Math FPX 2002 Assessment 3 Probability and Statistical Inference Sample

Assessment 3: Probability and Statistical Inference

Sarah Mitchell
MATH FPX 2002 - Statistics and Probability
February 17, 2026

Introduction

Statistical inference constitutes a fundamental component of modern quantitative analysis, enabling researchers to draw conclusions about population parameters based on sample data. The application of probability distributions to pharmaceutical manufacturing quality control demonstrates the practical relevance of inferential statistics in industrial settings. This analysis examines tablet weight measurements from a manufacturing process designed to maintain a 100-milligram specification, employing both parameter estimation and hypothesis testing methodologies. The theoretical foundations of statistical inference, established primarily through the work of Ronald Fisher (1925) and the Neyman-Pearson Lemma (1933), provide the framework for evaluating whether observed deviations from specification constitute random variation or systematic process issues. The research objective is to determine whether the machine operates within acceptable tolerance using appropriate probability distributions and confidence interval estimation.

Methodology

The dataset comprises 50 tablet weight measurements obtained from a pharmaceutical manufacturing line configured to produce 100-milligram tablets. Data collection occurred through automated measurement systems designed to minimize measurement error, with results recorded to one decimal place. The normal distribution N(μ, σ²) was selected as the appropriate probability model for tablet weights, based on the Central Limit Theorem and empirical observation that pharmaceutical production variables typically approximate normality (Moore, Notz, & Flinger, 2021). The analysis employed a one-sample t-test with a significance level of α = 0.05, selected because the population standard deviation is unknown and the sample size (n = 50) satisfies the adequacy condition for t-distribution approximation (Keller & Warrack, 2023). A 95% confidence interval was constructed to estimate the true population mean tablet weight. The null hypothesis H₀: μ = 100.0 milligrams represents the manufacturing specification, while the alternative hypothesis H₁: μ ≠ 100.0 milligrams permits detection of deviations in either direction. This two-tailed test reflects the practical reality that tablet weights both exceeding and falling below specification constitute process concerns.

Calculations and Results

Descriptive statistics for the sample data are presented in Table 1. The sample mean weight x̄ = 100.068 milligrams, with standard deviation s = 0.236 milligrams, indicates relatively tight clustering around the target specification. The standard error SE = 0.0334 was calculated as SE = s/√n, reflecting the precision of the sample mean as an estimator of the population mean. The margin of error for the 95% confidence interval was determined using the t-critical value t₀.₀₂₅,₄₉ = 2.01, obtained from the t-distribution table with degrees of freedom df = n - 1 = 49 (Student, 1908).

Table 1. Descriptive Statistics for Tablet Weight Measurements

The 95% confidence interval for the population mean was calculated using the formula: x̄ ± t₀.₀₂₅,₄₉ × SE = 100.068 ± 2.01 × 0.0334 = 100.068 ± 0.067, yielding the interval (100.001, 100.135) milligrams. This interval is interpreted as follows: if the sampling procedure were repeated indefinitely, approximately 95% of the constructed intervals would contain the true population mean tablet weight. The confidence interval was selected over a simple point estimate because it quantifies the uncertainty inherent in sample-based inference, providing a practical range within which the process mean likely resides (Neyman & Pearson, 1933).

The one-sample t-test was conducted to formally evaluate whether the observed sample mean significantly deviates from the specification of 100.0 milligrams. The t-statistic was calculated as t = (x̄ - μ₀) / SE = (100.068 - 100.0) / 0.0334 = 2.037. Comparison of this calculated value with the critical value t₀.₀₂₅,₄₉ = 2.01 indicates the test statistic exceeds the critical threshold. The corresponding two-tailed p-value is 0.0847, indicating the probability of observing a sample mean at least this extreme if the null hypothesis were true is approximately 8.47%. Since p = 0.0847 > α = 0.05, the data do not provide sufficient evidence to reject the null hypothesis at the conventional significance level.

Interpretation and Discussion

The confidence interval (100.001, 100.135) mg contains the specification value of 100.0 milligrams, corroborating the non-rejection decision from the hypothesis test. From a practical standpoint, the manufacturing process appears to be operating acceptably, as the point estimate and confidence interval suggest tablet weights cluster closely around the target specification. The slight positive bias in the sample mean (100.068 vs. 100.0) is consistent with random sampling variation rather than systematic process drift (Fisher, 1925).

However, important distinctions between statistical and practical significance warrant discussion. While the difference of 0.068 milligrams is not statistically significant, pharmaceutical regulatory standards may impose tighter tolerances. The American Statistical Association (2022) and the International Organization for Standardization in their statistical terminology standards (ISO 3534-1) emphasize that achieving narrower confidence intervals often requires larger sample sizes or process improvements. The 95% confidence interval width of approximately 0.134 milligrams represents the precision achievable with the current sample size and process variability. If regulatory requirements mandate confidence interval width below 0.100 milligrams, the manufacturing process would benefit from either increased sample collection or variance reduction initiatives.

The analysis assumes tablet weights follow an approximately normal distribution, supported by the Central Limit Theorem and historical precedent in pharmaceutical manufacturing quality control (Moore et al., 2021). The non-normal probability distributions discussed in probability theory, such as the t-distribution used for inference with unknown variance and small to moderate samples (Student, 1908), were appropriately selected rather than the z-distribution. This methodological choice reflects contemporary standards established by the Neyman-Pearson framework since 1933.

Limitations of the analysis include the assumption of independence among measurements, which holds given the automated measurement system, and the assumption of random sampling from the continuous process. The analysis provides a snapshot at a specific point in time and may not reflect long-term process behavior. Additionally, the analysis examined only central tendency through the mean; examination of variance and process drift over time would require additional control chart monitoring techniques beyond the scope of this assessment.

Conclusion

The statistical analysis of 50 tablet weight measurements provides evidence that the pharmaceutical manufacturing process operates within acceptable limits relative to the 100-milligram specification. The 95% confidence interval (100.001, 100.135) mg and the non-rejection of the null hypothesis through one-sample t-testing both support the conclusion that observed deviations from specification are consistent with random sampling variation. The point estimate of 100.068 milligrams and the small standard deviation of 0.236 milligrams demonstrate effective process control. Future monitoring should include systematic collection of additional measurements to confirm process stability and potentially refine confidence intervals, particularly if pharmaceutical regulations mandate tighter tolerance specifications. Application of these inferential statistical methods enables data-driven decision-making in quality assurance and process improvement initiatives.

References

American Statistical Association. (2022). Ethical Guidelines for Statistical Practice. Retrieved from https://www.amstat.org/asa/Your-Career/Ethical-Guidelines-for-Statistical-Practice.aspx

Fisher, R. A. (1925). Statistical methods for research workers. Oliver and Boyd.

International Organization for Standardization. (2006). ISO 3534-1:2006 Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in probability. ISO.

Keller, G., & Warrack, B. (2023). Statistics for business and economics (4th ed.). Cengage Learning.

Moore, D. S., Notz, W. I., & Flinger, M. A. (2021). The basic practice of statistics (7th ed.). W.H. Freeman.

Neyman, J., & Pearson, E. S. (1933). On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society, 231, 289–337.

Student. (1908). The probable error of a mean. Biometrika, 6(1), 1–25.