Confidence Intervals for Proportions

Introducing Confidence Intervals

• Confidence intervals provide an upper and lower estimate for predictions, written as (lower, upper).

• Example: A weatherman's confidence interval is (-40°, 200°). This interval is very likely to be correct but not precise due to its width, indicating a large margin of error.

• Polls often include a margin of error to account for sampling variation.

• Sample proportion is a guess for the population proportion.

• Example: A poll indicates 56% of voters disapprove of President Biden's job performance (p-hat = 0.56). The actual population proportion (p) is likely close to this.

• Margin of error is added and subtracted from the sample proportion to create a confidence interval.

• Example: A poll with a 56% disapproval rate and a 2.3% margin of error yields a confidence interval of (53.7%, 58.3%).

• Even if the true population proportion (e.g., 58% disapproval) is near the interval's edge, the confidence interval still captures the true proportion.

Level of Confidence

• Unless stated otherwise, assume a 95% confidence interval.

• Approximately 95% of observations in a Normal model fall within ~2 standard deviations of the mean.

• A 95% confidence interval uses 1.96 standard deviations (z* = 1.96).

Calculating Confidence Intervals

• The sampling distribution of proportions has a standard deviation of $\sqrt{\frac{p(1-p)}{n}}$$$\sqrt{\frac{p(1-p)}{n}}$$.

• Since 'p' (population proportion) is unknown, we estimate the standard deviation using sample statistics, which we call the standard error.

• The standard error is calculated as: $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$$\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$.

• The 68-95-99.7% Rule: Approximately 68% of samples have p-hats within 1 SE of p, 95% within 2 SEs, and 99.7% within 3 SEs.

• The formula for a confidence interval is: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$, where $\hat{p}$$$\hat{p}$$ is the sample proportion, and z* is the critical value.

• Critical values for common confidence levels:

  • 90% confidence interval: z* = 1.645

  • 95% confidence interval: z* = 1.96

  • 99% confidence interval: z* = 2.576

• Example: Margin of error calculation for a 95% confidence interval with a sample of 1856 people, where 56% disapprove of President Biden:

  • $MOE = 1.96 \sqrt{\frac{.56(1-.56)}{1856}} = 0.02258$$$MOE = 1.96 \sqrt{\frac{.56(1-.56)}{1856}} = 0.02258$$

  • Confidence Interval: $.56 \pm .02258 = (.5374, .5826)$$$.56 \pm .02258 = (.5374, .5826)$$

• Example: Margin of error calculation for a 99% confidence interval with the same sample:

  • $MOE = 2.576 \sqrt{\frac{.56(1-.56)}{1856}} = 0.02968$$$MOE = 2.576 \sqrt{\frac{.56(1-.56)}{1856}} = 0.02968$$

  • Confidence Interval: $.56 \pm .02968 = (.5303, .5897)$$$.56 \pm .02968 = (.5303, .5897)$$

• To decrease the margin of error:

  1. Decrease the confidence level.

  2. Increase the sample size; to halve the standard error/MOE, the sample size must be quadrupled.

Interpreting Confidence Intervals

• Confidence intervals vary because they are based on sample statistics.

• Confidence lies in the process of constructing the interval, not in any single interval itself.

• We expect 95% of all 95% confidence intervals to contain the true population parameter.

• Interpretation: "If the process were repeated many times, [confidence level]% of samples of this size will produce confidence intervals that capture the true proportion of [context]."

• For a single confidence interval: "We are [confidence level]% confident that the true proportion of [context] is within [lower%] and [upper%]."

• Example: Interpreting unemployment rate confidence intervals:

  • If confidence intervals for North Carolina and New Jersey do not overlap, their unemployment rates are considered different.

  • New Jersey: We are 95% confident the true unemployment rate is between 5.6% and 7%.

  • North Carolina: We are 95% confident the true unemployment rate is between 3.2% and 4.3%.

Conditions for Confidence Intervals

• Assumptions and Conditions:

  • Independence Assumption:

    • Check for Randomization Condition: Data should be sampled randomly or from a randomized experiment.

    • Check the 10% Condition: The sample size should be no more than 10% of the population.

  • Sample Size Assumption:

    • Success/Failure Condition: Expect at least 10 successes and 10 failures.

• Example: Rotten Tomatoes rating for Sing 2:

  • Conditions:

    1. Independent: Critics are assumed to be randomly chosen and not influencing each other.

    2. 10% Condition: 119 critics are less than 10% of all movie critics.

    3. Success/Failure: 82