Course: Introduction to Statistics (MATH 11) · Silver Oak University (fictional sample) · Prof. Rivera
This is a student-facing review page. Read it, work the fresh practice, and follow the dated plan. Then run the paired Exam-Prep Tutorial and take the Practice Exam for active recall. (This guide points to those two — it does not repeat them.)

Integrity note for students. Every practice item on this page is a fresh variant — new numbers and contexts — with a pre-computed, vetted answer. None of these are the live midterm questions. Working them builds the skill the midterm tests, the honest way.

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What the midterm covers (read this first)

Exam	Midterm — cumulative, Weeks 1–7, Objectives 1–4
Format	20 items, 100 points. Application-skewed: most items ask you to do something with data or interpret a result, not just recite a definition. Expect a mix of multiple-choice, short numeric answers, and a few "read this and explain" items.
Coverage (where the points are)	Obj 1 ≈ 4 items (foundations & data types) · Obj 2 ≈ 5 items (summarizing one variable) · Obj 3 ≈ 3 items (relationships between two variables) · Obj 4 ≈ 8 items (probability & random variables — the biggest slice). Study Objective 4 hardest.
Weight	The midterm is 20% of your course grade.
When it opens / where	Opens in the Week 8 module (the review-and-exam week). The exam window and the room/timing are posted with the exam itself in Canvas; this guide and the exam-prep tutorial post before it so you can prepare. There is no weekly quiz or assignment in Week 8 — the midterm replaces them.
What to bring	A calculator and the one-page formula list you build from this guide. Spreadsheets are for practice at home; the exam is do-it-and-interpret.

How to use this guide. Each objective below has the same four parts: (A) the key ideas in plain language, (B) the definitions / formulas / procedures, (C) the predictable mistakes and their cures, and (D) where to review in the module. After all four objectives come fresh worked examples + self-check questions (with answers), a dated study plan, and how it's graded + test strategy.

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Objective 1 — Foundations & Types of Data (Week 1) · ~4 items

(A) Key ideas, plain language

Statistics is about trusting a number that describes people you didn't count. Three questions decide whether a statistic deserves your trust: Who was measured? How were they picked? What was recorded? A population is everyone the question is about; a sample is the part you actually measured. The whole course is the bridge from a statistic (what you have) to a parameter (what you want).

(B) Definitions, formulas, procedures

• Population = everyone/everything the question is about. Sample = the part measured. Census = measuring the whole population.
• Parameter = a number describing the population. Statistic = a number from the sample. Memory hook: P*opulation → Parameter, Sample → Statistic — the letters line up.* Notation: population proportion p; sample proportion p̂ ("p-hat"). The hat means "measured," not "true."
• Levels of measurement — NOIR:
• Nominal = names, no order (major, eye color, zip code, jersey number, blood type, yes/no).
• Ordinal = ordered categories, gaps not equal/measurable (letter grade, S/M/L, 1–5 satisfaction, class standing, race finishing place).
• Interval = ordered, equal gaps, no true zero (°F, °C, calendar year, IQ).
• Ratio = ordered, equal gaps, true zero so ratios make sense (height, weight, age, income, counts, time).
• The test: Does zero mean "none"? yes → ratio. Equal gaps but zero is arbitrary? → interval. Ordered labels, fuzzy gaps? → ordinal. Just names? → nominal.
• Sampling methods: SRS (every individual equally likely — the gold standard) · Stratified (split into groups, random-sample within each) · Cluster (randomly pick whole groups, measure everyone) · Systematic (every k-th after a random start). The bad ones: convenience (whoever's easy) and voluntary response (people opt in) — usually biased.
• Bias = error baked into the method, pushing results the same wrong way no matter the sample size. Types: undercoverage, nonresponse, response, voluntary-response.
• Observational study = watch and record, change nothing. Experiment = impose a treatment and compare; only a randomized experiment supports a cause-and-effect claim. Confounding variable = a third variable tangled with both, so you can't tell which drives the outcome.

