Margin of Error and Confidence Intervals

Confidence Interval

An estimated range of plausible values for an unknown population parameter (such as the population mean mu) computed as the sample mean plus or minus the margin of error, giving a bounded range where researchers are 95% confident the true parameter lies. (Slide 15; Transcript 57:57, 58:19; Textbook pp. 172-174)

Margin of Error

Half the width of a confidence interval, calculated as the critical value multiplier times the standard error (Margin of Error = Critical Value * Standard Error), representing the maximum expected distance between sample means and the true population mean for 95% of samples. (Slide 11; Transcript 47:13, 56:32; Textbook p. 173)

Critical Value

The cutoff multiplier in standard error units (e.g., 1.96 for normal distribution, or t-critical values like 1.962 or 2.26 depending on degrees of freedom df = N - 1) that determines the exact margin of error needed to capture the middle 95% of sample estimates. (Slide 18, 19; Transcript 01:11:50, 01:12:37; Textbook pp. 173-174)

Estimated Standard Error

The practical approximation of standard error calculated when the population standard deviation sigma is unknown, substituting the sample standard deviation s into the formula s / sqrt(N) (e.g., 1.11 / sqrt(1071) = 0.034). (Slide 16; Transcript 31:39, 43:44, 01:07:48; Textbook p. 167)

t-Distribution in Confidence Intervals

A symmetric, bell-shaped distribution with heavier tails than a standard normal curve, used when the population standard deviation is unknown and estimated using s; as sample size N (or degrees of freedom df = N - 1) decreases, the t-distribution becomes flatter and more stretched out, requiring larger critical values to maintain a 95% confidence level. (Slide 17, 18; Transcript 01:08:31, 01:09:09, 01:09:52; Textbook p. 173)

Frequentist Interpretation of 95% Confidence

A long-run concept asserting that if an experiment is replicated infinitely across independent random samples of size N drawn from the same population, exactly 95% of the constructed confidence intervals will successfully capture/cover the true population mean mu, while 5% will miss it. (Slide 15; Transcript 58:43, 59:03, 01:00:30; Textbook pp. 174-175)

Unbiased Estimator

A sample statistic whose expected value across infinite repeated draws equals the true population parameter (e.g., sample mean x-bar estimating population mean mu), contrasting with biased estimators that systematically over- or under-estimate parameters. (Textbook pp. 167-170)

Bayesian Credible Interval

A Bayesian statistical interval that makes a direct probability statement regarding the chance (e.g., 95% probability) that a population parameter falls between two specific numerical bounds based on observed data and prior distributions. (Textbook p. 175)

70% Confidence Interval Multiplier

Using 1.04 as the critical value multiplier instead of 1.96 when constructing a 70% confidence interval for a normally distributed variable. (Textbook p. 173)

jamovi Manual CI Calculation from Descriptives

Computing a 95% confidence interval manually in jamovi by checking 'Mean' and 'S.E. Mean' under Exploration -> Descriptives and calculating Lower 95% CI = Mean - (1.96 * S.E. Mean) and Upper 95% CI = Mean + (1.96 * S.E. Mean). (Textbook p. 174)

Sampling Distribution Equality Rule

To belong to the exact same sampling distribution, multiple samples must address the same research question, draw from the exact same population, and share the exact same sample size N. (Slide 4; Transcript 03:30, 08:14, 08:50)

Effect of Population Standard Deviation on Standard Error

Comparing two populations with different standard deviations (e.g., sigma = 0.5 vs sigma = 1.0), the population with the smaller standard deviation generates a smaller standard error (0.015 vs 0.034 for N = 1071) and produces sample means that cluster tighter around the true population mean. (Slide 4; Transcript 04:12, 06:34, 08:02)

Good Buddies Hide and Seek Metaphor

A conceptual analogy explaining that sample mean and population mean are 'good buddies' bounded within margin of error 95% of the time; the lowest possible population mean location is sample mean minus margin of error, and the highest is sample mean plus margin of error. (Transcript 51:42, 53:01, 53:33, 55:29)

distrACTION Module in jamovi

A specialized jamovi module used to calculate exact central interval quantiles and critical values (e.g., CV = 1.962 for N = 1071, df = 1070) by selecting T-Distribution -> Compute quantiles -> central interval quantiles p = 0.95. (Slide 19; Transcript 01:13:13)

SECTION 4: Low Priority (Questions)

Does a 95% confidence interval always contain the sample mean?

Yes, 100% of confidence intervals contain the sample mean because the interval is constructed directly by adding and subtracting the margin of error from the sample mean, placing the sample mean at the exact midpoint. (Transcript 01:05:21, 01:05:57)

Does a 95% confidence interval always contain the true population mean?

No, only 95% of sample confidence intervals across repeated sampling will successfully cover the true population mean, while 5% of samples yield intervals that fail to capture the population mean. (Slide 15; Transcript 59:58, 01:01:07)

Do 95% of all samples produce the exact same confidence interval bounds?

No, every sample yields a distinct sample mean and estimated standard error, resulting in different numerical upper and lower bounds for each sample. (Transcript 01:03:02, 01:03:15)

What happens to the width of a 95% confidence interval when sample size N decreases?

The width of the confidence interval increases because a smaller sample size increases the standard error and increases the t critical value, directly inflating the margin of error. (Slide 20; Transcript 01:08:12, 01:11:02)

What happens to the width of a 95% confidence interval when sample standard deviation s increases?

The width of the confidence interval increases because a larger sample standard deviation increases the estimated standard error (s / sqrt(N)), making the margin of error wider. (Slide 21; Transcript 01:07:54, 01:08:18)

Why does a smaller sample size require a larger critical value when using the t-distribution?

Because smaller sample sizes (lower degrees of freedom df = N - 1) cause the t-distribution to be flatter and more stretched out with heavier tails, requiring cutoffs further away from 0 (e.g., 2.26 for N = 10 vs 1.962 for N = 1071) to capture 95% of the distribution. (Slide 18; Transcript 01:10:41, 01:11:39, 01:12:00)

If 1,000 independent studies each recruit 100 participants to evaluate a depression treatment, how many deviation scores, sample means, sample standard deviations, sampling errors, and true standard errors exist?

There are 100,000 deviation scores (100 per sample * 1,000 samples), 1,000 sample means, 1,000 sample standard deviations, 1,000 sampling errors, and exactly 1 true standard error. (Slide 8, 9; Transcript 24:01, 25:10, 25:41, 27:01)