Hypothesis Testing: formula sheet

Key formulas

Test statistic for a mean

t = (x̄ − μ0) ÷ (s ÷ √n)

Use t with n − 1 degrees of freedom when the population variance is unknown. Use z when σ is known or n is large.

Standard error of the sample mean

SE = s ÷ √n

Divide the sample standard deviation by the square root of the sample size.

General test statistic

(estimate − hypothesized value) ÷ standard error of estimate

Works for means, regression coefficients and differences.

Decision rule, two-tailed

Reject H0 if |test statistic| > critical value

Critical values use α ÷ 2 in each tail. For z at 5%: ±1.96. At 1%: ±2.58.

Decision rule, one-tailed

Right tail: reject if statistic > critical value. Left tail: reject if statistic < −critical value

For z at 5%: 1.645. At 1%: 2.33.

Confidence interval link

x̄ ± critical value × SE

A two-tailed test at α rejects H0 when μ0 lies outside the (1 − α) confidence interval.

Standard error (σ known)

SE = σ ÷ √n

Use when the population standard deviation is given.

Standard error (σ unknown)

SE = s ÷ √n

s is the sample standard deviation, with n − 1 in the denominator.

Confidence interval for the mean

x̄ ± c × SE

c is the z critical value or the t critical value with n − 1 degrees of freedom.

Common z critical values (two-sided)

90%: 1.645; 95%: 1.96; 99%: 2.576

Memorise these. Each leaves α/2 in each tail.

Test statistic

(x̄ − μ0) ÷ SE

Compare with the same critical value used for the interval.

Degrees of freedom for t

df = n − 1

Use the t table for small samples with unknown variance.

Test statistic for a mean

t = (X̄ − μ₀) ÷ (s ÷ √n)

Use s when σ is unknown. If σ is known, replace s with σ and use z. Degrees of freedom = n − 1.

Standard error of the mean

SE = s ÷ √n

Falls as the sample grows. Quadrupling n halves SE.

t-statistic for a regression coefficient

t = (β̂ − β₀) ÷ SE(β̂)

Usually β₀ = 0. Degrees of freedom = n − k − 1, where k is the number of slope coefficients.

Confidence interval

Estimate ± critical value × standard error

Use t critical value with the right degrees of freedom, or z if the normal applies.

Difference between two means (independent samples, equal variances)

t = (X̄₁ − X̄₂) ÷ √(sp² × (1/n₁ + 1/n₂)), where sp² = [(n₁−1)s₁² + (n₂−1)s₂²] ÷ (n₁ + n₂ − 2)

Degrees of freedom = n₁ + n₂ − 2. Assumes independent samples and equal population variances.

Decision rule

Reject H₀ if |statistic| > critical value (two-tailed)

Equivalent to p-value < significance level. For a one-tailed test, use the one-tail critical value.

Common two-tailed 5% critical values

z = 1.96; t(df=10) = 2.228; t(df=20) = 2.086; t(df=30) = 2.042

The t value shrinks toward 1.96 as degrees of freedom grow.

Decision rule using p-value

Reject H0 if p-value ≤ α

Set α before looking at the data. A smaller p-value means stronger evidence against H0.

Type I error probability

P(reject H0 | H0 true) = α

Equals the significance level.

Type II error probability

β = P(fail to reject H0 | H0 false)

Depends on the true parameter value, sample size and α.

Power of a test

Power = 1 − β

Probability of rejecting H0 when it is false.

Confidence level and significance

Confidence level = 1 − α

A two-sided test at α rejects H0 exactly when the hypothesised value lies outside the (1 − α) confidence interval.

Two-sided p-value for a z statistic

p = 2 × P(Z > |z|)

For a one-sided test, use just one tail: p = P(Z > z) or P(Z < z).

Test statistic for a mean

t = (x̄ − μ0) ÷ (s ÷ √n)

Use the z critical values when the population standard deviation is known or n is large.

Chi-square statistic for one variance

χ² = (n − 1) s² ÷ σ₀²

Degrees of freedom = n − 1. Assumes a normal population. σ₀² is the variance under the null.

F-statistic for two variances

F = s₁² ÷ s₂²

Degrees of freedom: numerator n₁ − 1, denominator n₂ − 1. Samples must be independent and from normal populations.

Hypotheses for a single variance

H₀: σ² = σ₀² versus H₁: σ² ≠ σ₀² (or > or <)

One-tailed alternatives use one critical value; two-tailed uses a lower and an upper value.