(C) Predictable mistakes → cures

• ❌ "If it's a number, it's quantitative." → ✅ Zip codes, jersey numbers, and student IDs are numbers that label. Ask does arithmetic mean anything? You can't average area codes — nominal.
• ❌ "A bigger sample is automatically better." → ✅ Size never fixes bias. The 1936 Literary Digest poll got 2.4 million replies and still called the wrong winner. Method beats size.
• ❌ "Population means a lot of people." → ✅ The population is whoever the question is about — it could be 25 students. It's a role, not a size.
• ❌ "Strong correlation, so X causes Y." → ✅ Ask was anything randomly assigned? If no, it's observational → a link, not a cause. Hunt the confounder.
• ❌ Mixes stratified and cluster. → ✅ Stratified = sample within every group; cluster = sample whole groups.

(D) Review in the module

Week 1 → Lecture Outline (Segments 2–7), Slides (Deck 1), Readings (population/sample, NOIR, sampling & bias, correlation ≠ causation), and Lecture Tutorial 1.

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Objective 2 — Summarizing One Variable (Weeks 2–3) · ~5 items

(A) Key ideas, plain language

First see the data (a table, then a histogram); then summarize it with one number for the center and one for the spread — and match those numbers to the shape. The drama of the whole objective: under skew or outliers, the mean and SD lie, and the median and IQR tell the truth. Shape decides the summary.

(B) Definitions, formulas, procedures

Displays (Week 2):
- Frequency = a count. Relative frequency = frequency ÷ total (a share; all classes sum to 1). Cumulative frequency = a running total.
- Histogram = the picture of a frequency table for quantitative data; bars touch. Bar chart = for categorical data; bars have gaps. Touching → histogram (numbers); apart → bar chart (categories).
- Shapes: symmetric, skewed right (long tail to the right / big values), skewed left (long tail to the left / small values), uniform, bimodal (two humps). Skew is named for the tail, not the hump.
- Describe any distribution in order — S-C-S-O: Shape, Center, Spread, Outliers.

Center & spread (Week 3):
- Mean x̄ = (Σx) ÷ n (add all, divide by count — the balance point). Median = middle value once sorted (even count → average the two middles). Mode = most frequent (the only center for categorical data).
- Variance & SD (sample): deviation = (value − mean); square the deviations; variance s² = (sum of squared deviations) ÷ (n − 1); standard deviation s = √variance (back in the data's units — typical distance from the mean). (Whole population → divide by n; a sample → n − 1.)
- Five-number summary: Min · Q1 · Median · Q3 · Max. Q1 = median of the lower half; Q3 = median of the upper half. IQR = Q3 − Q1 (the middle 50%). Range = Max − Min.
- Resistance: median & IQR are resistant (one outlier can't move them — they depend on position). Mean & SD are non-resistant (they use every value, so an outlier drags them). Pair them: mean rides with SD; median rides with IQR — don't mix partners.
- Skew-vs-center rule: right-skew → mean > median; left-skew → mean < median; symmetric → about equal.

(C) Predictable mistakes → cures

• ❌ "Skewed left means it points/leans left." → ✅ Skew is named for the tail. Skewed left = long thin tail to the left (small values); the tall bulk sits on the right. Find the tail first.
• ❌ "Switching to relative frequency changes the shape." → ✅ It only relabels the vertical axis (every height ÷ total). Same shape; it just lets you compare different-sized groups.
• ❌ Adds the deviations and expects a nonzero spread. → ✅ Plain deviations always sum to 0 — that's why we square them.
• ❌ Divides the sample variance by n. → ✅ Sample variance/SD divides by n − 1.
• ❌ Reports the mean with the IQR (or median with SD). → ✅ Keep the couple together: mean ↔ SD, median ↔ IQR.
• ❌ Reports the mean for obviously skewed/outlier data. → ✅ The mean chases the outlier; report the median under skew. Match the summary to the shape.
• ❌ Confuses range and IQR. → ✅ Range uses the two extremes (an outlier wrecks it); IQR is the middle 50% (resistant).

