Don't memorize the formulas — compute them yourself.
Every unit below pairs a deliberately simple dataset with a hand-calculation walkthrough and the exact TI-84 button sequence. Check that your calculator's answer matches the hand calculation, and these procedures will never trip you up on test day.
Normal Distribution & Z-scores
Unit 1.9 / 5.2 — When data follow a normal distribution, standardize a value's position to turn it into a probability.
Core Formula Concept
z tells you "how many standard deviations away from the mean" a value is. Once you know z, the standard normal table (or your calculator) gives you the cumulative probability directly.
Work It With Simple Numbers Example
Student A scored 85. What proportion of students scored lower than A?
- Compute z: z = (85 − 70) / 10 = 1.5
- Look up z = 1.50 in the standard normal table → cumulative probability ≈ 0.9332
- Conclusion: about 93.3% of students scored lower than A.
💡 For "negative infinity," type -1E99 on the calculator: (-) → 1 → EE (2nd, comma key) → 99.
Working Backwards: Probability → Score Inverse
- Top 10% = the point below which 90% of the data falls → invNorm(0.90)
- From the table, cumulative probability 0.90 corresponds to z ≈ 1.28
- x = μ + zσ = 70 + 1.28(10) = 82.8 points
Binomial Distribution
Unit 4.10–4.11 — When each independent trial has only two outcomes, "success" or "failure," repeated n times.
Conditions & Formulas Concept
Simple Example Example
- C(5,2) = 10
- P(X=2) = 10 × (0.4)² × (0.6)³ = 10 × 0.16 × 0.216 = 0.3456
- Mean: μ = np = 5(0.4) = 2 questions / σ = √(5×0.4×0.6) = √1.2 ≈ 1.095
💡 For "3 or more correct," use 1 − binomcdf(n,p,2) (the complement rule).
Sampling Distributions & the Central Limit Theorem (CLT)
Unit 5.3, 5.5, 5.7 — If you repeatedly draw samples, the sample mean/sample proportion each have their own distribution.
Distribution of Sample Proportion p̂ Concept
σ(p̂) = √(0.5×0.5/100) = √0.0025 = 0.05 → p̂ will typically vary around 0.50 by about ±0.05.
Distribution of Sample Mean x̄ Concept
Confidence Interval for a Proportion (1-PropZInt)
Unit 6.2 — Use a sample proportion to estimate a range for the population proportion p.
Formula Concept
Simple Example Example
- p̂ = 30/50 = 0.6
- SE = √(0.6×0.4/50) = √0.0048 ≈ 0.0693
- z* (95%) = 1.96 → ME = 1.96 × 0.0693 ≈ 0.1358
- Confidence interval: 0.6 ± 0.136 → (0.464, 0.736)
Interpretation sentence (memorize!): "We are 95% confident that the true population proportion is between 0.464 and 0.736."
Hypothesis Test for a Proportion (1-PropZTest)
Unit 6.4–6.7 — Where p-value, significance level α, and Type I/II errors first appear.
Procedure & Formula Concept
Simple Example Example
H0: p = 0.5, Ha: p ≠ 0.5
- p̂ = 60/100 = 0.6
- z = (0.6 − 0.5) / √(0.5×0.5/100) = 0.1 / 0.05 = 2.0
- Two-sided p-value = 2 × P(Z > 2.0) = 2 × 0.0228 = 0.0456
- 0.0456 < 0.05(α) → reject H0. There is significant evidence the coin is not fair.
What Are Type I and Type II Errors? Must-Know
Memory tip: "α is the number I set in advance (usually 0.05)", "β shrinks as sample size n grows", "a larger n leaves α unchanged but reduces β → power increases."
Confidence Interval for a Mean (TInterval)
Unit 7.2 — When σ (the population standard deviation) is unknown, use the t-distribution instead of z.
Formula Concept
Simple Example Example
- x̄ = (4+5+5+6+7+9)/6 = 36/6 = 6
- Sum of squared deviations = (−2)²+(−1)²+(−1)²+0²+1²+3² = 4+1+1+0+1+9 = 16
- s = √(16/(6−1)) = √3.2 ≈ 1.789
- df = 5, 90% confidence level → t* ≈ 2.015
- ME = 2.015 × (1.789/√6) ≈ 2.015 × 0.730 ≈ 1.471
- Confidence interval: 6 ± 1.47 → (4.53, 7.47)
Hypothesis Test for a Mean (T-Test)
Unit 7.4–7.5 — Use a sample to test whether a population mean differs from a claimed value.
