Estimating a Population Proportion and Confidence Intervals

Estimating a Population Proportion

Point Estimate

When estimating a population proportion with a single value, the best estimate is the sample proportion, denoted as \( \hat{p} \). Because \( \hat{p} \) consists of a single value that is equivalent to a point on a line, it is called a point estimate.

• Definition: A point estimate is a single value used to estimate a population parameter.

• Unbiased Estimator: The sample proportion \( \hat{p} \) is the best point estimate of the population proportion \( p \) because it is unbiased and has the smallest standard deviation among all unbiased estimators of \( p \).

Confidence Intervals for a Proportion

Definition and Interpretation

A confidence interval (CI) is a range of values used to estimate the true value of a population parameter. The confidence interval provides an estimated range that is likely to include the population proportion \( p \) with a certain level of confidence.

• Correct Interpretation: "We are 95% confident that the interval from 0.405 to 0.455 actually does contain the true value of the population proportion \( p \)." This means that if we were to select many different random samples and construct confidence intervals, about 95% of them would contain the true population proportion.

• Incorrect Interpretations: It is incorrect to say there is a 95% chance that \( p \) will fall within a specific interval, or that 95% of sample proportions will fall within the interval.

Confidence Level

The confidence level is the probability \( 1 - \alpha \) (such as 0.95 or 95%) that the confidence interval actually contains the population parameter, assuming the estimation process is repeated a large number of times.

• The confidence level is also called the degree of confidence or confidence coefficient.

Critical Value

A critical value is the number on the borderline separating sample statistics that are significantly high or low from those that are not significant. The critical value \( z_{\alpha/2} \) is a z-score that separates an area of \( \alpha/2 \) in the right tail of the standard normal distribution.

• For a 95% confidence level, \( z_{\alpha/2} = 1.96 \).

• Other common critical values are shown in the table below:

Requirements for Constructing a Confidence Interval for a Proportion

• The sample is a simple random sample.

• The conditions for the binomial distribution are satisfied: fixed number of trials, independent trials, two categories of outcomes, and constant probability for each trial.

• There are at least 5 successes and at least 5 failures (i.e., \( np \geq 5 \) and \( nq \geq 5 \)).

Margin of Error

The margin of error (E) is the maximum likely amount of error between the sample proportion \( \hat{p} \) and the population proportion \( p \). It is calculated as:

$E = z_{\alpha/2} \sqrt{\frac{\hat{p}\hat{q}}{n}}$

• \( \hat{q} = 1 - \hat{p} \)

• The margin of error is used to construct the confidence interval around the point estimate.

Confidence Interval Formula

The confidence interval for a population proportion \( p \) is given by:

$\hat{p} - E < p < \hat{p} + E$

or equivalently,

$\hat{p} \pm E$

Sampling Distribution of Sample Proportions

When certain requirements are met, the sampling distribution of sample proportions can be approximated by a normal distribution. This allows us to use z-scores and critical values to construct confidence intervals.

• Sample proportions tend to have a normal distribution if the sample size is large enough.

Determining Sample Size for Estimating a Proportion

To achieve a specific margin of error and confidence level, the required sample size can be calculated using the following formulas:

• If an estimate \( \hat{p} \) is known:

$n = \left[ \frac{z_{\alpha/2}}{E} \right]^2 \hat{p}\hat{q}$

• If no estimate is known, use \( \hat{p} = 0.5 \) for maximum variability:

$n = \left[ \frac{z_{\alpha/2}}{E} \right]^2 0.25$

Applications and Examples

Media Example: Margin of Error in Polls

Polls often report a margin of error to indicate the uncertainty in their estimates. For example, a poll of likely voters may report a margin of error of ±3 percentage points at the 95% confidence level. This means the true proportion is likely within 3 points of the reported value, 95% of the time.

Visualizing Confidence Intervals

Confidence intervals can be visualized as error bars around point estimates. In repeated sampling, about 95% of confidence intervals constructed from different samples will contain the true population proportion if the confidence level is 95%.

Example: Coin Flips

Suppose you flip a coin 10 times and calculate the sample proportion of heads. You can compute a 90% confidence interval for this point estimate using the formulas above. If your interval contains the true parameter \( p = 0.50 \), your sample is consistent with the expected probability for a fair coin.

Summary Table: Common Confidence Levels and Critical Values

Confidence Level	\( \alpha \)	Critical Value, \( z_{\alpha/2} \)
90%	0.10	1.645
95%	0.05	1.96
99%	0.01	2.575

Key Terms and Definitions

• Point Estimate: A single value used to estimate a population parameter.

• Confidence Interval (CI): A range of values used to estimate the true value of a population parameter.

• Confidence Level: The probability that the CI contains the population parameter.

• Critical Value: The z-score that separates the area \( \alpha/2 \) in the tails of the normal distribution.

• Margin of Error (E): The maximum likely difference between the sample estimate and the population parameter.