Paired Samples T-Test

Hypothesis Testing (Statistical Significance Testing)

An inferential statistical procedure where an unknown population parameter (such as population mean mu) is replaced with a hypothesized value or prediction, and sample data (such as sample mean x-bar) is used to evaluate whether the evidence supports or refutes the hypothesis. In lecture, the professor illustrated this using the Humor Styles Questionnaire, hypothesizing that the population mean rating for using humor when depressed is mu = 3 ("sometimes true"), and assessing whether an observed sample mean of 3.26 supports or refutes this guess. (Slide 2, 4, 18; Transcript 10.27 33:28, 36:07, 36:40; Transcript 10.29 07:28, 08:01; Textbook pp. 178-182)

Null Hypothesis (H0)

A specific statistical hypothesis regarding the population parameter that opposes the researcher's prediction and typically embodies the idea that "nothing is going on" or that no real effect exists (e.g., H0: mu = 3, meaning people are neutral with no humor tendency). The goal of hypothesis testing is to show that the null hypothesis is probably false; it is treated as innocent until proven guilty, meaning researchers can only fail to reject it or refute it, but can never prove it is true. (Slide 7, 8; Transcript 10.27 01:01:05, 01:01:48, 01:02:04, 01:04:46; Transcript 10.29 07:55, 13:52, 1:00:39; Textbook pp. 181-183)

Alternative Hypothesis (Ha or H1)

The statistical hypothesis reflecting the researcher's interest, prediction, or theoretical belief that "something is happening" (e.g., that a treatment has an effect or a group difference exists). The professor noted that in academic writing, Ha and H1 are interchangeable notations denoting the alternative hypothesis. (Slide 5, 8; Transcript 10.27 55:44, 56:20, 58:16; Transcript 10.29 49:24, 51:21; Textbook pp. 181-182)

One-Tailed (Directional) Hypothesis

An alternative hypothesis where the researcher expects or predicts a specific directional change, predicting either an increase or a decrease, but not both. Illustrated in lecture by anticipating that the population mean humor score is strictly higher than neutral (Ha: mu > 3) or strictly lower than neutral (Ha: mu < 3), and in the textbook by predicting ESP performance strictly above chance (theta > 0.5). (Slide 5, 6; Transcript 10.27 56:32, 57:49, 59:15; Transcript 10.29 49:24, 50:07, 51:52; Textbook pp. 181, 189-190)

Two-Tailed (Non-Directional) Hypothesis

An alternative hypothesis where the researcher predicts that the population parameter differs from the null value in either direction (scores can increase or decrease). Expressed mathematically as Ha: mu != 3 (or mu > 3 or mu < 3). Illustrated in lecture by testing whether people maintain either a humorous or humorless perspective in the face of stress. (Slide 6, 7; Transcript 10.27 59:23, 01:00:29; Transcript 10.29 41:46, 43:58; Textbook pp. 181, 189)

Test Statistic (t-Statistic)

A standardized mathematical quantity calculated from sample data that guides the decision to reject or retain the null hypothesis, representing the difference between the sample estimate and the hypothesized population mean in units of standard error: t = (x-bar - mu0) / SE. In lecture, the sample mean 3.26 yielded t = (3.26 - 3) / 0.094 = 2.77, meaning the observed sample mean was 2.77 standard errors above the hypothesized population mean. (Slide 10, 11, 19; Transcript 10.29 16:53, 19:35, 20:19, 21:34; Textbook pp. 184-186)

Critical Value

The cutoff value in standard error units (e.g., 1.96 for a normal distribution, or quantiles like 1.979 for a t-distribution with df = 128) that defines the boundary of the margin of error needed to capture the middle 95% of sample estimates under the null hypothesis. (Slide 3, 4, 10; Transcript 10.27 40:50, 52:39; Transcript 10.29 21:15, 23:28, 32:34; Textbook pp. 187-188)

Critical Region (Rejection Region)

