Inference About a Population Proportion: Hypothesis Testing, Confidence Intervals, and Sample Size

Inference About a Population Proportion

Overview

Statistical inference about a population proportion involves using sample data to estimate and test hypotheses about the proportion of individuals in a population with a specific characteristic. This is fundamental in analyzing categorical data, especially in fields such as biology, medicine, and social sciences.

Conditions for Inference on Proportions

Assumptions

• Random Sample: The data used for the estimate must be a random sample from the population studied.

• Population Size: The population should be at least 20 times as large as the sample.

• Sample Size: The sample size n must be large enough so that the sampling distribution of the sample proportion is approximately normal.

The Sample Proportion p̂

Definition and Calculation

• Categorical Data: Inference is drawn on the proportion or percentage of the population with a specific characteristic.

• Success: If a categorical characteristic is defined as "success," the sample proportion of successes is: $\hat{p} = \frac{\text{count of successes in the sample}}{\text{count of observations in the sample}}$

• Example: In a group of 120 Herpes patients treated with a new drug, 30 get better: $\hat{p} = \frac{30}{120} = 0.25$

The Sampling Distribution of p̂

Normal Approximation

• When n is large and p is not close to 0 or 1, the sampling distribution of p̂ is approximately normal:

$\hat{p} \sim N \left( \mu_{\hat{p}} = p, \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \right)$

• The mean and standard deviation (width) of the sampling distribution are determined by p and n.

Significance Test for a Proportion

Hypothesis Testing

• Null Hypothesis: $H_0: p = p_0$ (where $p_0$ is the value being tested)

• If $H_0$ is true, the sampling distribution is: $\hat{p} \sim N \left( p_0, \sqrt{\frac{p_0(1-p_0)}{n}} \right)$

• Test Statistic: The standardized value of p̂: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

p-value for One or Two-Sided Alternatives

• The p-value is the probability, if $H_0$ is true, of obtaining a test statistic as extreme or more extreme than the observed value, in the direction of the alternative hypothesis $H_1$.

Alternative Hypothesis	p-value Calculation
$H_1: p > p_0$	$P(Z \geq z)$
$H_1: p < p_0$	$P(Z \leq z)$
$H_1: p \neq p_0$	$2P(Z \geq |z|)$

Example: Genetic Inheritance

Mendel's Law and Hypothesis Test

• Mendel's first law states that crossing dominant and recessive homozygote parents yields a second generation (F2) with 75% dominant-trait individuals.

• Data: 5474 smooth peas, 1850 wrinkled peas ($n = 5474 + 1850 = 7324$).

• Test: Is the proportion of smooth peas in F2 different from 75%?

• Sample proportion: $\hat{p} = \frac{5474}{7324}$

• Test statistic: $z = \frac{\hat{p} - 0.75}{\sqrt{\frac{0.75 \times 0.25}{7324}}}$

• Calculate p-value and draw conclusion based on significance level.

Confidence Interval for p

Estimating the Population Proportion

• When p is unknown, both the center and spread of the sampling distribution are unknown and must be estimated.

• Two main methods:

  1. Use the sample proportion $\hat{p}$

  2. Use an improved $\hat{p}$ via the "plus four method"

Large-Sample Confidence Interval for p

Formula and Calculation

• For a simple random sample (SRS) of size n from a large population, with sample proportion $\hat{p}$:

$\hat{p} \pm m$

• Where: $m = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ $SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• $z^*$ is the critical value for the desired confidence level (e.g., $z^* = 1.645$ for 90% confidence).

Example: Medication Side Effects

• Of 440 arthritis patients taking a pain reliever, 23 suffered adverse symptoms.

• Sample proportion: $\hat{p} = \frac{23}{440}$

• 90% confidence level: $z^* = 1.645$

• Margin of error: $m = 1.645 \sqrt{\frac{\hat{p}(1-\hat{p})}{440}}$

• Confidence interval: $\hat{p} \pm m$

Confidence Level C	z*
0.90	1.645
0.95	1.960
0.99	2.576

"Plus Four" Confidence Interval for p

Improved Estimation Method

• Add four imaginary observations: two successes and two failures.

• New sample size: $n + 4$

• New count of successes: $X + 2$

• "Plus four" estimate: $\hat{p} = \frac{\text{count of successes} + 2}{\text{count of all observations} + 4}$

• Margin of error: $m = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n+4}}$

• Confidence interval: $\hat{p} \pm m$

Example: Medication Side Effects (Plus Four Method)

• Apply the plus four method to the arthritis pain reliever data.

• Calculate $\hat{p}$ and $m$ as above, using $n+4$ and $X+2$.

• Construct the 90% confidence interval.

Sample Size for a Desired Margin of Error

Planning Studies

• To estimate a population proportion with a specified margin of error m at a given confidence level:

$m = z^* \sqrt{\frac{p^*(1-p^*)}{n}} \implies n = \left( \frac{z^*}{m} \right)^2 p^*(1-p^*)$

• $p^*$ is a guessed likely value for the population proportion (use $p^* = 0.5$ for the most conservative estimate).

Example: Medication Side Effects (Sample Size Calculation)

• What sample size is needed to achieve a margin of error no more than 0.01 (1 percentage point) with 90% confidence?

• Assume $p^* = 0.1$ (since no more than 10% of patients should suffer adverse symptoms).

• For $z^* = 1.645$: $n = \left( \frac{1.645}{0.01} \right)^2 (0.1)(0.9) = 2434.4 \approx 2435$

Confidence Level C	z*
0.90	1.645
0.95	1.960
0.99	2.576

Summary Table: Key Formulas for Inference About a Proportion

Concept	Formula (LaTeX)
Sample Proportion	$\hat{p} = \frac{\text{count of successes}}{\text{sample size}}$
Standard Error	$SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
Test Statistic (z)	$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$
Confidence Interval (Large Sample)	$\hat{p} \pm z^* SE$
Confidence Interval (Plus Four)	$\hat{p} = \frac{\text{successes} + 2}{n + 4}$ $m = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n+4}}$
Sample Size for Margin of Error	$n = \left( \frac{z^*}{m} \right)^2 p^*(1-p^*)$

Additional info: These notes expand on the original slides by providing full definitions, formulas, and step-by-step examples for each concept, ensuring a self-contained study guide for exam preparation.