Mind on Statistics (6th. Ed) Chapter 10 - Estimating Proportions with Confidence

Introduction

• Explain the meaning of a confidence level
• Recognise the conditions required for using a confidence interval for a population proportion and a difference in two proportions
• Calculate conservative and approximate margins of error for a condfidence interval for one proportion
• Calculate a confidence interval for a population proportion
• Calculate a confience interval for the difference in two proportions
• Interpret confidence intervals for one proportion and the difference in two proportions
• Find bootstrap confidence intervals for one proportion and the difference in two proportions

Overview of Confidence Intervals

Confidence interval: is an interval of values computed from sample data that is likely to include the unknown value of a population parameter

\[\textrm{sample statistic } \pm \textrm{ multiplier } \cdot \textrm{ standard error}\]

The sammple size, confidence level, and natural variability among units affect the width of the interval. The first two are the only ones controllable.

Confidence Interval for Population Proportion

Conditions (arise when data are collected similar to binomial experiment):

• sample is randomly selected from population
• $n\hat{p}$ and $n(1-\hat{p})$ are at least 10.

The first condition changes if parameter $p$ is a long-run probability.

alternative condition: The sample proportion $\hat{p}$ is the proportion of times a specified outcome occurs in n repeated independent trials with fixed probability $p$.

Margin of Error: describes value of multiplier x s.e. in a 95% confidence interval estimate of a population proportion

95% Margin of Error (2 standard errors): \(= 2\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) Margin of error is the same as conservative margin of error when $\hat{p}$ is 0.5.

Confidence Interval: \(\hat{p} \pm z*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)

Confidence Interval for Difference in Two Population Proportions

Confidence Interval:

\[\hat{p}_1 - \hat{p}_2 \pm z*\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}\]

Using Simulation to Calculate Confidence Intervals: Bootstrapping

A sampling distribution is constructed by assuming we can draw an infinite number of samples of size $n$ from a population, and compute the sample statistic from each sample. The distribution of the sample statistics for large number of random samples is the bootstrap sampling distribution or we use resampling.

Using Confidence Intervals to Guide Decisions

• A value that is not in a confidence interval can be rejected as a likely value of the population proportion. Any value that is in a confidence interval is an “acceptable” probability for the value of the population proportion.
• When a confidence interval for a difference in two proportions does not cover 0, it is reasonable to conclude that the two population proportions have different values
• When the confidence intervals for proportions in two different populations do not overlap, it is reasonable to conclude that the the population proportions are different. However, if the intervals overlap, no conclusion can be made, and a confidence interval for the difference should be computed.