The set of extreme test statistic values (located in the tails of the sampling distribution) that would lead researchers to reject the null hypothesis. To maintain a significance level of alpha = .05, the critical region must encompass exactly 5% of the sampling distribution under the null hypothesis (split into 2.5% in each tail for a two-tailed test, or 5% in one tail for a one-tailed test). (Slide 10, 14, 15; Transcript 10.27 44:16, 47:55; Transcript 10.29 43:51, 50:30; Textbook pp. 186-190)

Statistical Significance (Rejecting the Null Hypothesis)

The decision reached when the test statistic falls in the critical region (outside the margin of error) and the p-value is less than or equal to .05 (p <= .05), indicating that the observed data is very unlikely to have been produced by a population where the null hypothesis is true, allowing researchers to reject/refute the null hypothesis. (Slide 10, 11, 17; Transcript 10.27 44:27, 54:47, 55:04; Transcript 10.29 16:08, 24:19, 46:02, 1:01:12; Textbook pp. 187-188)

Non-Significance (Failing to Reject the Null Hypothesis)

The decision reached when the test statistic falls within the non-critical region (inside the margin of error) and the p-value is greater than .05 (p > .05), indicating that the hypothesized population could reasonably have produced the data. Researchers conclude that the null hypothesis is plausible, but never claim that the null hypothesis is proven or accepted. (Slide 11, 12, 17; Transcript 10.27 01:04:26, 01:04:46; Transcript 10.29 45:53, 1:00:39, 1:09:39; Textbook pp. 183, 187)

p-Value (Probability Value)

The conditional probability, calculated under the assumption that the null hypothesis is true, of obtaining a sample test statistic (or sample mean) at least as extreme as the one observed in the actual data. In lecture, the two-tailed p-value of 0.006 indicated that if the population mean is truly 3, there is only a 0.6% chance of obtaining a sample t-statistic equal to or beyond +/-2.77. (Slide 13, 14, 17; Transcript 10.27 44:38; Transcript 10.29 35:41, 36:20, 44:28; Textbook pp. 190-192)

Significance Level Alpha (.05 Conventional Cutoff)

The maximum acceptable Type I error rate tolerated by researchers, conventionally set at .05 (5%). In lecture and textbook, this threshold was traced to Sir Ronald A. Fisher, who argued that a 1-in-20 chance (.05) is a convenient standard to conclude that either a real effect exists or an improbable coincidence has occurred. (Slide 16; Transcript 10.27 44:38; Transcript 10.29 46:59, 47:17, 1:00:29; Textbook pp. 183-184, 190)

Research Hypothesis versus Statistical Hypothesis

A research hypothesis is a substantive scientific claim about the world expressed in conceptual language (e.g., "ESP exists"), whereas a statistical hypothesis translates that scientific claim into a precise mathematical statement about the data-generating probability distribution or population parameter (e.g., theta != 0.5). (Textbook pp. 179-181)

Type I Error (Alpha)

The error of incorrectly rejecting the null hypothesis when the null hypothesis is actually true in reality (a false positive, analogous to wrongfully convicting an innocent defendant in a criminal trial). (Textbook pp. 183-184)

Type II Error (Beta)

The error of failing to reject (retaining) the null hypothesis when the null hypothesis is actually false in reality (a false negative, analogous to letting a guilty defendant go free). (Textbook pp. 183-184)

Power of a Statistical Test (1 - Beta)

The probability of correctly rejecting the null hypothesis when the null hypothesis is truly false in the population. (Textbook pp. 184, 196)

Blackstone's Legal Maxim Analogy

The classical English legal principle articulated by William Blackstone stating that it is "better that ten guilty persons escape than that one innocent suffer", used as the conceptual foundation for why statistical tests strictly prioritize controlling the Type I error rate (alpha) over the Type II error rate (beta). (Textbook pp. 183-184)

Neyman's Interpretation of the p-Value

An operational interpretation that defines the p-value as the smallest Type I error rate (alpha) that a researcher must be willing to tolerate in order to reject the null hypothesis for a given dataset. (Textbook pp. 190-191)

