FRM Exam Part I · Linear Regression

Hypothesis Testing and Confidence Intervals for Regression Coefficients

Q: What is the difference between a p-value and a t-statistic in regression?

The t-statistic measures how many standard errors the estimate lies from the null value. The p-value converts that statistic into a probability of getting a result at least that extreme if the null is true. A larger absolute t gives a smaller p-value.

Q: How do I construct a confidence interval for a regression slope?

Take the estimated slope and add and subtract the critical t value times the slope's standard error. Use n − k − 1 degrees of freedom to find the critical value. For large samples at 95%, the critical value is about 1.96.

Q: What is the formula for the standard error of a regression coefficient?

In simple regression, SE(β̂₁) = s ÷ √Σ(Xᵢ − X̄)², where s = √(SSR ÷ (n − 2)). The exam usually gives you the standard error directly. Know that it falls when residual variation is smaller or when X is more spread out.

Q: When should I use the normal critical value instead of t?

Use the t distribution when degrees of freedom are small or when the question gives a t table. With a large sample the two are close, and 1.96 is a fair approximation at 95% two-sided. Follow what the question states.

Updated 11 October 2026 · Fact-checked

A regression coefficient test asks whether a slope differs from a hypothesized value, usually zero. Compute t = (estimate − hypothesized value) ÷ standard error, then compare it with the critical t value with n − k − 1 degrees of freedom. A confidence interval is estimate ± critical t × standard error.

Understand Hypothesis Testing and Confidence Intervals for Coefficients

An OLS slope is calculated from a sample, so it changes from sample to sample. It is an estimate of the true slope, not the slope itself. The standard error measures how much the estimate would typically vary across samples.

Hypothesis testing asks whether the data are consistent with a stated value of the true slope. The null hypothesis is usually H0: β = 0, meaning the variable has no linear effect. The alternative is H1: β ≠ 0 for a two-sided test. You measure how many standard errors the estimate lies from the hypothesized value. That count is the t-statistic.

If the t-statistic is large in absolute value, the estimate is far from the null value relative to its noise, so you reject H0. The cutoff is the critical value from the Student's t distribution. The degrees of freedom are n − k − 1, where k is the number of slope coefficients. With a simple regression, that is n − 2. For large samples, the two-sided 5% critical value is close to 1.96.

The p-value is the smallest significance level at which you could reject H0. It is the probability, assuming H0 is true, of getting a t-statistic at least as extreme as the one observed. Reject H0 when the p-value is below your significance level. The t-statistic and the p-value carry the same information in different forms.

A confidence interval gives a range of plausible values for the true slope. A 95% interval is the estimate ± critical t × standard error. If the interval excludes the hypothesized value, a two-sided test at 5% rejects it. If it includes it, you fail to reject. Note that failing to reject does not prove the null is true.

Key formulas to remember

t-statistic for a coefficient: t = (β̂ − β₀) ÷ SE(β̂); β₀ is the hypothesized value, often 0. Degrees of freedom = n − k − 1.
Confidence interval for a slope: β̂ ± t_c × SE(β̂); t_c is the two-sided critical value for the chosen confidence level, with n − k − 1 degrees of freedom.
Standard error of slope (simple regression): SE(β̂₁) = s ÷ √Σ(Xᵢ − X̄)², where s = √(SSR ÷ (n − 2)); s is the standard error of the regression. A larger spread in X gives a smaller SE.
Decision rule (two-sided): Reject H0 if |t| > t_c, or if p-value < α; Both rules give the same decision.
Confidence interval and test link: Reject H0: β = β₀ at level α if β₀ lies outside the (1 − α) confidence interval; Applies to two-sided tests.

How to solve Hypothesis Testing and Confidence Intervals for Coefficients questions

Use this sequence for any question on testing or building intervals for a coefficient.

