Course: Introduction to Statistics (MATH 11) · Silver Oak University (fictional sample) · Prof. Rivera
Module: Week 14 of 16 · Fall 2026 · in-person, two 75-minute sessions
Objective covered: Objective 7 — Conduct and interpret hypothesis tests for means and proportions (this week: the mechanics — the one-sample t-test, the two-sample idea, and the one-proportion z-test).

This file holds two pieces: (A) the Module 14 Overview page ("Start Here") and (B) the Welcome Announcement that drips out when the module opens. Week 14 is a normal two-session week: we meet Tuesday Dec 1 and Thursday Dec 3, and end-of-week work is due Sunday Dec 6, 11:59 p.m. — with the Discussion initial post due Friday Dec 4 so classmates have something to reply to. Adjust day-of-week and times to match your section.

(A) Module 14 Overview — Start Here

Welcome to Week 14: Tests for Means & Proportions

This is your home base for the week. Read it first, then work the checklist below from top to bottom. Everything you need is linked inside the module.

Last week you learned the logic of a hypothesis test — state the hypotheses, read the p-value, compare it to α, conclude in context. But all week, one question hung in the air: where does that p-value actually come from? This week is the answer. We build the evidence ourselves with two formulas — a t-test for a claim about a mean, a z-test for a claim about a proportion — plus the idea of comparing two groups to each other. The four beats never change — State → Compute → Compare → Conclude — we're just installing the engine that produces the p-value.

The week's big question

"Where does the p-value come from? How do we compute the evidence for a claim about a mean or a proportion — and how do we pick the right test for the question?"

By the end of the week you'll take a claim about a mean or a proportion, decide which test it needs, plug friendly numbers into one formula, get a test statistic, compare its p-value to α, and write the conclusion as a sentence about the real world.

By the end of this week, you can…

Use this as a checklist. If you can do all four out loud, you're ready for the quiz.

• [ ] Run a one-sample t-test for a mean end to end — state H₀/Hₐ, compute t = (x̄ − μ₀)/(s/√n) (keep the √n!), compare the given p to α, and conclude in context.
• [ ] Run a one-proportion z-test — state, compute z = (p̂ − p₀)/√(p₀(1−p₀)/n) (standard error uses p₀; decimals, not percents), compare, and conclude.
• [ ] Interpret a two-sample comparison of two group means — H₀ is "the two means are equal"; read a supplied p-value and say plainly whether the means differ (without sliding "different" into "better/big").
• [ ] Choose the right test from the question — mean → t; proportion → z; one group vs. a number → one-sample; two groups vs. each other → two-sample. (Count the groups.)

What's due this week, and when

Work these in order — each one gets you ready for the next.

Heads-up on the AI tutorial: you'll use a chatbot to draft, and then you judge its work against what we cover in class. Chatbots routinely drop the √n (reporting the wrong standard error and t), use p̂ instead of p₀ in a proportion test, or slide from "significant" to "large/important." Catching and rewriting those is the point.

Late policy reminder: 10% off per day late. If life happens, reach out before the deadline — I'd much rather hear from you early.

How to succeed this week

• Pick the test first, every time. Before any arithmetic, ask the two questions: Mean or proportion? (average → t, share/rate → z) and One group or two? (vs. a fixed number → one-sample, vs. another group → two-sample). The test you choose is the part students miss most.
• Keep the √n. The t denominator is the standard error s/√n, not s — write the standard error as its own step so you never drop it. (In our worked example, 10/√25 = 2.)
• Proportions: p₀ and decimals. A proportion test uses p₀ inside √(p₀(1−p₀)/n) — because everything is computed assuming H₀ is true — and the numbers go in as decimals (0.60, not 60).
• The decision rule hasn't changed. Once you have the statistic and its p-value, it's the same call as Week 13: p ≤ α → reject; p > α → fail to reject. Then finish the sentence: "At the 0.05 level, there is significant evidence that [the real-world claim]."
• "Different" is not "better" (or "big"). A two-sample test that rejects H₀ says the two means differ — not that one is better or that the gap is important. Size is a separate, practical question.

You don't need to memorize a table this week — every number is friendly and the test statistics come out clean (they all land on t = 2 or z = 2 by design). Come to class Tuesday ready to choose the right test for a scenario on the board.

(B) Welcome Announcement — Module 14

Release setting: post on the module's start day (offset = 0 days), i.e., Tue Dec 1, 2026 — not before. If your platform won't preserve the scheduled date on import, post this as a draft labeled "Release: Tue Dec 1."

Subject: Week 14 — where the p-value actually comes from 🧮

Hi everyone,

Quick callback to start the week: last week I kept saying "technology hands us p = 0.03," and I dodged the obvious question every time — where does that number come from? This week is the answer.

This week — Tests for Means & Proportions — we turn last week's logic into real tests. You'll learn to compute the test statistic yourself: a t for a claim about a mean (t = (x̄ − μ₀)/(s/√n)), a z for a claim about a proportion (z = (p̂ − p₀)/√(p₀(1−p₀)/n)), plus the idea of a two-sample comparison when two groups are measured against each other. The four beats are exactly the same as Week 13 — State → Compute → Compare → Conclude — we just install the engine in Step 2. Good news: the numbers are friendly, every statistic comes out clean, and the p-values are handed to you. The real skill is choosing the right test and interpreting the result.

Two things not to miss:
1. Lecture Tutorial 14 — work the one-sample t, the one-proportion z, and the two-sample idea with one approved chatbot (Gemini, Claude, or ChatGPT) and submit the share link. You'll catch the model's classic mistakes — it loves to drop the √n or slide "significant" into "important." Due Sun Dec 6.
2. Discussion 14 — "Two groups, one claim: is it justified?" — find a real claim that compares two groups (an A/B test, a drug vs. a placebo, a before/after study) and figure out which test fits and whether the conclusion holds up. Initial post Fri Dec 4, replies Sun Dec 6.

One callback: Week 13 was the logic of a test; this week is the machinery that produces the p-value. Same courtroom, two new engines — a t for means, a z for proportions.

Open the Start Here / Module Overview page first — it lays out everything in order with due dates. Bring a question to class Tuesday: for the scenario on the board, which test do we run, and why?

See you soon,
Prof. Rivera