FRM Exam Part I · Hypothesis Testing

t-Test and z-Test for Means and Regression Coefficients

Updated 11 October 2026 · Fact-checked

A t-test or z-test checks whether a sample mean or regression coefficient differs from a hypothesized value. Compute (estimate − hypothesized value) ÷ standard error. Compare it with a critical value. Use t when the population variance is unknown and estimated from the sample, with degrees of freedom n − 1 for a mean, or n − k − 1 in regression.

Understand t-Test and z-Test for Means and Regression Coefficients

A hypothesis test asks whether the gap between what you measured and what you assumed is too big to blame on sampling luck. You measure the gap in units of standard error. That ratio is the test statistic.

For a mean, the statistic is (sample mean − hypothesized mean) ÷ standard error, where the standard error is the standard deviation divided by √n. If the statistic lands far out in the tail, you reject the null hypothesis.

The z-test uses the standard normal distribution. It applies when the population standard deviation is known, or when the sample is large and the normal approximation is accepted. The t-test uses Student's t distribution. It applies when you estimate the standard deviation from the sample. The t distribution has fatter tails, because estimating the variance adds uncertainty. Its shape depends on the degrees of freedom. As degrees of freedom rise, t approaches z.

For a mean, degrees of freedom = n − 1. You lose one because the sample mean is estimated from the data. In a regression with k slope coefficients, degrees of freedom = n − k − 1, because k + 1 parameters are estimated, including the intercept.

For a regression coefficient, the logic is the same. The t-statistic is (estimated coefficient − hypothesized value) ÷ standard error of the coefficient. The usual null is that the true coefficient equals zero, meaning the variable has no linear effect. You can also build a confidence interval: estimate ± critical value × standard error. If the interval excludes the hypothesized value, you reject at that confidence level.

Key formulas to remember

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.

How to solve t-Test and z-Test for Means and Regression Coefficients questions

Use this sequence for any t-test or z-test question on a mean or a regression coefficient.

1State H₀ and H₁. Decide whether the test is two-tailed (≠) or one-tailed (> or <).
2Identify the estimate, the hypothesized value and the standard error. If only the standard deviation is given for a mean, compute SE = s ÷ √n.
3Choose the distribution. Population σ known: z. σ estimated from the sample, or a regression coefficient: t.
4Compute the statistic: (estimate − hypothesized value) ÷ standard error.
5Find the degrees of freedom: n − 1 for a mean, n − k − 1 for a regression coefficient, n₁ + n₂ − 2 for two pooled means.
6Pick the critical value for the stated significance level and tail type.
7Compare and conclude: reject H₀ if the absolute statistic exceeds the critical value. State the conclusion in words about the variable or mean.
8Sanity check: a statistic near 2 in absolute value is borderline at 5%. A confidence interval excluding the hypothesized value should agree with rejection.

Quickest way: Rule-of-2 shortcut

When to use it: Use when the question gives a coefficient and its standard error and you need a fast significance call at the 5% level with a reasonably large sample.

Divide the coefficient by its standard error.
If the absolute ratio is clearly above about 2, the coefficient is significant at 5% two-tailed. If it is clearly below, it is not.
If the ratio is close to the cutoff, or the sample is small, look up the exact t critical value with the right degrees of freedom.
Check the options: if the answer choices turn on exact degrees of freedom, do not rely on 1.96.

Common mistakes in t-Test and z-Test for Means and Regression Coefficients

Using z when the sample standard deviation is used and n is small
Students remember that z is simpler and forget that estimating σ changes the distribution.
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
The standard deviation is given in the question, so it gets plugged in directly.
Fix: Always compute the standard error first. Testing a mean needs s ÷ √n.
Wrong degrees of freedom in regression
Students use n − 1 from the mean case.
Fix: Use n − k − 1, with k the number of slope coefficients. A simple regression has n − 2.
Using a two-tailed critical value for a one-tailed test, or the reverse
The alternative hypothesis is not read carefully.
Fix: Look at H₁. A directional claim (greater than, less than) is one-tailed. A claim of difference is two-tailed.
Testing against zero when the null is a different value
Zero is the default null in regression output, so students apply it automatically.
Fix: Subtract the hypothesized value from the estimate. For a null that beta equals 1, use (β̂ − 1) ÷ SE.
Saying that failing to reject proves the null is true
The wording of the conclusion is loose.
Fix: Say there is insufficient evidence to reject H₀. The test does not prove it.

Worked examples

Example 1

A risk analyst tests whether the mean daily return of a fund is zero. Over 25 days, the sample mean is 0.12% and the sample standard deviation is 0.50%. Compute the t-statistic and decide at the 5% level (two-tailed). The critical t value with 24 degrees of freedom is 2.064.

Show the solution

H₀: μ = 0. H₁: μ ≠ 0.
Standard error = 0.50% ÷ √25 = 0.50% ÷ 5 = 0.10%.
t = (0.12% − 0) ÷ 0.10% = 1.20.
Degrees of freedom = 25 − 1 = 24, critical value = 2.064.
|1.20| < 2.064, so do not reject H₀.

Answer: t = 1.20. Do not reject H₀. There is not enough evidence at 5% that the mean daily return differs from zero.

Example 2

A regression of a stock's excess returns on market excess returns uses 32 monthly observations. The estimated beta is 1.30 with a standard error of 0.12. Test H₀: β = 1 against H₁: β ≠ 1 at the 5% level. The critical t value with 30 degrees of freedom is 2.042.

Show the solution

Simple regression, so k = 1. Degrees of freedom = 32 − 1 − 1 = 30.
The null value is 1, not zero.
t = (1.30 − 1) ÷ 0.12 = 0.30 ÷ 0.12 = 2.50.
Critical value = 2.042.
|2.50| > 2.042, so reject H₀.
Check with an interval: 1.30 ± 2.042 × 0.12 = 1.30 ± 0.245 = (1.055, 1.545). It excludes 1, which agrees.

Answer: t = 2.50, which exceeds 2.042. Reject H₀. The beta is significantly different from 1 at the 5% level.

Exam tips

Read whether the standard deviation is for the population or the sample. That decides z or t.
Check the null value before computing. Questions about beta often test against 1, not 0.
Compute degrees of freedom from the number of slope variables. Multiple regression questions often change k.
Use a confidence interval as a cross-check. If it excludes the null value, you reject.
Show the sign. A negative statistic in a one-tailed test for a lower tail compares with the negative critical value.

Practice questions from Hypothesis Testing

t-Test and z-Test for Means and Regression Coefficients in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

t-Test and z-Test for Means and Regression Coefficients: frequently asked questions

When do I use a t-test instead of a z-test?

Use a t-test when the population standard deviation is unknown and you estimate it from the sample. This is the usual case for sample means and all regression coefficients. Use a z-test when the population standard deviation is given as known.

How do I test the significance of a regression coefficient?

Divide the estimated coefficient by its standard error to get the t-statistic, with n − k − 1 degrees of freedom. Compare its absolute value with the critical t value for your significance level. If it is larger, reject the null that the coefficient is zero.

How are degrees of freedom calculated?

For a single mean, degrees of freedom are n − 1. For a regression with k slope coefficients, they are n − k − 1. For two independent means with pooled variance, they are n₁ + n₂ − 2.

Does the t distribution ever equal the normal distribution?

It approaches the normal as degrees of freedom grow large. With many observations, the t critical value is close to 1.96 at 5% two-tailed. With small samples, t is noticeably larger, so the test is more cautious.