(D) Review in the module

Week 2 → Lecture Outline (frequency tables, histograms, shape, outliers), Slides (Deck 2), Lecture Tutorial 2. Week 3 → Lecture Outline (mean/median/mode, variance & SD, five-number summary & IQR, resistance), Slides (Deck 3), Lecture Tutorial 3.

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Objective 3 — Relationships Between Two Variables (Week 4) · ~3 items

(A) Key ideas, plain language

When two things move together, you can picture it (a scatterplot), measure it (correlation r), or tabulate it (a two-way table) — and you still cannot draw the causal arrow without an experiment or a ruled-out lurking variable. Correlation is a handshake, not a push.

(B) Definitions, formulas, procedures

• Scatterplot: two quantitative variables, one dot per individual. Explanatory (x) explains; response (y) responds. Describe with D-F-S: Direction (positive/negative), Form (linear/curved/none), Strength (tight/moderate/loose) — plus any outliers?
• Correlation r: one number for the direction and strength of a linear relationship. Always between −1 and +1. r = +1 perfect up line; r = −1 perfect down line; r = 0 no linear relationship. Closer |r| to 1 → tighter line. Rough guide: |r| 0.8–1.0 strong, 0.5–0.8 moderate, < 0.3 weak. r is unitless, symmetric (corr(x,y)=corr(y,x)), not a percent.
• Two-way (contingency) table: cross-classifies two categorical variables. Marginal proportion = a margin total ÷ grand total (one variable, ignoring the other). Conditional proportion = restrict to one group first, then divide by that group's total (the word "given" tells you the denominator). Conditional proportions are how you compare groups.
• Lurking / confounding variable = a third variable, not among your two, that drives both and manufactures a misleading association. Correlation ≠ causation. Two-question cure: (1) Could a third variable explain both? (2) Was anything randomly assigned?

(C) Predictable mistakes → cures

• ❌ "r = 0 means the variables aren't related." → ✅ Only the linear part is 0. A clean U-curve can have r ≈ 0 and still be tightly related. Picture before number.
• ❌ "Stronger r means a steeper line." → ✅ Strength = how tight the dots hug the line; slope = how steep. r measures scatter, not slope.
• ❌ Divides a two-way cell by the grand total when the question says "of the exercisers…". → ✅ "Given"/"of" names the group → divide by that group's row/column total. Restrict first.
• ❌ "r = 0.6 means 60%." → ✅ r is not a percent. Report sign + size in words.
• ❌ Sees strong r and says "X causes Y." → ✅ Name a plausible lurking variable and ask whether anything was randomized.

(D) Review in the module

Week 4 → Lecture Outline (scatterplots D-F-S, correlation r, two-way tables, lurking variables), Slides (Deck 4), Lecture Tutorial 4.

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Objective 4 — Probability & Random Variables (Weeks 5–7) · ~8 items — STUDY HARDEST

(A) Key ideas, plain language

Put an honest number on what you don't yet know, and learn how new information changes it. Then turn outcomes into numbers (random variables) and ask: what should I expect on average (E(X)), and how much will it swing (σ)? Finally, the binomial — counts of yes/no successes — and when the bell curve can stand in for it. This is the biggest slice of the exam; budget the most time here.

(B) Definitions, formulas, procedures

Probability rules (Week 5):
- Sample space S = the complete list of outcomes. Event = a subset. Equally-likely probability: P(event) = favorable ÷ total. Every probability is between 0 and 1; all outcomes in S sum to 1.
- Complement: P(not A) = 1 − P(A). "At least one" → 1 minus "none."
- Addition (OR): P(A or B) = P(A) + P(B) − P(A and B). If mutually exclusive (can't both happen), the overlap is 0 → P(A or B) = P(A) + P(B).
- Multiplication (AND): if independent (one doesn't change the other's odds), P(A and B) = P(A) × P(B).
- Conditional: P(A | B) = P(A and B) ÷ P(B) — of the times B happened, how often did A? On a two-way table, "given" shrinks the world to one row/column; the denominator becomes the subgroup total.
- Independence test: A and B are independent exactly when P(A | B) = P(A). Mutually exclusive ≠ independent (different ideas). P(A | B) ≠ P(B | A) in general (confusion of the inverse; base-rate trap).

