Project 2: Simulation And P-Value STAT 35000 Summer 2017

Project 2: Simulation and p-value STAT35000 Summer 2017

This project involves studying the sampling distribution and calculating p-values for Bernoulli-distributed data using R. The specific tasks include generating samples, computing sample proportions, comparing these with hypothesized probabilities, and evaluating the p-value through simulation, exact calculations, and normal approximation methods, including a continuity correction for improved accuracy.

Paper For Above instruction

Understanding the sampling distribution of a Bernoulli process and the calculation of p-values are foundational elements in statistical inference. This project provides a comprehensive exercise in applying these concepts through simulation, analytical calculations, and normal approximation techniques, utilizing R programming.

Introduction

The Bernoulli distribution models binary outcomes, such as success or failure, with a probability p of success. Analyzing data derived from Bernoulli trials allows statisticians to estimate the underlying probability, test hypotheses, and understand variability through sampling distributions. The p-value plays a critical role in hypothesis testing by quantifying how extreme observed data are under a given null hypothesis. Using R, students can simulate data, visualize distributions, and perform calculations to deepen understanding of these statistical concepts.

Sampling from Bernoulli Distribution

The first step involves sampling 50 points from a Bernoulli distribution with a known success probability p0 = 0.2. This process models real-world experiments where binary outcomes are observed repeatedly. In R, this can be achieved using the rbinom function, which generates random Binomial counts, and dividing by the sample size to estimate the proportion of successes:

sample_successes 

p_hat_star

This sample proportion (\(\hat{p}^*\)) provides an estimate of the true probability p0 based on the sample data. The adequacy of this estimate depends on the sample size and inherent variability, which can be assessed visually and statistically.

Assessing the Sample Estimate

To determine the reasonableness of the sample proportion, it's essential to consider the variability expected under the true p0. Since the sample size is 50, the standard error can be computed, and confidence intervals or visual assessments, such as histograms, can provide insights into the stability of \(\hat{p}^*\). A proportion close to 0.2, within a reasonable margin considering random fluctuations, indicates an adequately representative sample.

Hypothesis Testing with an Incorrect Guess

Suppose that the true success probability is unknown, and the guess is p0 = 0.4. To evaluate whether this guess is reasonable, two approaches can be considered:

  • Approach (a): Compare the observed \(\hat{p}^*\) with the guess 0.4. If they are close enough, the guess might be accepted. Closeness can be judged by examining confidence intervals or the magnitude of the difference relative to the variability expected under p0 = 0.4.
  • Approach (b): Use simulation to generate 10,000 samples assuming p0 = 0.4, calculate the sample proportions for each, and visualize the distribution with a histogram. If \(\hat{p}^\) falls within the bulk of this distribution, it suggests that p0 = 0.4 is plausible. If \(\hat{p}^\) appears in the tails, the guess may be rejected.

Simulation of Sampling Distribution

In R, generating 10,000 synthetic samples under p0 = 0.4 and calculating their sample proportions can be performed as follows:

set.seed(123)

N 

sample_proportions 

successes 

successes / 50


})


hist(sample_proportions, main = "Distribution of Sample Proportions under p=0.4",


xlab = "Sample Proportion", col = "lightblue")

This histogram visualizes the distribution under the null hypothesis p=0.4. The p-value can be approximated by the proportion of simulated \(\hat{p}\) less than the observed \(\hat{p}^*\).

Calculating the p-value

The p-value is estimated as the proportion of simulated \(\hat{p}_k 

p_value

This approach provides an empirical estimate based on simulation, reflecting the likelihood of observing a sample proportion as extreme as \(\hat{p}^*\) under the null hypothesis.

Exact Calculation of the Probability

For the binomial distribution, the exact probability that \(\hat{p}\) is less than \(\hat{p}^*\) corresponds to the cumulative probability:

P(X \leq x) = pbinom(x, size=50, prob=0.4)

where x is the number of successes corresponding to the observed \(\hat{p}^\). For example, if \(\hat{p}^ = 0.5\) with 25 successes:

P(Successes ≤ 25) = pbinom(25, 50, 0.4)

This computes the exact probability without approximation.

Normal Approximation with Continuity Correction

The central limit theorem suggests that, for n=50, the distribution of \(\hat{p}\) can be approximated by a normal distribution with mean \(\mu = p_0 = 0.4\) and variance \(\sigma^2 = p_0 (1 - p_0)/n\). The probability that \(\hat{p}\) is less than a certain observed value can then be approximated as:

P(\hat{p} ) ≈ \Phi\left(\frac{\hat{p}^ - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}\right)\

where \(\Phi\) is the standard normal cumulative distribution function. Applying a continuity correction, the probability becomes:

P(\hat{p} \leq \hat{p}^ + 1/(2n)) ≈ \Phi\left(\frac{\hat{p}^ + 1/(2n) - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}\right)

This correction improves the approximation's accuracy, especially for moderate sample sizes.

Comparison of Methods

Finally, comparing the p-values obtained via simulation, exact binomial calculations, and the normal approximation (with continuity correction) demonstrates their relative accuracies. Typically, for sample sizes like 50, the CLT approximation closely matches the simulation-based p-value, affirming its utility in practical scenarios.

Conclusion

This comprehensive exercise highlights the importance of multiple methods to evaluate statistical hypotheses. Simulation offers flexibility and intuitive understanding but can be computationally intensive. Exact methods provide precise probabilities but may be complex for larger data. The normal approximation, especially with a continuity correction, balances accuracy and simplicity. Together, these approaches form a robust toolkit in statistical inference, exemplified through the Bernoulli experiment explored in this project.

References

  • Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). W.W. Norton & Company.
  • Rice, J. A. (2006). Mathematical Statistics and Data Analysis (3rd ed.). Thomson Brooks/Cole.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
  • McCullagh, P., & Nelder, J. A. (1989). Generalized Linear Models (2nd ed.). Chapman and Hall.
  • Agresti, A. (2002). Categorical Data Analysis. Wiley.
  • Langville, A. N., & Meyer, C. D. (2006). Google's PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press.
  • Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). Chapman and Hall/CRC.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Conover, W. J. (1999). Practical Nonparametric Statistics (3rd ed.). Wiley.
  • Schervish, M. J. (1995). Theory of Statistics. Springer.

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