Alpha, beta, p-value, z, t, binomial, normal — the names alone are confusing. This workbook walks through each one using deliberately simple numbers you can compute by hand. Every unit follows the same path: formula → hand calculation → TI-84 keystrokes → try it yourself.
Three rules for this workbook
1. Every dataset uses simple, easy-to-compute numbers — nothing messy.
2. Every formula is calculated by hand first, then shown as TI-84 keystrokes.
3. Each unit ends with a box where you type in your own answer and get it checked instantly.
When data follows a bell-shaped (normal) distribution, a z-score tells you how many standard deviations a value is from the mean. Once you know the z-score, you can find the proportion of data above or below it using a z-table or a calculator.
By hand: z = (85 − 70) / 10 = 1.5
From the standard normal table, z = 1.5 → about 0.9332
Find the probability that "success" happens exactly x times out of n trials. Conditions: trials must be independent, and the success probability p must stay the same every trial.
By hand: C(10,3) = 120, 120 × (0.3)³ × (0.7)⁷ = 120 × 0.027 × 0.0823 ≈ 0.2668
Mean and SD: μ = 10×0.3 = 3, σ = √(10×0.3×0.7) = √2.1 ≈ 1.449
Models "on which trial does the first success occur?" Unlike the binomial, the number of trials n is not fixed in advance.
By hand: (0.75)³ × 0.25 = 0.4219 × 0.25 ≈ 0.1055
Average number of trials: μ = 1/0.25 = 4 trials
If you repeatedly draw samples of size n from a population, the distribution of the sample means x̄ is called the "sampling distribution." When n is large enough (roughly n≥30), the distribution of x̄ becomes approximately normal regardless of the population's shape — this is the Central Limit Theorem (CLT).
By hand: σx̄ = 12/√36 = 12/6 = 2, z = (53−50)/2 = 1.5
Use the sample proportion p̂ to estimate the population proportion p (confidence interval), or to test a claim about p (hypothesis test).
By hand: SE = √(0.6×0.4/200) = √0.0012 ≈ 0.0346
0.6 ± 1.96×0.0346 = 0.6 ± 0.0679
By hand: SE₀ = √(0.5×0.5/200) = √0.00125 ≈ 0.03536
z = (0.6 − 0.5) / 0.03536 ≈ 2.828
p-value = P(Z > 2.828) ≈ 0.0023
When you don't know the population standard deviation σ (true in most real situations), you use the sample standard deviation s instead, and use the t-distribution instead of the normal distribution. Degrees of freedom df = n − 1.
By hand: SE = 6/√15 ≈ 1.549, margin of error = 2.145×1.549 ≈ 3.32
By hand: t = (82−80)/1.549 ≈ 1.291, df = 14
p-value = P(T > 1.291) ≈ 0.109
Tests whether the observed counts of a categorical variable differ from the expected counts.
Claim: 4 candy colors occur "equally often" (n=100, expected 25 each)
By hand: (30−25)²/25 + (20−25)²/25 + 0 + 0 = 1 + 1 + 0 + 0 = 2
df = 4 − 1 = 3, p-value = P(χ² > 2, df=3) ≈ 0.572
Tests whether there is real evidence of a linear relationship (slope ≠ 0) between two quantitative variables.
By hand: slope b₁ = 1.5, SEb1 ≈ 0.379
t = 1.5 / 0.379 ≈ 3.96, df = 5−2 = 3
two-sided p-value ≈ 0.028
The six most commonly confused terms in hypothesis testing, side by side. This is where most students lose points on the exam.
| Term | Meaning | One-line example |
|---|---|---|
| p-value | Assuming H₀ is true, the probability of getting a result this extreme (or more extreme) | p = 0.0023 → "if H₀ were true, data like this would be very rare" |
| α (significance level) | The threshold you set in advance. If p-value < α, you reject H₀ | Usually α = 0.05 (chosen by the researcher before testing) |
| β | The probability of failing to reject H₀ when H₀ is actually false (probability of a Type II error) | A large β means low power (1−β) |
| Type I error | Rejecting H₀ when H₀ is actually true. Probability of this error = α | Concluding "there is an effect" when there really isn't one |
| Type II error | Failing to reject H₀ when H₀ is actually false. Probability of this error = β | Concluding "not enough evidence" when there really is an effect |
| Power | The probability of correctly rejecting H₀ when H₀ is false = 1 − β | Increasing the sample size n increases power |
Decision rule: p-value < α → reject H₀ (statistically significant) / p-value ≥ α → fail to reject H₀ (insufficient evidence — this does NOT "prove" H₀ is true)
Click a question to reveal the answer and explanation. For anything you miss, jump back to that unit above and review it.