Chapter 7 – Estimating Parameters & Determining Sample Sizes

Estimating a Population Proportion

In statistics, estimating a population proportion is a fundamental process for making inferences about a population based on sample data. This section introduces the key concepts, notations, and methods used to estimate population proportions and determine appropriate sample sizes.

Point Estimate

A point estimate is a single value used to estimate a population parameter. For proportions, the sample proportion is the best point estimate of the population proportion because it is unbiased and consistent.

• Notation for Proportions:

  • x: Number of successes in the sample

  • n: Sample size

  • N: Population size

  • \( \hat{p} = \frac{x}{n} \): Sample proportion

  • \( p = \frac{x}{N} \): Population proportion

• Key Point: \( \hat{p} \) is used as the point estimate for \( p \).

Confidence Interval

A confidence interval (or interval estimate) is a range of values used to estimate the true value of a population parameter. It provides a measure of reliability for the estimate.

• Confidence Interval (CI): Often written as \( \hat{p} - E < p < \hat{p} + E \) or \( (\hat{p} - E, \hat{p} + E) \), where E is the margin of error.

Confidence Level

The confidence level is the probability that the confidence interval actually contains the population parameter, assuming the estimation process is repeated many times. It is denoted as \( 1 - \alpha \), where \( \alpha \) is the significance level.

• Common confidence levels: 90%, 95%, 99%

• \( \alpha \): The probability that the interval does not contain the parameter

Critical Value

The critical value for the standard normal distribution is a z-score that separates the area \( \alpha/2 \) in the tails. It is used to construct confidence intervals.

• Critical value (\( z_{\alpha/2} \)): The z-score corresponding to the desired confidence level.

Margin of Error

The margin of error (E) is the maximum likely amount of error between the sample statistic and the population parameter. It quantifies the uncertainty in the estimate.

• Formula for Margin of Error (for proportions):

• Where \( \hat{q} = 1 - \hat{p} \)

• \( z_{\alpha/2} \): Critical value

• n: Sample size

Constructing a Confidence Interval for a Population Proportion

To construct a \((1-\alpha)\times100\%\) confidence interval for a population proportion, use the following formulas:

• Lower Bound:

• Upper Bound:

Alternatively, the confidence interval can be expressed as \( \hat{p} \pm E \).

Interpreting a Confidence Interval

It is important to interpret confidence intervals correctly. The correct interpretation is that we are confident the interval contains the true population proportion, not that the probability the parameter is in the interval is 95% (for a 95% CI).

Determining Sample Size

Before collecting data, it is often necessary to determine the sample size required to estimate a population proportion with a specified margin of error and confidence level.

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

• When no estimate \( \hat{p} \) is known:

• If the computed sample size is not a whole number, round up to the next whole number.

Summary Table: Common Confidence Levels, Alpha, and Critical Values

Examples and Applications

• Example: If a survey of 1144 Americans finds that 51% approve of a nominee, and the margin of error is ±3.5%, the confidence interval is (0.475, 0.545).

• Application: Determining the sample size needed for a survey to estimate the proportion of e-cigarette users with a margin of error of 1.5% at 95% confidence.

Additional info: These methods are foundational for inferential statistics and are widely used in survey analysis, quality control, and scientific research.