3.5.7 One and Two Sample Proportion

One and Two Sample Proportion Foundations of Proportion Analysis What is a Proportion? A proportion is a special kind of average used when data are in categories such as pass/fail, yes/no, or defect/acceptable. - Definition: A proportion is the fraction of observations in a category of interest. - Notation: - ( p ) = true population proportion (unknown) - ( \hat{p} ) = sample proportion (estimate of ( p )) - Formula for sample proportion: [ \hat{p} = \frac{x}{n} ] where - ( x ) = number of “successes” (events of interest) - ( n ) = total sample size Examples: - Defect rate in a process: “defective” vs “non-defective” - Response to a survey: “satisfied” vs “not satisfied” - Compliance: “meets spec” vs “does not meet spec” When to Use Proportion Methods Use proportion-based methods instead of mean-based methods when: - The outcome is binary (two categories). - The metric of interest is a percentage, rate, or fraction. - You count how many items fall into a category, rather than measuring a continuous quantity. Examples of appropriate questions: - Has the defect rate improved after a process change? - Is the current nonconformance proportion higher than the specified target? - Is the defect rate different between two machines, shifts, or suppliers? Data Structure and Assumptions Bernoulli and Binomial Context Binary data can be modeled using: - Bernoulli trial: a single trial with two possible outcomes (success/failure). - Binomial distribution: number of successes ( X ) in ( n ) independent Bernoulli trials with success probability ( p ). Key properties: - ( X \sim \text{Binomial}(n, p) ) - Sample proportion ( \hat{p} = X/n ). Core Assumptions For standard one- and two-sample proportion methods: - Independence: - Each observation is independent of others. - The outcome of one item does not affect another. - Binomial structure: - Fixed number of trials ( n ). - Each trial has the same probability ( p ) of success. - Only two outcomes per trial. - Sample size condition for normal approximation: - For one-sample: - ( n\hat{p} \ge 5 ) and ( n(1 - \hat{p}) \ge 5 ) (some sources use 10 instead of 5; be conservative in borderline cases). - For two-sample: - ( n1\hat{p}1,, n1(1-\hat{p}1),, n2\hat{p}2,, n2(1-\hat{p}2) ) all sufficiently large. When these conditions hold, the sampling distribution of ( \hat{p} ) (or ( \hat{p}1 - \hat{p}2 )) can be approximated by a normal distribution, which enables z-tests and confidence intervals. One-Sample Proportion: Concepts and Testing One-Sample Proportion Scenario A one-sample proportion situation compares a single observed proportion to a reference value (target or specification). Typical questions: - Is the nonconformance rate greater than the customer requirement? - Has the defect rate fallen below a specified goal? - Is the proportion of late deliveries different from a historical rate? Components: - Observed sample proportion ( \hat{p} = x/n ) - Hypothesized population proportion ( p_0 ) (target or baseline) Hypotheses for One-Sample Proportion Formulate hypotheses using the population proportion ( p ): - Two-sided test (difference in either direction): - ( H0: p = p0 ) - ( Ha: p \ne p0 ) - One-sided test (increase of concern): - ( H0: p \le p0 ) - ( Ha: p > p0 ) - One-sided test (decrease of concern): - ( H0: p \ge p0 ) - ( Ha: p < p0 ) Choose: - Two-sided when any difference matters. - One-sided when only higher or only lower values are practically important. Test Statistic for One-Sample Proportion Assuming the normal approximation is valid: - Standard error under ( H_0 ): [ SE0 = \sqrt{\frac{p0(1 - p_0)}{n}} ] - z-test statistic: [ z = \frac{\hat{p} - p0}{SE0} ] Interpretation: - Large positive ( z ): sample proportion much higher than ( p_0 ). - Large negative ( z ): sample proportion much lower than ( p_0 ). - Near-zero ( z ): sample proportion close to ( p_0 ). The p-value is calculated from the standard normal distribution using the chosen alternative hypothesis. One-Sample Proportion Confidence Interval To estimate the true proportion ( p ), construct a confidence interval (CI): - Standard error using sample estimate: [ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} ] - Confidence interval: [ \hat{p} \pm z_{\alpha/2} \times SE ] where: - ( z_{\alpha/2} ) is the critical value for the desired confidence level (for example, 1.96 for 95%). - The interval provides a plausible range for the true proportion. Interpretation: - If the hypothesized proportion ( p_0 ) lies outside the CI, it is inconsistent with the data at that confidence level. Sample Size for One-Sample Proportion Tests To plan a study, determine the sample size needed to detect a meaningful difference with acceptable risk. Key ingredients: - Baseline or target proportion ( p_0 ). - Minimum detectable difference ( \Delta = |p - p_0| ) considered practically important. - Significance level ( \alpha ) (probability of Type I error). - Power ( 1 - \beta ) (probability of detecting the difference, where ( \beta ) is Type II error). Approximate formulas use the normal approximation and involve both: - ( z_\alpha ): critical value for significance level. - ( z_\beta ): critical value for power. Conceptual relationships: - Larger ( \Delta ) → smaller sample size. - Lower ( \alpha ) or higher power → larger sample size. - Proportions closer to 0.5 require larger samples (maximum variability). Statistical software is typically used to compute the final sample size. Two-Sample Proportion: Concepts and Testing Two-Sample Proportion Scenario Two-sample proportion analysis compares proportions from two independent groups. Examples: - Defect rate on Line A vs Line B. - Error rate before vs after a process change (if treated as independent samples). - Proportion of on-time deliveries from Supplier 1 vs Supplier 2. Components: - Group 1: ( x1 ) successes in ( n1 ) trials → ( \hat{p}1 = x1/n_1 ). - Group 