Confidence Intervals and Binomial Distributions

Introduction to Confidence Intervals

• Opening quote by Richard Feynman: "The first principle is that you must not fool yourself – and you are the easiest person to fool."

Key Learning Objectives

• Understand the difference between probability distribution and cumulative binomial distribution.

• Learn how to use binomial distributions to make inferences about a population.

• Identify the assumptions that underpin confidence intervals.

• Calculate 95% confidence intervals (CIs).

• Comprehend the meaning behind the 95% confidence interval.

Binomial Distribution Example Questions

• If you flip a coin 10 times, what’s the probability of observing exactly 7 heads?

• Given that, on average, 5% of patients undergoing an operation get an infection, what’s the chance that 10 out of the next 30 patients will get an infection?

• If 40% of voters are Democrats, what is the chance that a random sample of 600 voters will comprise 45% Democrats?

Cumulative Binomial Distribution

• Question: If you flip a coin 10 times, what’s the probability of observing 7 or more heads?

• Scenario: If 5% of patients get infections, what’s the chance that 10 or more of the next 30 patients will contract infections?

• Query: If 40% of voters are GOP, what is the chance that a random sample of 600 people will be at least 45% GOP?

Probability vs Cumulative Density Function

• Data from the U.S. National Lightning Detection Network shows the number of lightning strikes for the year 2022, peaking in late May:

  • Daily lightning counts peaked at 3.3 million strikes on May 21.

  • Cumulative lightning counts were presented in millions throughout the months from January to December 2022.

Binomial Distribution Characteristics

• Binomial probability relies on a known probability for the population to predict the likelihood of observing various outcomes in random samples.

• In scientific data, results are often represented as a single proportion.

Example I: Calculating Successes and Success Proportions

• It’s crucial to determine how many successes occurred and the proportions thereof.

• Consider if calculations will help make broader conclusions about a phenomenon.

Example I Continued: Estimating Ranges

• Questions:

  • How far away from those calculated success proportions can the actual population percentages likely be?

  • How certain can we be of these estimates?

• Confidence intervals provide a range of values that are expected to contain the overall population percentage.

Confidence Interval (CI)

• Definition: A confidence interval (CI) represents a range of percentages in which we are 95% confident the true population value lies.

• Queries posed: How do we derive this interval from our data?

Steps for Calculating CIs

• First step: Plotting the data to visualize observed successes versus missed attempts.

• Observations must be clear and unambiguous to avoid uncertainty about what we observed.

CI Calculation Methods

• CF Clopper and Pearson method (1934).

• Standard Wald method.

• Modified Wald method.

Example II: Polling Voters

• Scenario: 100 randomly selected voters are polled before an election, with 33 indicating a vote for a specific candidate.

• Queries arise on what this implies about the proportion of voters who will support the candidate in the larger population.

Sample Representativity and Sampling Error

• First Issue: Ensuring that the sample is genuinely representative of the population.

• Second Issue: Accounting for potential sampling error in our results.

Limitations in Estimating Proportions

• There is no absolute certainty about the proportion of voters in the population.

• The best course of action is to calculate a range of possible values for this proportion, known as the 95% confidence interval.

Assumptions of Confidence Intervals

1. Random representative sample: In polling, if the sample isn't randomly selected, this assumption is violated.

2. Independent observations: Each observation must be selected independently of one another.

3. Accurate data: Quality of data must be assured, as poor recording leads to unrelated results (garbage in, garbage out).

Understanding 95% CI Calculation

• Confidence Interval Formula: CI = ar{p} ext{ ± } z imes SE Where:

  • ar{p} is the sample proportion.

  • SE = rac{ar{p}(1-ar{p})}{n}

  • z is the critical value drawn from the standard normal distribution.

Interpretation of Standard Errors

• Reason behind multiplying the standard error by the z-value: To ascertain how many standard errors must be extended from the mean in order to obtain our desired confidence level (e.g., 1.96 for a 95% CI).

Insights from the Monty Hall Problem

• The CI represents a range of acceptable values instead of a single expected value, showcasing assessment variability.

• Illustrative example using probability bars calculated from multiple simulated experiments regarding confidence intervals.

Proportion Estimation Techniques

• Explanation of 95% CI: When performing repeated experiments, 95% of confidence intervals will encompass the true population value.

• True or False Question: "There is a 95% chance that the population value lies within this 95% CI" - understanding the semantics of confidence intervals is key.

Special Significance of 95%

• Understanding what makes 95% a standard choice for confidence levels in statistical applications.

Best Practices for Calculating Binomial CIs

1. Calculate adjusted proportion:
   p' = rac{s + 2}{n + 4}

2. Compute Margin of Error (W):
   W = 2 imes ext{sqrt} rac{p'(1-p')}{n+4}

3. Establish 95% confidence interval: From ( p' - W ) to ( p' + W ).

Derivation of Equation Constants

• The constants “2” and “4” within the adjusted proportion formula: p' = rac{s + 2}{n + 4} are explained in context.

Confidence Interval Comparisons: Wald vs. Agresti-Coull

• Comparative analysis of confidence intervals for varying sample sizes (n) and proportions (p) within produced graphical representations.

Key Reminders

• Distinctive note on the variable p' not being equivalent to the p-value, where p' signifies proportion.

• Explanation of margin of error and its respective direction in CI spans, and how the CI’s length correlates with margin of error.

Practical Case Studies

Case Study 1: Plant Growth Under Different Light Conditions

• Research on plant species growth under various light types:

  • 75 had exposure to red light.

  • 45 exhibited significant growth improvement.

Case Study 2: Antibiotic Resistance in Bacteria

• Study conducted by microbiologists into bacterial populations:

  • 120 colonies assessed.

  • 80 found to exhibit antibiotic resistance to specific treatments.

Case Study 3: Frog Population Sex Ratio in a Polluted Pond

• Investigation by ecologists examining frog populations in habitats affected by pollution:

  • Of 60 frogged caught in a pond, 25 were determined to be female.