Random variables (Week 6):
- Random variable X attaches a number to a chance outcome. Discrete = countable/listable (counts); continuous = fills an interval (measurements). Count vs. ruler.
- Valid probability distribution: every P in [0, 1] and ΣP = 1. (A missing probability = 1 − the others.)
- Expected value E(X) = Σ [ x · P(X=x) ] — the long-run weighted average (need not be an attainable value).
- Variance Var(X) = E(X²) − [E(X)]², where E(X²) = Σ [ x² · P(X=x) ]. σ = √Var(X) (real units, typical swing). Mean of the squares minus the square of the mean.
- Decisions: judge a bet/warranty/insurance by E(net gain). Negative E favors the other side.

Binomial & normal model (Week 7):
- Binomial setting — BINS (all four): Binary outcome, Independent trials, N fixed number of trials, Same p each trial. (Without-replacement or "until I get one" breaks it.)
- Binomial probability: P(X = k) = C(n, k) · pᵏ · (1−p)ⁿ⁻ᵏ. Small "choose" values: n=2 → 1,2,1; n=3 → 1,3,3,1; n=4 → 1,4,6,4,1.
- Binomial mean & SD: μ = np, σ² = np(1−p), σ = √(np(1−p)).
- Normal approximation license: use it only when np ≥ 10 AND n(1−p) ≥ 10 (both must clear 10). Then a normal curve centered at np with SD √(np(1−p)) stands in for the bars.

(C) Predictable mistakes → cures

• ❌ "After a streak, the other outcome is 'due.'" → ✅ Independent trials have no memory (gambler's fallacy). Past results don't change the next probability.
• ❌ "For 'A or B,' just add." → ✅ Only if mutually exclusive. Otherwise subtract the overlap (King-or-Heart = 16/52, not 17/52). A probability over 1 means you double-counted.
• ❌ Confuses mutually exclusive and independent. → ✅ Exclusive = can't both happen; independent = one doesn't change the other's odds.
• ❌ Swaps P(A|B) and P(B|A), or assumes a "99% accurate" positive means 99% sick. → ✅ The thing after the bar is the world you're in. With a rare disease the base rate dominates — a positive can still be ~17% likely true.
• ❌ Treats any table as a distribution. → ✅ Must pass both gates: each P in [0,1] and ΣP = 1. (The x-values do not have to sum to 1 — the probabilities do.)
• ❌ "E(X) is the most likely outcome." → ✅ That's the mode. E(X) is the weighted average (a die's E(X) = 3.5, which can't occur).
• ❌ Reports E(X²) as the variance. → ✅ Subtract [E(X)]². Write μ² down first so you don't forget. (And there's no n to divide by — outcomes are weighted by probability.)
• ❌ "Any yes/no thing is binomial." → ✅ Run all four BINS conditions. Without replacement breaks independence and constant p.
• ❌ "'Success' means something good." → ✅ "Success" is just the outcome you're counting (a defect can be a "success").
• ❌ Uses √(np) for the SD. → ✅ SD = √(np(1−p)) — the (1−p) factor is essential.
• ❌ Applies the normal curve to small n. → ✅ Check np ≥ 10 and n(1−p) ≥ 10 first; if either fails, use the exact binomial.

(D) Review in the module

Week 5 → Lecture Outline (sample spaces, complement/addition, multiplication/independence, conditional), Slides (Deck 5), Lecture Tutorial 5. Week 6 → Lecture Outline (discrete vs. continuous, valid distributions, E(X), variance & SD), Slides (Deck 6), Lecture Tutorial 6. Week 7 → Lecture Outline (BINS, binomial probabilities, np & √(np(1−p)), normal-approximation check), Slides (Deck 7), Lecture Tutorial 7.