Binomial Sampling Distribution for ESP Test

The exact theoretical discrete probability distribution Binomial(theta = 0.5, N = 100) used under the null hypothesis in the textbook's card-guessing experiment, yielding critical boundaries of X

Statistical Significance Stars Notation

The conventional reporting standard where p > .05 is labeled non-significant (n.s.), p < .05 is marked with one asterisk (), p < .01 with two asterisks (), and p < .001 with three asterisks (). (Textbook pp. 192-194)

Three Equivalent Ways to Test the Null Hypothesis

The three mathematical methods that always yield identical conclusions regarding H0: (1) checking whether the calculated t-statistic falls outside the critical value boundaries, (2) checking whether the p-value is less than or equal to .05, and (3) checking whether the hypothesized population mean falls outside the 95% confidence interval. (Slide 18; Transcript 10.29 45:18, 1:08:11, 1:11:25)

Testing the Null Hypothesis via 95% Confidence Intervals

An alternative testing method where a 95% confidence interval is constructed around the sample mean; if the hypothesized population mean (mu = 3) falls inside the interval [3.07, 3.45], H0 is plausible (fail to reject), but if the hypothesized mean falls outside the interval, H0 is refuted (statistically significant). (Slide 17, 18; Transcript 10.29 1:11:25)

jamovi Quirk for One-Sample t-Test Confidence Intervals

A software operational quirk where jamovi users must leave the 'Test value' set to 0 under the Hypothesis options to obtain the true 95% confidence interval for the sample mean, because entering the actual null value (e.g., 3) causes jamovi to output a confidence interval for the mean difference rather than the raw score scale. (Slide 18)

Humor Styles Scale Question 2 Dataset

The specific empirical example analyzed throughout lecture (Q2: "If I am feeling depressed, I can usually cheer myself up with humor", N = 129, mean = 3.26, SD = 1.07, SE = 0.0944) evaluated against H0: mu = 3, producing t(128) = 2.77, two-tailed p = 0.006, one-tailed p = 0.003 (for mu > 3), and one-tailed p = 0.997 (for mu < 3). (Slide 9-18; Transcript 10.27 1:05:36; Transcript 10.29 11:05, 44:28)

Conditional Probability Representation of p-Value

Representing the p-value mathematically using conditional probability notation: p(data like ours | H0 is true in the population), where the vertical bar represents "assuming that" or "conditional on". (Slide 16; Transcript 10.29 43:09, 58:53)

The Ghost Metaphor for the Null Hypothesis

Professor Du's classroom analogy explaining that finding a single ghost immediately proves the null hypothesis ("ghosts do not exist") is false, but failing to find a ghost over 100 years can never prove that ghosts do not exist, demonstrating why researchers can reject a null hypothesis but can never accept or prove it. (Transcript 10.27 1:02:56-1:04:08)

Hide-and-Seek Metaphor for Estimation vs. Testing

The conceptual comparison where confidence intervals (Neyman) provide a bounded region where the true population mean is hiding, whereas hypothesis testing (Fisher) makes an initial guess for the hiding spot and checks whether the empirical data refutes that guess. (Transcript 10.27 33:57-35:43)

p-Hacking and Pre-Registration

The unethical academic practice of testing multiple alternative hypotheses or models until obtaining a significant p-value (p < .05), which drastically inflates the familywise false positive rate; combated in modern social sciences through pre-registration where hypotheses and analysis plans are formally declared prior to data collection. (Transcript 10.29 56:23-58:09)

APA-Style Summary Format for One-Sample t-Test

The standardized reporting structure consisting of four elements: (1) statement of the analysis used and research question, (2) reference to a table of descriptive statistics (M, SD, N), (3) formal statistical report string: t(df) = test statistic, p = value, and 95% CI, and (4) substantive conclusion stating whether the null hypothesis was rejected. (Slide 17; Transcript 11.10 08:44-11:05)

Evaluation of Skewness via t-Statistic

Applying the t-statistic logic to distributional shape by dividing the skewness estimate by the standard error of skewness; values exceeding +/-1.96 indicate that the distribution is statistically significantly skewed. (Transcript 10.29 27:32-28:49)