Hypotheses for two variances

H₀: σ₁² = σ₂² versus H₁: σ₁² ≠ σ₂² (or >)

Convention: put the larger sample variance on top so F ≥ 1, then use the upper critical value.

Decision rule

Reject H₀ if the statistic falls in the rejection region

Equivalent to rejecting when the p-value is below the significance level.

Chebyshev's inequality

P(|X − μ| ≥ kσ) ≤ 1/k²

Valid for any distribution with finite mean and variance, k > 0. Gives an upper bound only.

One-sided tail bound (Cantelli)

P(X − μ ≥ kσ) ≤ 1 ÷ (1 + k²)

Tighter when you need only one tail. Do not just halve 1/k²; that is not a valid general rule.

Minimum mass within k standard deviations

P(|X − μ| < kσ) ≥ 1 − 1/k²

Complement of the Chebyshev bound. At k = 2 at least 75%.

Exception probability

p = 1 − c

For a 99% VaR, p = 1%. Expected exceptions = p × T.

Binomial probability of x exceptions

P(X = x) = C(T, x) × p^x × (1 − p)^(T − x)

Use for exact backtest probabilities.

Normal approximation z-score

z = (x − pT) ÷ √(p(1 − p)T)

Compare with the critical z (e.g. 1.96 two-sided at 5%). Less reliable when pT is small.

Kupiec unconditional coverage (POF) test statistic

LR = −2 ln[(1 − p)^(T − x) × p^x] + 2 ln[(1 − x/T)^(T − x) × (x/T)^x]

Compared with a chi-squared distribution with 1 degree of freedom; 5% critical value is 3.84.

Quick revision

• H₀ is the claim you try to reject; failing to reject does not prove it true.
• Standard error of a sample mean = s ÷ √n.
• Test statistic = (estimate − hypothesised value) ÷ standard error.
• Confidence interval = estimate ± critical value × standard error.
• Reject H₀ when |statistic| exceeds the critical value, or when the p-value is below the significance level.
• The p-value is the probability of a result at least this extreme if H₀ is true; it is not the probability H₀ is true.
• Type I error: rejecting a true H₀; its probability is the significance level α.
• Type II error: not rejecting a false H₀; power = 1 − probability of Type II error.
• Lowering α reduces Type I errors but raises Type II errors for a fixed sample size.
• A larger sample reduces standard error and raises power.
• Chi-square tests one variance; F tests the ratio of two variances; both are non-negative and skewed.
• Chebyshev: at most 1 ÷ k² of observations lie more than k standard deviations from the mean, for any distribution with finite variance and k > 1.

Common mistakes

• Saying 'accept H0' when the statistic is not significant Fix: Always write 'fail to reject H0'. The test cannot prove H0 true.
• Using the two-tailed critical value for a one-tailed test Fix: Read H1 first. If it has > or <, use the one-tailed value (1.645 at 5%).
• Using the standard deviation instead of the standard error in the interval. Fix: Always divide by √n first. The interval for the mean uses SE.
• Forgetting to take the square root of n. Fix: Write SE = s ÷ √n on paper before substituting numbers.
• Using z when the sample standard deviation is used and n is small Fix: If σ comes from the sample, use t with the correct degrees of freedom. Use z only when σ is stated as known.
• Dividing by s instead of s ÷ √n Fix: Always compute the standard error first. Testing a mean needs s ÷ √n.
• Saying the p-value is the probability that H0 is true. Fix: The p-value is calculated assuming H0 is true. It is the probability of data this extreme, not the probability of H0.
• Swapping Type I and Type II errors. Fix: Type I = reject a true null (false positive, α). Type II = fail to reject a false null (false negative, β).
• Using standard deviations instead of variances in the statistic. Fix: Square every standard deviation before computing χ² or F.
• Using n degrees of freedom instead of n − 1. Fix: Always write df = n − 1 before looking at the table.

Exam tips

• Read the wording for direction. 'Different from' is two-tailed. 'Exceeds' or 'is less than' is one-tailed.
• Know the z values 1.645, 1.96, 2.33 and 2.58 by heart. Questions often provide t tables, but z values save time.
• Wrong options are often built from the wrong tail count or from 'accept H0'. Check both before choosing.
• If a question gives a confidence interval, you can test a two-tailed hypothesis by checking whether the hypothesized value lies inside it.
• Questions may ask for the effect of changing α. A smaller α makes the critical value larger and rejection harder.
• Check whether the question gives a population or sample standard deviation. It decides z or t.
• Many questions test direction, not arithmetic: a higher confidence level widens the interval, a larger n narrows it.
• Use the interval to answer a hypothesis test question. If the hypothesised mean is outside, reject.