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Representative practice (all fresh — vetted answers)

None of these are live midterm items. New numbers, new contexts. Each answer is pre-computed; the one-line why names the idea it tests. Cover the answers, work each one, then check.

Objective 1 practice

Worked example 1 — population/sample, statistic/parameter, NOIR.
A campus dining service emails all 6,000 meal-plan holders and 480 reply; 312 rate the food "good or better." They also record each respondent's class standing (Fr/So/Jr/Sr) and meals eaten last week (a count).
- (a) Population? Sample? (b) Is 312/480 a statistic or a parameter, and what is it as a percent? (c) Classify class standing and meals eaten.
Answer. (a) Population = all 6,000 meal-plan holders; sample = the 480 who replied. (b) Statistic (it's from the sample): 312 ÷ 480 = 0.65 = 65%. (c) Class standing = ordinal (ordered, unequal gaps); meals eaten = ratio (a count, true zero). Why: a sample number is a statistic estimating the unseen parameter; NOIR is settled by order and whether zero means "none."

Worked example 2 — sampling method + bias.
To estimate average sleep for all 6,000 meal-plan holders, a researcher posts a link in a campus forum and uses whoever replies.
- (a) Name the method. (b) Name the most likely bias. (c) What would a trustworthy method be?
Answer. (a) Voluntary response (people opt in). (b) Voluntary-response / nonresponse bias — the motivated reply, not a typical cross-section. (c) Pull a simple random sample from the registrar's full list (or stratify by class standing). Why: how you pick beats how many; only a chance-based method earns "random."

Self-check (Obj 1).
1. True/false: a census is a kind of sample. → False — a census measures the whole population. (role vs. size)
2. A study finds students who use the library more have higher GPAs (nothing assigned). Cause or link? → Link — observational; a lurker like conscientiousness could drive both.
3. Classify calendar year a car was made. → Interval (equal gaps, no true zero).
4. Which guarantees every class standing is represented: cluster or stratified? → Stratified (sample within every group).

Objective 2 practice

Worked example 1 — frequency/relative-frequency table + shape.
Twelve quiz scores (out of 20): 18, 15, 17, 16, 19, 14, 16, 20, 13, 17, 16, 11. Use classes 11–13, 14–16, 17–20.
- Build frequency and relative frequency; name the modal class; describe the shape.
Answer. Sorted: 11, 13, 14, 15, 16, 16, 16, 17, 17, 18, 19, 20.
| Class | Frequency | Relative frequency |
|---|---|---|
| 11–13 | 2 | 2/12 ≈ 0.167 (16.7%) |
| 14–16 | 5 | 5/12 ≈ 0.417 (41.7%) |
| 17–20 | 5 | 5/12 ≈ 0.417 (41.7%) |
| Total | 12 | 1.00 |
Modal class = 14–16 and 17–20 tie (both 5). With most scores high and a thin tail toward 11, the shape is slightly skewed left. Why: relative frequency = count ÷ total (sums to 1); skew is named for the tail, which here trails toward the low scores.

Worked example 2 — mean/median/SD + which to trust.
Five wait times (minutes): 4, 6, 7, 8, 25.
- (a) Mean and median. (b) Sample SD. (c) Which center is honest here, and why?
Answer. (a) Mean = (4+6+7+8+25) ÷ 5 = 50 ÷ 5 = 10; median = 3rd of sorted five = 7. (b) Deviations from 10: −6, −4, −3, −2, +15; squares: 36, 16, 9, 4, 225; sum = 290; variance = 290 ÷ (5−1) = 72.5; SD = √72.5 ≈ 8.51. (c) The median (7) — the lone 25 is an outlier that drags the mean above four of the five values. Why: mean & SD are non-resistant; under an outlier, report the median (and IQR).