2: ( x2 ) successes in ( n2 ) trials → ( \hat{p}2 = x2/n_2 ). Hypotheses for Two-Sample Proportion Use the population proportions ( p1 ) and ( p2 ): - Two-sided test: - ( H0: p1 = p_2 ) - ( Ha: p1 \ne p_2 ) - One-sided test (greater): - ( H0: p1 \le p_2 ) - ( Ha: p1 > p_2 ) - One-sided test (less): - ( H0: p1 \ge p_2 ) - ( Ha: p1 < p_2 ) The choice depends on the practical question: - Are the rates different at all? - Is one rate specifically higher or lower than the other? Test Statistic for Two-Sample Proportion When testing equality, a pooled estimate of the common proportion under ( H_0 ) is used: - Pooled proportion: [ \hat{p} = \frac{x1 + x2}{n1 + n2} ] - Standard error under ( H_0 ): [ SE0 = \sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{n1} + \frac{1}{n_2}\right)} ] - z-test statistic: [ z = \frac{\hat{p}1 - \hat{p}2}{SE_0} ] Interpretation: - Large positive ( z ): sample suggests ( p1 > p2 ). - Large negative ( z ): sample suggests ( p1 < p2 ). - Near-zero ( z ): sample proportions are similar. The p-value is obtained from the standard normal distribution based on the alternative hypothesis. Two-Sample Proportion Confidence Interval To estimate the difference ( p1 - p2 ), use an unpooled standard error: - Unpooled standard error: [ SE = \sqrt{\frac{\hat{p}1(1 - \hat{p}1)}{n1} + \frac{\hat{p}2(1 - \hat{p}2)}{n2}} ] - Confidence interval for difference: [ (\hat{p}1 - \hat{p}2) \pm z_{\alpha/2} \times SE ] Interpretation: - If the CI for ( p1 - p2 ) includes 0, there is no statistically significant difference at that confidence level. - The sign of the CI (all positive or all negative) shows which group has the higher proportion. - The width of the CI reflects precision (narrower = more precise). Note the distinction: - Hypothesis test: uses pooled standard error under null. - Confidence interval: uses unpooled standard error based on separate estimates. Sample Size for Two-Sample Proportion Tests Planning a two-sample comparison requires: - Baseline proportion ( p_1 ) (for example, current rate). - Expected or targeted new proportion ( p_2 ). - Difference of interest ( \Delta = |p1 - p2| ). - Significance level ( \alpha ). - Desired power ( 1 - \beta ). - Allocation of sample sizes (often ( n1 = n2 ) for simplicity). Conceptual effects: - Smaller target difference → larger sample sizes. - More extreme proportions (near 0 or 1) usually allow smaller samples than near 0.5 for the same difference and error levels. - Unequal sample sizes can be used, but balanced designs are typically more efficient for a given total sample size. Software uses normal approximations to solve for required ( n1 ) and ( n2 ). Interpreting and Applying Proportion Results Statistical Versus Practical Significance It is important to distinguish: - Statistical significance: - Based on p-value and confidence interval. - Addresses whether the observed difference is unlikely under the null hypothesis. - Practical (or practical business) significance: - Based on the size of the difference in proportion. - Addresses whether the difference matters in real-world terms. Examples: - A tiny reduction in defect rate (for example, 0.2%) might be statistically significant with a very large sample but might not justify a major process change. - A moderate change (for example, from 4% to 2% defects) may be both statistically and practically meaningful. Error Types and Risks Key error concepts in proportion testing: - Type I error (α): - Concluding there is a difference when none exists. - Controlled by choosing the significance level (for example, 0.05). - Type II error (β): - Failing to detect a true difference of interest. - Controlled by sample size and chosen power. In planning and interpreting tests: - Lower ( \alpha ) protects against false alarms. - Higher power protects against missing important changes. - There is a trade-off: reducing both errors simultaneously typically requires larger samples. Checking Assumptions and Common Pitfalls Practical checks before relying on proportion results: - Independence check: - Avoid using multiple observations from the same item as separate trials. - Be cautious with data from time sequences where dependence may exist (for example, time-series clustering). - Sample size condition: - Verify that expected numbers of successes and failures in each group are large enough. - If any expected count is very small, consider exact tests (such as exact binomial or Fisher’s exact) rather than normal approximations. - Misclassification risk: - Ensure that the definition of “success” is consistent and that classification errors are minimized. - Mislabeling can bias estimated proportions and distort conclusions. - Multiple testing: - When testing many proportions (for example, many segments or suppliers), the chance of at least one false positive increases. - Recognize that without adjustment, some significant results may arise by chance alone. Summary One- and two-sample proportion methods are used when outcomes are binary and the key metric is a proportion or percentage. For one-sample problems, the focus is on comparing a single observed proportion ( \hat{p} ) with a reference value ( p0 ), using a z-test and confidence interval based on the binomial distribution and its normal approximation. For two-sample problems, the goal is to compare proportions from two independent groups, using pooled standard error for hypothesis testing and unpooled standard error for confidence intervals on ( p1 - p_2 ). Sound application requires understanding and checking assumptions (independence, binomial structure, adequate sample size for the normal approximation), setting clear hypotheses, interpreting p-values and confidence intervals, and distinguishing statistical from practical significance. Thoughtful sample size planning ensures that tests have adequate power to detect meaningful differences in proportions, while controlling error risks.