BUAD 310 FORMULAS (Midterm 2)

Sample mean

x̄ = Σxᵢ / n

Sample proportion

p = x / n, where x is the number of successes.

Expected value of the sample mean

E(x̄) = μ

Standard error of x̄ when σ is known

SE = σ / √n

Estimated standard error of x̄ when σ is unknown

SE = s / √n

Standard error of p for a confidence interval

SE = √[p(1 − p) / n]

General confidence-interval formula

Point estimate ± (critical value)(standard error)

Margin of error

ME = (critical value)(standard error)

Confidence interval for μ when σ is known

x̄ ± z(α/2)(σ / √n)

Confidence interval for μ when σ is unknown

x̄ ± t(α/2, n−1)(s / √n)

Degrees of freedom for a one-sample t procedure

df = n − 1

Confidence interval for population proportion π

p ± z(α/2)√[p(1 − p) / n]

Common z-critical values

90%: 1.645; 95%: 1.960; 99%: 2.576.

One-mean z-test statistic

z = (x̄ − μ₀) / (σ / √n)

One-mean t-test statistic

t = (x̄ − μ₀) / (s / √n)

One-proportion z-test statistic

z = (p − π₀) / √[π₀(1 − π₀) / n]; use π₀, not p, in the standard error.

Normality check for a one-proportion z-test

nπ₀ ≥ 10 and n(1 − π₀) ≥ 10

Standard error of x̄₁ − x̄₂ when population variances are known

SE = √(σ₁²/n₁ + σ₂²/n₂)

Two-mean z-test with known variances

z = [(x̄₁ − x̄₂) − D₀] / √(σ₁²/n₁ + σ₂²/n₂); usually D₀ = 0.

Pooled variance

sₚ² = [(n₁ − 1)s₁² + (n₂ − 1)s₂²] / (n₁ + n₂ − 2)

Pooled standard error

SE = √[sₚ²(1/n₁ + 1/n₂)]

Pooled two-sample t-test

t = [(x̄₁ − x̄₂) − D₀] / √[sₚ²(1/n₁ + 1/n₂)]

Degrees of freedom for a pooled two-sample t-test

df = n₁ + n₂ − 2

Paired t-test statistic

t = (d̄ − μd₀) / (sd / √n); usually μd₀ = 0 and df = n − 1.

Expected count for chi-square independence

e = (row total × column total) / n

Chi-square test statistic

χ² = Σ[(f − e)² / e]

Degrees of freedom for chi-square independence

df = (r − 1)(c − 1)

Cramér’s V

V = √[χ² / (n × minimum of (r − 1) and (c − 1))]

Expected count for goodness-of-fit

e = n × stated proportion

Degrees of freedom for goodness-of-fit

df = c − 1