Worked example 3 — five-number summary + IQR.
Seven daily step-counts (thousands): 5, 7, 8, 10, 12, 13, 22. (sorted)
- Give the five-number summary, the IQR, and the range.
Answer. Min = 5, Max = 22. Median (4th of 7) = 10. Lower half {5, 7, 8} → Q1 = 7; upper half {12, 13, 22} → Q3 = 13. Five-number summary = 5 · 7 · 10 · 13 · 22. IQR = 13 − 7 = 6; range = 22 − 5 = 17. Why: Q1/Q3 are the medians of the halves; IQR (middle 50%) ignores the outlier that inflates the range.

Self-check (Obj 2).
1. Histogram or bar chart for students' declared majors? → Bar chart (categorical, bars apart).
2. A distribution has mean 70 and median 58. Likely shape? → Skewed right (mean > median).
3. For the set 3, 3, 5, 9, 10, what is the mode? → 3 (most frequent).
4. You report the median as the center. Which spread pairs with it? → IQR (keep the couple together).

Objective 3 practice

Worked example 1 — scatterplot D-F-S + interpret r.
Six employees' years of experience (x) and monthly sales in $1,000s (y): (1, 12), (2, 14), (3, 15), (5, 19), (6, 20), (8, 26). The computed correlation is r ≈ +0.99.
- (a) Describe the relationship (D-F-S). (b) Read r in words. (c) Does this prove experience causes higher sales?
Answer. (a) As experience rises, sales rise, dots near a straight line, tightly: strong, positive, linear, no outliers. (b) r ≈ +0.99 → a very strong positive linear relationship (near-perfect upward line). (c) No — it's observational; a lurker (e.g., territory size or motivation) could drive both. Why: r measures linear direction/strength only; correlation is not causation.

Worked example 2 — two-way table conditional proportions.
200 students answered Commute by bike? and Arrive on time?

	On time	Late	Row total
Bikes	54	6	60
Does not bike	98	42	140
Column total	152	48	200

• (a) P(on time given bikes). (b) P(on time given does not bike). (c) What does the comparison say?
  Answer. (a) 54 ÷ 60 = 0.90 = 90%. (b) 98 ÷ 140 = 0.70 = 70%. (c) Bikers are more likely to arrive on time (90% vs. 70%, a 20-point gap) — an association, but not proof bikes cause punctuality. Why: a conditional proportion divides by the group's total ("given" picks the denominator); a gap is association, not causation.

Self-check (Obj 3).
1. r for a perfect U-shaped scatter? → ≈ 0 (no linear relationship, though clearly related).
2. From the table above, the marginal P(bikes)? → 60 ÷ 200 = 0.30 = 30%.
3. If you swap x and y, does r change? → No — r is symmetric.
4. "Cities with more police have more crime." Lurking variable? → City population/size (drives both).

Objective 4 practice — largest section; work all of these

Worked example 1 — addition & complement.
Draw one card from a standard 52-card deck.
- (a) P(Queen or Diamond). (b) P(not a face card). (Face cards: J, Q, K — 12 total.)
Answer. (a) Queen and Diamond overlap (Q♦): P = 4/52 + 13/52 − 1/52 = 16/52 = 4/13 ≈ 0.308. (b) P(face) = 12/52; P(not face) = 1 − 12/52 = 40/52 = 10/13 ≈ 0.769. Why: "or" subtracts the overlap unless mutually exclusive; complement = 1 − P.

Worked example 2 — multiplication & independence.
A fair coin is flipped and a fair six-sided die is rolled.
- (a) P(heads and a 6). (b) P(at least one 6 in two die rolls).
Answer. (a) Independent: (1/2)(1/6) = 1/12 ≈ 0.083. (b) Complement: P(no 6 either roll) = (5/6)(5/6) = 25/36; P(at least one 6) = 1 − 25/36 = 11/36 ≈ 0.306. Why: "and" with independence multiplies; "at least one" = 1 − "none."

