Set 1 – Core Ideas (Ch. 8–10)

Z‑score

Z = (value − mean) / SD; tells how many standard deviations a value is above or below the mean.

Confidence Interval (CI)

An interval (two numbers) that likely contains the true parameter (p, μ, σ, or σ²) with a certain confidence level.

α (alpha)

The significance level; α = 1 − confidence level (e.g., 95% confidence → α = 0.05).

p‑value

The probability (between 0 and 1) of getting results as extreme as your sample (or more) if H₀ is true.

Decision rule using p and α

If p ≤ α → Reject H₀ (statistically significant). If p > α → Fail to reject H₀.

p vs p̂

p = true population proportion; p̂ = sample proportion = x/n.

μ vs x̄

μ = true population mean; x̄ = sample mean (average of sample).

σ vs s

σ = population standard deviation; s = sample standard deviation.

df (degrees of freedom) for t and χ²

df = n − 1.

1‑PropZInt

Use for a confidence interval for a proportion p.

TInterval

Use for a confidence interval for a mean μ when σ is unknown.

1‑PropZTest

Use for a hypothesis test about a proportion p.

TTest

Use for a hypothesis test about a mean μ with sample data.

Chi-square (χ²)

Used for confidence intervals or tests about σ or σ² (standard deviation / variance).

Button for CI for proportion p

STAT → TESTS → 1‑PropZInt.

Button for CI for mean μ

STAT → TESTS → TInterval (Data or Stats).

Button for test of proportion p

STAT → TESTS → 1‑PropZTest.

Button for test of mean μ

STAT → TESTS → TTest (never ZTest in this class).

Button to get Z_{α/2}

2nd → VARS → 3:invNorm (use left‑side area).

Button to get t_{α/2,df}

2nd → VARS → 4:invT (use left‑side area and df = n−1).

Button for P(Z > a)

2nd → VARS → 2:normalcdf( a, 1E99 ) for standard normal.

Sample proportion p̂ formula

p̂ = x / n, where x = # successes, n = sample size.

General CI form for p

p̂ ± Z_{α/2}·√[p̂(1−p̂)/n]; use 1‑PropZInt in practice.

General CI form for μ

x̄ ± t_{α/2,df}·(s/√n); use TInterval in practice.

95% CI for p meaning

We are 95% confident that the true proportion p is between the two CI bounds.

95% CI for μ meaning

We are 95% confident that the true mean μ is between the two CI bounds.

Sample size n for proportion formula

n = \dfrac{p̂(1 - p̂)(Z_{α/2})^2}{E^2}; if no p̂ given, use 0.5 and round n UP.

CI for variance σ² using chi‑square structure

Lower: (n−1)s² / χ²{1−α/2,df}; Upper: (n−1)s² / χ²{α/2,df}.

Getting CI for σ from CI for σ²

Take square roots of lower and upper bounds.

Hypothesis test must include

1) H₀ and H₁, 2) Name of test (1‑PropZTest or TTest), 3) p‑value, 4) Conclusion sentence in context.

Rule for H₀ and H₁ symbols

H₀ uses '=' (p = p₀ or μ = μ₀); H₁ uses

Wording → symbol: "less than 10"

H₁: parameter < 10.

Wording → symbol: "greater than 0.4"

H₁: parameter > 0.4.

Wording → symbol: "different from 50"

H₁: parameter ≠ 50.

Statistically significant meaning

p ≤ α, so we reject H₀.

Written p‑value not between 0 and 1

It’s wrong; professor gives 0 credit for that p‑value.

Should I use ZTest for a mean?

No, use TTest (σ is unknown).

α not given in a test question

Default α = 0.05 (5%).

Rounding for sample size n

Always round n UP to the next whole number.