Poll Simulation Dashboard

You can access the dashboard on Streamlit servers:

.

This repository contains a Streamlit-based dashboard that simulates poll results based on user-defined parameters. The aim is to predict the number of 1s in the population based on a poll results with a given sample size. The simulation demonstrates how sample size affects the accuracy of poll results, confidence intervals, and statistical significance.

Overview

The dashboard allows users to simulate a poll by specifying:

• True Percentage of 1 Among Population ($P_{true}$): The actual proportion of the population with a specific characteristic. This value is unknown in real-world scenarios.
• Max Sample Size: The maximum number of individuals to sample.
• Confidence Level ($CL$): The desired confidence level for the confidence interval.
• Random Seed: A seed value for reproducibility of random sampling.

The dashboard generates three interactive graphs:

1. Sample Mean and Confidence Interval: Visualizes the sample mean and its confidence interval as the sample size increases.
2. Confidence Level Probability: Shows the probability that the sample mean falls within the specified confidence level of the true percentage.
3. P-Value Analysis: Displays the p-value for testing the null hypothesis ($P_{true} = 0.5$) and indicates whether the null hypothesis can be rejected.
4. Biased Pollsters: Demonstrates the effect of biased pollsters on the accuracy of aggregated poll results. Uses Monte Carlo simulation to generate biased poll results and a data-driven approach to aggregate the results.

Key Features

1. Sample Mean and Confidence Interval

• Sample Mean: Calculated as: $$ \bar{X} = \frac{1}{n} \sum_{i=1}^{n} x_i $$
• Confidence Interval: $$ CI = \bar{X} \pm 1.96 \times SE $$ where $SE$ (Standard Error) is: $$ SE = \sqrt{\frac{\bar{X} \times (1 - \bar{X})}{n}} $$

2. Confidence Level Probability

• The probability that the sample mean is within the confidence level of the true percentage: $$ P(\left|\bar{X} - P_{true}\right| < 1-CL) $$ Simplified to: $$ P((1-CL) / SE(\bar{X})) - P(-(1-CL) / SE(\bar{X})) $$

3. P-Value Analysis

• Tests the null hypothesis ($P_{true} = 0.5$) using: $$ \text{p-value} = 2 \times \left(1 - \Phi\left(\left|\frac{\bar{X} - 0.5}{SE(\bar{X})}\right|\right)\right) $$ If the p-value is less than 0.05, the null hypothesis is rejected.

How to Use

1. Clone this repository:

   git clone https://github.com/HamidAbbasi-R/Poll-Simulation.git

2. Install the required dependencies:

   pip install streamlit numpy scipy plotly

3. Run the Streamlit app:

   streamlit run Poll_simulation.py

4. Adjust the parameters in the sidebar and observe the changes in the graphs.

Graphs

1. Sample Mean and Confidence Interval

2. Confidence Level Probability

3. P-Value Analysis

4. Biased Pollsters

Dependencies

• streamlit: For building the interactive dashboard.
• numpy: For numerical computations.
• scipy.stats: For statistical functions like the cumulative distribution function (CDF).
• plotly: For creating interactive visualizations.

Contributing

Contributions are welcome! Please feel free to submit a pull request or open an issue for any improvements or bug fixes.