E

Psychological Statistics - Confidence Intervals

Quick Review: Sampling Error

  • Sampling error is the difference between an estimate and the true population statistic.

  • Each hypothetical sample has a different amount of error.

Quick Review: Sampling Distribution

  • A sampling distribution is the distribution of estimates from many hypothetical samples.

  • With large N, sample means follow a normal curve around the true mean.

Quick Review: Standard Error

  • Standard error is the average distance from a sample mean to the true mean.

  • It reflects expectations across hypothetical samples.

Quick Review: Normal Curve Rule

  • Standard error is the standard deviation of hypothetical sample means.

  • Apply normal curve rules of thumb.

  • 95% of means from large samples are within ± 1.96 standard errors of the true mean.

Confidence Interval Overview

  • Confidence intervals use standard error to estimate plausible values for the true population statistic.

  • They define a range of μ values that could have reasonably produced the sample mean.

Most Extreme Parameter Values

  • A confidence interval gives the two most extreme values of the population mean that could have reasonably produced the data.

  • X could be from a sampling distribution with a population mean as high as μ_H

  • X could be from a sampling distribution with a population mean as low as μ_L

Terminology: Critical Value

  • A 95% critical value (CV_{95%}) is the number of standard error units from the true mean that captures 95% of sample means.

  • For large samples, CV_{95%} = ± 1.96

  • We may increase CV_{95%} for smaller samples (t distribution).

Terminology: Margin of Error

  • Margin of error is ± 1.96 × SE in large samples.

  • Adjust CV_{95%} for smaller samples (t distribution).

Smoking and Drinking Cessation Trial

  • Trial compared varenicline and naltrexone against varenicline alone for smoking cessation and drinking reduction among heavy-drinking smokers.

Key Variables

  • Breath carbon monoxide: Biomarker of smoking behavior.

  • Medication arm: Participants received varenicline plus naltrexone or varenicline plus placebo.

Standard Error

  • SE = 0.46 reflects expectation across hypothetical samples.

Margin of Error

  • Margin of error = ± 1.96 × SE = ± 0.90

  • 95% of hypothetical samples have a mean within ± 0.90 of μ

  • The CV_{95%} from the t distribution depends on N.

5% Confidence Interval

  • The 95% confidence interval [4.6, 6.4] defines a range of μ values that could have reasonably produced the sample mean.

Most Extreme Parameter Values

  • Confidence interval gives extreme values of the population mean that could have reasonably produced these data.

  • X could be from a sampling distribution with μ_H = 6.4

  • X could be from a sampling distribution with μ_L = 4.6

Interpretation

  • The 95% confidence interval was [4.6, 6.4].

  • The true mean could be as low as 4.6 and as high as 6.4.

Hypothesis Testing with 95% Intervals

  • A population with μ = 5 could have reasonably produced the sample mean.

Confusion Over Confidence and Probability

  • Confidence and probability unfold over many hypothetical random samples.

  • They are properties of data, not the population statistic.

Frequentist Framework Revisited

  • Estimates vary across hypothetical random samples.

  • 95 out of 100 sample means should fall within the margin of error of the true mean.

  • The true mean doesn’t change.

Estimates Vary Across Samples

  • Estimates and Margin of Error

Confidence Intervals Vary

  • 95% of confidence intervals will contain the true mean across many hypothetical samples.

Confidence Intervals Properties

  • 95% probability refers to a long-run process over many random samples.

  • There is a 95% chance that the confidence interval contains the true mean.

Statistical History: T Distribution

  • 1908: William Sealy Gosset - t distribution

William Sealy Gosset

  • Gossett derived the t-distribution for small samples.

The T-Distribution

  • The t-distribution stretches out as N decreases.

  • Software uses the t-distribution for confidence intervals.

T-Distribution vs. Normal Curve

  • T-distribution predicts critical values, adjusting for sample size.

Clinical Trial T-Distribution

  • Degrees of freedom adjustment: shape of t-distribution and CV_{95%} depend on N – 1

Jamovi Output

  • Note: CI of the mean assumes sample means follow a t-distribution with N - 1 degrees of freedom.

RStudio Output

  • One Sample t-test

Study Questions

  • Practice interpreting standard errors and confidence intervals.