Formula Concept
Simple Example Example
H0: μ = 10, Ha: μ ≠ 10, α = 0.05
- x̄ = (9+9+10+11+11)/5 = 50/5 = 10
- Sum of squared deviations = 1+1+0+1+1 = 4 → s = √(4/4) = 1
- t = (10 − 10)/(1/√5) = 0
- t = 0 → p-value = 1 (nothing could be less extreme) → fail to reject H0. No evidence battery life differs from 10 hours.
Chi-Square Tests
Unit 8.2–8.6 — Test the difference between "observed" and "expected" counts for categorical data.
Formula Concept
Simple Goodness-of-Fit Example Example A
| Face | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Observed | 8 | 9 | 12 | 11 | 10 | 10 |
| Expected | 10 | 10 | 10 | 10 | 10 | 10 |
- χ² = (8−10)²/10 + (9−10)²/10 + (12−10)²/10 + (11−10)²/10 + (10−10)²/10 + (10−10)²/10
- = 0.4 + 0.1 + 0.4 + 0.1 + 0 + 0 = χ² = 1.0
- df = 6 − 1 = 5 → p-value(χ²=1.0, df=5) ≈ 0.963 (very large)
- p-value > α → fail to reject H0. No evidence the die is unfair.
Simple Independence Test Example Example B
| Member | Not a member | |
|---|---|---|
| Male | 12 | 8 |
| Female | 8 | 12 |
- Row/column totals: Male 20, Female 20 / Member 20, Not a member 20 → total 40
- Expected count (same for all cells) = (row total × column total)/total = (20×20)/40 = 10 (all 4 cells)
- χ² = (12−10)²/10 × 4 cells = 4×(4/10) = 1.6
- df = (2−1)(2−1) = 1 → p-value(χ²=1.6, df=1) ≈ 0.206
- p > α(0.05) → not enough evidence of an association (gender and club membership can be treated as independent)
α · β · p-value · Type I/II Error — Side by Side
The five most commonly confused terms on the exam, laid out in one table.
| Term | Definition | Who/When It's Set | Effect of Sample Size n | Memory Sentence |
|---|---|---|---|---|
| α (significance level) | The maximum acceptable probability of rejecting H0 when it's actually true. Chosen before the test (commonly 0.05, 0.01, 0.10) | Set by the researcher before running the test | Independent of n — it's a number you choose, so it doesn't change | "The biggest mistake I'm willing to risk" |
| p-value | Assuming H0 is true, the probability of observing a result at least as extreme as the one obtained | Calculated from the sample data (a result of the test) | All else equal, larger n tends to make the p-value smaller for the same effect size | "How surprising is this result, if it were really just chance?" |
| Type I Error | Rejecting H0 when it is actually true (a false alarm) — probability = α | Lowering α lowers the probability of a Type I error | Independent of n | "Convicting an innocent person" |
| Type II Error (β) | Failing to reject H0 when it is actually false (a miss) — probability = β | Determined indirectly by α, effect size, and n (not chosen directly) | As n increases, β decreases (power increases) | "Letting a guilty person go free (missed it)" |
| Power (= 1−β) | The probability of correctly rejecting H0 when it is false | Increases with larger α, larger effect size, or larger n | Power increases as n increases | "The test's ability to catch a real difference" |
The One-Line Decision Rule Must Memorize
10 Practice Multiple-Choice Questions
Click "Show answer" under each question to check your work. Try solving without a calculator first, then verify.
Exam scores follow a normal distribution with μ=70, σ=10. What is the z-score for a score of 60?
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For a binomial distribution with n=5, p=0.4, which is the correct calculator command for P(X=2)?
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For a population with σ=20, if the sample size increases from n=25 to n=100, how does the standard error (SE) change?
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For a sample with n=50, x=30, what value of p̂ does the calculator automatically compute when running 1-PropZInt?
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A 1-PropZTest gives p-value = 0.0456 with α = 0.05. What is the correct conclusion?
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A new drug actually works (H0: "no effect" is false), but the test concluded "no effect." What is this error called?
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If sample size n increases (all else equal), which of the following is true?
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When the population standard deviation σ is unknown and you're estimating a population mean from sample data, which distribution is used?
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For a 2×3 contingency table, what is the degrees of freedom (df) for a chi-square independence test?
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Which of the following is the correct interpretation of a "p-value"?