Worked example 3 — conditional probability from a table.
A survey of 200 students records housing and whether they own a car:

	Owns car	No car	Total
On campus	30	70	100
Off campus	80	20	100
Total	110	90	200

• (a) P(owns car). (b) P(owns car given off campus). (c) Are owning a car and living off campus independent?
  Answer. (a) 110 ÷ 200 = 0.55. (b) 80 ÷ 100 = 0.80. (c) No — P(car | off campus) = 0.80 ≠ P(car) = 0.55, so they're dependent. Why: independence is a number check — does the condition move the probability? Here it does.

Worked example 4 — expected value & SD of a random variable.
A discrete RV X takes values 0, 1, 2, 3 with probabilities 0.2, 0.4, 0.3, 0.1.
- (a) Confirm it's valid. (b) E(X). (c) Var(X) and σ.
Answer. (a) Each P in [0,1] and 0.2+0.4+0.3+0.1 = 1.00 ✓. (b) E(X) = 0(0.2)+1(0.4)+2(0.3)+3(0.1) = 0 + 0.4 + 0.6 + 0.3 = 1.3. (c) E(X²) = 0(0.2)+1(0.4)+4(0.3)+9(0.1) = 0 + 0.4 + 1.2 + 0.9 = 2.5; Var = 2.5 − (1.3)² = 2.5 − 1.69 = 0.81; σ = √0.81 = 0.9. Why: E(X) is the weighted average; variance = E(X²) − [E(X)]², then square-root for the SD.

Worked example 5 — expected value of a bet.
A raffle ticket costs $3. With probability 0.05 you win $50; otherwise you win $0.
- Find the expected net value to you. Good deal?
Answer. Net outcomes: win = $50 − $3 = +$47 (p = 0.05); lose = −$3 (p = 0.95). E(net) = 47(0.05) + (−3)(0.95) = 2.35 − 2.85 = −$0.50. Bad deal — you lose about 50¢ per ticket on average. Why: judge a bet by the expected value of the net; negative favors the seller.

Worked example 6 — binomial probability (small case).
A free-throw shooter makes 60% of her shots and takes 4 (independent).
- P(she makes exactly 3)?
Answer. X ~ B(4, 0.6). C(4,3) = 4. P(X=3) = 4 · (0.6)³ · (0.4)¹ = 4 · 0.216 · 0.4 = 0.3456. Why: P(X=k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ; exponents must add to n (3 makes, 1 miss).

Worked example 7 — binomial mean/SD + normal-approximation check.
An ad is shown to 150 independent users; each clicks with probability 0.4. X = number of clicks.
- (a) μ and σ. (b) Is the normal approximation licensed?
Answer. (a) μ = np = 150 × 0.4 = 60; σ² = np(1−p) = 150 × 0.4 × 0.6 = 36; σ = √36 = 6. (b) np = 60 ≥ 10 ✓ and n(1−p) = 90 ≥ 10 ✓ → yes, a normal curve centered at 60 with SD 6 is fine. Why: μ = np, σ = √(np(1−p)); the normal approximation needs both np and n(1−p) ≥ 10.

Self-check (Obj 4).
1. P(rolling an even number on one fair die)? → 3/6 = 0.5.
2. Is "draw 5 cards from a deck without replacement, count kings" binomial? → No — trials aren't independent and p changes.
3. A distribution lists P = 0.3, 0.25, ?, 0.15. The missing probability? → 1 − (0.3+0.25+0.15) = 0.30.
4. For X ~ B(50, 0.1), is the normal approximation allowed? → np = 5 < 10 → No; use the exact binomial.
5. A test is "98% accurate" for a disease 1% of people have, and you test positive. Is your chance of being sick about 98%? → No — the base rate makes it far lower; P(positive | sick) ≠ P(sick | positive).

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Study plan — a dated countdown (sized to 2 sessions/week)

Built for the Week 8 midterm. Adjust the exact dates to your section's posted exam day; the rhythm is what matters. Do a little every day rather than one long cram.

