Estimates and Sample Sizes: Confidence Intervals and Sample Size Determination

Chapter 7: Estimates and Sample Sizes

7-1 Review and Preview

This chapter marks the transition from descriptive statistics to inferential statistics. Inferential statistics involve using sample data to estimate population parameters and to test hypotheses about those parameters.

• Descriptive statistics summarize data using graphs and statistics such as the mean and standard deviation.

• Inferential statistics use sample data to estimate population values and test claims about populations.

• Key parameters estimated: proportions, means, variances.

• Methods for determining necessary sample sizes are also introduced.

7-2 Estimating a Population Proportion

This section presents methods for using a sample proportion to estimate the value of a population proportion, including constructing confidence intervals and determining required sample sizes.

• Sample proportion (\( \hat{p} \)) is the best point estimate of the population proportion (\( p \)).

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

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

• Confidence level: The probability (\( 1-\alpha \)) that the CI contains the population parameter. Common levels: 90%, 95%, 99%.

Critical Values

• A critical value (\( z_{\alpha/2} \)) is a z-score that separates likely from unlikely sample statistics under the normal distribution.

• For a 95% confidence level, \( \alpha = 0.05 \), so \( \alpha/2 = 0.025 \), and \( z_{0.025} = 1.96 \).

Margin of Error for Proportions

• The margin of error (\( E \)) is the maximum likely difference between the sample proportion and the true population proportion.

Formula:

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

where \( \hat{q} = 1 - \hat{p} \).

Confidence Interval for a Population Proportion

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

or equivalently, \((\hat{p} - E, \hat{p} + E)\).

• Round CI limits for \( p \) to three significant digits.

Example

Given: \( n = 1501, \hat{p} = 0.70 \), 95% confidence level.

• \( E = 1.96 \sqrt{\dfrac{0.70 \times 0.30}{1501}} = 0.023183 \)

• CI: $0.70 - 0.023183 < p < 0.70 + 0.023183$

• So, $0.677 < p < 0.723$

Sample Size for Estimating a Population Proportion

• To determine the required sample size \( n \) for a desired margin of error \( E \):

When \( \hat{p} \) is known:

$n = \left( \dfrac{z_{\alpha/2}^2 \hat{p} \hat{q}}{E^2} \right)$

When \( \hat{p} \) is unknown, use \( 0.25 \) for \( \hat{p}\hat{q} \):

$n = \left( \dfrac{z_{\alpha/2}^2 \times 0.25}{E^2} \right)$

• Always round \( n \) up to the next whole number.

Example

• Suppose \( \hat{p} = 0.73, \hat{q} = 0.27, z_{\alpha/2} = 1.96, E = 0.03 \):

• $n = \dfrac{(1.96)^2 (0.73)(0.27)}{(0.03)^2} = 842$

• If \( \hat{p} \) unknown: $n = \dfrac{(1.96)^2 (0.25)}{(0.03)^2} = 1068$

Summary Table: Key Formulas for Estimating a Population Proportion

Concept	Formula	Description
Margin of Error (E)	$E = z_{\alpha/2} \sqrt{\dfrac{\hat{p}\hat{q}}{n}}$	Maximum likely difference between sample and population proportion
Confidence Interval	$\hat{p} - E < p < \hat{p} + E$	Interval estimate for population proportion
Sample Size (\( \hat{p} \) known)	$n = \dfrac{z_{\alpha/2}^2 \hat{p}\hat{q}}{E^2}$	Required sample size for desired margin of error
Sample Size (\( \hat{p} \) unknown)	$n = \dfrac{z_{\alpha/2}^2 \times 0.25}{E^2}$	Conservative estimate for required sample size

Additional info:

• Critical values are found using the standard normal (z) distribution table.

• Confidence intervals provide a range of plausible values for the population parameter, not a guarantee that the parameter lies within the interval for a specific sample.