Chapter 7 Study Guide

Estimating a Population Proportion

From Sample Proportion to Population Proportion

Estimating a population proportion involves using a sample proportion to infer the true proportion in the entire population. This process is foundational in inferential statistics, which allows us to make conclusions about populations based on sample data.

• Population Proportion (p): The true proportion of interest in the population, usually unknown.

• Sample Proportion (\hat{p}): Calculated as \( \hat{p} = \frac{x}{n} \), where x is the number of successes and n is the sample size.

• Point Estimate: The sample proportion \( \hat{p} \) serves as the best single estimate for p.

Example: In a survey of 400 students, 248 regularly use a reusable water bottle. \( \hat{p} = \frac{248}{400} = 0.620 \).

Why Use Confidence Intervals?

A point estimate does not account for sampling variability. Confidence intervals provide a range of plausible values for the population proportion, quantifying uncertainty.

• Confidence Interval (CI): An interval estimate for p, typically written as \( \hat{p} \pm E \), where E is the margin of error.

• Margin of Error (E): Indicates the maximum likely error in estimating p.

Confidence Level and Critical Values

The confidence level (e.g., 90%, 95%, 99%) describes the long-run success rate of the interval-building procedure. The critical value \( z_{\alpha/2} \) is determined by the chosen confidence level.

• Confidence Level: The proportion of intervals that will contain the true parameter in repeated sampling.

• Alpha (\( \alpha \)): The total area outside the confidence region; \( \alpha = 1 - \text{confidence level} \).

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

Common Critical Values:

Confidence Level	\( \alpha \)	\( z_{\alpha/2} \)
90%	0.10	1.645
95%	0.05	1.960
99%	0.01	2.576

Margin of Error Formula

The margin of error for a population proportion is calculated as:

$E = z_{\alpha/2} \sqrt{ \frac{ \hat{p} (1 - \hat{p}) }{ n } }$

Where \( \hat{p} \) is the sample proportion, n is the sample size, and \( z_{\alpha/2} \) is the critical value.

Constructing the Confidence Interval

To construct a confidence interval for p:

• Calculate \( \hat{p} \) and \( q = 1 - \hat{p} \).

• Find \( z_{\alpha/2} \) for the desired confidence level.

• Compute the margin of error E.

• CI: \( \hat{p} - E < p < \hat{p} + E \) or \( (\hat{p} - E, \hat{p} + E) \).

Example: For \( n = 400 \), \( \hat{p} = 0.620 \), 95% confidence, \( z_{\alpha/2} = 1.96 \):

$SE = \sqrt{ 0.620 \times 0.380 / 400 } \approx 0.02427$ $E = 1.96 \times 0.02427 \approx 0.04757$ $CI: (0.572, 0.668)$

Requirements for the Wald Interval

• Simple random sample

• Binomial setting: fixed n, independent trials, two outcome categories, constant p

• At least 5 successes and 5 failures

Sample Size Planning

To achieve a desired margin of error E at a selected confidence level, use:

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

If no prior estimate is available, use \( \hat{p} = \hat{q} = 0.50 \):

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

Always round up to the next whole number.

Estimating a Population Mean

From Sample Mean to Population Mean

Estimating a population mean involves using the sample mean to infer the true mean in the population. The confidence interval provides a range of plausible values for the population mean.

• Population Mean (\mu): The unknown average for all members of a population.

• Sample Mean (\bar{x}): The average measured in a sample.

• Sample Standard Deviation (s): Describes variation among individual observations.

Confidence Interval for the Mean

The confidence interval for the mean depends on whether the population standard deviation (\( \sigma \)) is known or unknown.

• Known \( \sigma \): Use z critical value. $E = z_{\alpha/2} \frac{ \sigma }{ \sqrt{n} }$

• Unknown \( \sigma \): Use sample standard deviation s and t critical value. $E = t_{\alpha/2} \frac{ s }{ \sqrt{n} }$

Most research settings use the t interval, as \( \sigma \) is rarely known.

Student's t Distribution

When \( \sigma \) is unknown, the t distribution accounts for extra uncertainty. It has heavier tails than the normal distribution, especially for small sample sizes.

• Degrees of Freedom (df): For a one-sample t procedure, \( df = n - 1 \).

• Critical t Value (\( t_{\alpha/2} \)): Depends on confidence level and df.

Example: For n = 25, df = 24, 95% confidence, \( t_{\alpha/2} = 2.064 \).

Requirements for a t Interval

• Simple random sample

• Independent observations

• Population is approximately normal OR n is sufficiently large for the Central Limit Theorem (CLT) to apply

Formula: t Confidence Interval for \( \mu \)

$\bar{x} \pm E$ $E = t_{\alpha/2} \frac{ s }{ \sqrt{n} }$ $\bar{x} - E < \mu < \bar{x} + E$

Sample Size Planning for Means

To achieve a desired margin of error E at a selected confidence level:

$n = \left( \frac{ z_{\alpha/2} \sigma }{ E } \right)^2$

Use a reasonable planning value for \( \sigma \) from prior research, pilot studies, or the range rule of thumb (\( \sigma \approx \text{range}/4 \)). Always round up.

Interpretation of Confidence Intervals

• The confidence interval estimates the population parameter (proportion or mean).

• The confidence level describes the procedure's long-run coverage, not the probability for a fixed parameter.

• Wider intervals result from higher confidence levels, smaller sample sizes, or greater variability.

Visual Representation: Two-Tailed Confidence Interval

The shaded central region of the normal curve represents the confidence interval, while the tails represent the area outside the confidence level (\( \alpha \)). For a 95% confidence interval, \( \alpha = 0.05 \) is split between the two tails.

Summary Table: Confidence Interval Components

Parameter	Point Estimate	Critical Value	Standard Error	Margin of Error	Interval
Proportion (p)	\( \hat{p} \)	\( z_{\alpha/2} \)	\( \sqrt{ \hat{p}(1-\hat{p})/n } \)	\( z_{\alpha/2} \sqrt{ \hat{p}(1-\hat{p})/n } \)	\( \hat{p} \pm E \)
Mean (\mu)	\( \bar{x} \)	\( t_{\alpha/2} \)	\( s/\sqrt{n} \)	\( t_{\alpha/2} \times s/\sqrt{n} \)	\( \bar{x} \pm E \)

Key Takeaways

• Use sample statistics to estimate population parameters.

• Confidence intervals quantify uncertainty in estimation.

• Critical values depend on confidence level and sample size.

• Check requirements before applying interval formulas.

• Plan sample size to achieve desired precision.

Additional info: The notes emphasize the importance of proper sampling methods, the limitations of the Wald interval, and the use of alternative methods (such as Wilson score or bootstrap) when assumptions are questionable.