When	Do this (≈45–75 min)
~7 days out (Week 7, after class)	Read this guide's Objectives 1–3 sections. Work the Obj 1 & 2 practice. Build your one-page formula sheet (S-C-S-O, mean/SD, five-number summary, the probability rules).
~5 days out	Read Objective 4 carefully (it's ~8 of 20 items). Work Obj 4 worked examples 1–4 (probability + E(X)/σ). Re-derive any you missed.
~3 days out	Finish Obj 4 worked examples 5–7 (bets + binomial + normal check) and the Obj 3 practice. Then run the paired Exam-Prep Tutorial (N-exam-prep-tutorial-week-08) in an approved chatbot — it diagnoses your weak spots across the whole midterm and drills them.
~2 days out	Take the Practice Exam (the paired practice exam in this module) under timed, closed-note conditions. Score it; list every missed idea.
~1 day out	Re-teach only the topics you missed on the practice exam (use this guide's mistake-cures and the relevant Lecture Tutorial). Re-do those specific self-checks. Sleep.
Exam day	Skim your one-page formula sheet. Arrive early. Read each item twice.

Two paired tools — use both (don't skip):
- Exam-Prep Tutorial (N-exam-prep-tutorial-week-08) — a copy/paste chatbot tutor that diagnoses, re-teaches, and drills you across all of Objectives 1–4, ending with a readiness summary. Best for active recall and shoring up weak spots.
- Practice Exam (the paired practice exam in the Week 8 module) — a full, fresh, timed run that mirrors the real format. Best for pacing and a final readiness check.

(This guide points to both on purpose — it doesn't duplicate them.)

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How the midterm is graded + test-taking strategy

How it's graded.
- 100 points across 20 items, weighted toward application (doing/interpreting, not reciting). Partial credit is available on multi-step numeric items where shown — show your work so you can earn it.
- The midterm is 20% of your course grade. It replaces Week 8's quiz and assignment (there are none that week).
- Coverage matches this guide: Obj 1 ≈ 4 · Obj 2 ≈ 5 · Obj 3 ≈ 3 · Obj 4 ≈ 8. Time is dominated by Objective 4, so practice it until the procedures are automatic.

Honest test-taking strategies for this material.
1. Name the shape before you pick a summary. If a distribution is skewed or has an outlier, the answer wants the median/IQR, not the mean/SD. The skew is your signal.
2. Circle the keyword in probability items: and / or / given / at least / exactly / independent / mutually exclusive. Each points to a specific rule. "At least one" almost always means 1 − P(none).
3. Run BINS before any binomial formula. If "independent" or "same p" fails (e.g., without replacement), it's not binomial — don't force the formula.
4. For E(X) and variance, write μ² down first so you remember to subtract it: Var = E(X²) − [E(X)]². And remember there's no n to divide by.
5. Check the normal-approximation license every time: np ≥ 10 and n(1−p) ≥ 10. If either fails, say "use the exact binomial."
6. Watch the denominator on two-way tables. "Given/of the [group]" → divide by that group's total, never the grand total.
7. Sanity-check every probability: it must land in [0, 1]. A negative or >1 answer means a missed overlap or a mis-listed sample space.
8. Do the easy items first, flag the hard ones, and budget your time — with 20 items in the period, that's a few minutes each. Don't sink ten minutes into one item while four quick ones wait.
9. Read each item twice and answer the question actually asked (the count vs. the proportion, P(A|B) vs. P(B|A) — the exam rewards reading carefully).

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Canvas placement block

canvas_object   = Page
title           = "Midterm Study Guide — Weeks 1–7 (Objectives 1–4)"
module          = "Week 8 — Midterm Review & Exam"
grading_type    = not_graded
available_from  = 2026-10-15      # posts before the Week 8 exam window opens
published       = true
provenance      = "~ Prof. Rivera's edition · Fall 2026 · built with thecoursemaker.com"

Term-update note: each term's $39 update regenerates fresh practice variants from this same scope — the live midterm is never reproduced here.