% confidence interval, , , .

Review of the Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion describes the behavior of sample proportions from repeated samples.

• Shape: Approximately normal if and and the sample size is less than 5% of the population.

• Mean: The mean of the sampling distribution is the population proportion .

• Standard Deviation:

Key Ideas Regarding Confidence Intervals

• A confidence interval for an unknown parameter consists of an interval of numbers based on a point estimate.

• The level of confidence represents the expected proportion of intervals that will contain the parameter if many samples are taken.

• Whether a confidence interval contains the population parameter depends only on the value of the sample statistic, not the population parameter itself.

• As the level of confidence increases, the critical value increases, resulting in a wider interval.

Critical Values for Common Confidence Levels

Margin of Error

The margin of error quantifies the maximum expected difference between the true population proportion and the sample estimate.

• Formula:

• As sample size increases, the margin of error decreases.

• As the level of confidence increases, the margin of error increases.

• Example: For a sample proportion of 0.72, , and , the margin of error is .

Objective 3: Determine the Sample Size Necessary for Estimating a Population Proportion within a Specified Margin of Error

Determining the required sample size is essential for planning studies to achieve a desired margin of error at a given confidence level.

• Formula for Sample Size:

• If no prior estimate of is available, use for maximum sample size.

• Example: To estimate a proportion with a margin of error of 0.02 at 95% confidence, use , , .

Summary Table: Confidence Interval Construction for Population Proportion

Applications and Examples

• Interpreting Confidence Intervals: A 95% confidence interval for the proportion of Americans who pay tax might be (0.69, 0.75). This means we are 95% confident the true proportion is between 0.69 and 0.75.

• Determining Sample Size: An economist wants to estimate the proportion of U.S. population who commutes to work. If a prior estimate is available, use it in the sample size formula; otherwise, use 0.5.

Additional info: These notes expand on the brief points in the original file, providing full definitions, formulas, and examples for clarity and completeness.