1Write H0 and H1. The null is usually β = 0. Note whether the test is one-sided or two-sided.
2Identify the estimate β̂, its standard error, the sample size n and the number of slopes k.
3Compute degrees of freedom: n − k − 1.
4Compute t = (β̂ − β₀) ÷ SE. Keep the sign.
5Find the critical value for the stated significance level and degrees of freedom. Use 1.96 or 2.58 only for large samples or when the question says to use the normal.
6Compare |t| with the critical value, or the p-value with α, and state reject or fail to reject.
7For an interval, compute β̂ ± t_c × SE and check whether it contains the hypothesized value.
8State the conclusion in words about the variable, not just the statistic.

Quickest way: Rule-of-two shortcut

When to use it: When the sample is reasonably large and the question asks only whether a coefficient is significant at about 5%.

Divide the estimate by its standard error.
If the absolute ratio exceeds about 2, it is significant at roughly 5% two-sided.
For an approximate 95% interval, use estimate ± 2 × SE.
If the sample is small, look up the t value for the degrees of freedom instead, because it will be larger than 2.
Check the answer options: if one is much closer to the exact value, compute exactly.

Common mistakes in Hypothesis Testing and Confidence Intervals for Coefficients

Using n − 1 or n as degrees of freedom
The rule for a sample mean is n − 1 and gets applied by habit.
Fix: For regression use n − k − 1. In simple regression this is n − 2.
Dividing by the estimate instead of its standard error, or forgetting to subtract a nonzero hypothesized value
Most examples use H0: β = 0, so the subtraction is skipped.
Fix: Always write (β̂ − β₀) ÷ SE. Read the null value from the question.
Interpreting a 95% interval as a 95% probability that the true slope lies in this one interval
The wording sounds like a probability statement about the parameter.
Fix: Say that the method captures the true slope in 95% of repeated samples. The true slope is fixed, and the interval is random.
Treating the p-value as the probability that the null is true
A small p-value feels like evidence for the alternative with a stated probability.
Fix: The p-value is the probability of data this extreme if H0 were true. It is not the probability that H0 is true.
Concluding that failing to reject proves β = 0
Students read 'not significant' as 'no effect'.
Fix: Say there is insufficient evidence to reject. A wide standard error or small sample can hide a real effect.
Using a two-sided critical value for a one-sided test
The tables show two-sided values by default.
Fix: For a one-sided test at α, use the critical value with α in one tail. For example, one-sided 5% matches two-sided 10%.

Worked examples

Example 1

A regression of a fund's excess returns on market excess returns uses 32 monthly observations. The estimated beta is 1.15 with a standard error of 0.10. Test H0: β = 1 against H1: β ≠ 1 at the 5% level. The two-sided 5% critical t value with 30 degrees of freedom is 2.042.

Show the solution

Degrees of freedom = n − 2 = 32 − 2 = 30.
t = (1.15 − 1) ÷ 0.10 = 0.15 ÷ 0.10 = 1.5.
Critical value = 2.042.
|1.5| < 2.042, so the statistic lies inside the acceptance region.

Answer: Fail to reject H0. There is not enough evidence at the 5% level that beta differs from 1.

Example 2

In a simple regression with 27 observations, the slope estimate is 0.80 and its standard error is 0.25. Construct a 95% confidence interval for the slope, given the two-sided critical t value with 25 degrees of freedom is 2.060. Does the interval support rejecting H0: β = 0?

Show the solution

Degrees of freedom = 27 − 2 = 25, so t_c = 2.060.
Margin = 2.060 × 0.25 = 0.515.
Lower bound = 0.80 − 0.515 = 0.285.
Upper bound = 0.80 + 0.515 = 1.315.
Zero is not inside (0.285, 1.315).
Check: t = 0.80 ÷ 0.25 = 3.2, which exceeds 2.060, so the test agrees.

Answer: The 95% interval is (0.285, 1.315). It excludes 0, so reject H0: β = 0 at the 5% level.

Exam tips

Check what the question gives: if you are given a t-statistic or p-value, do not recompute from scratch.
Watch the null value. Questions often test H0: β = 1 for a beta, not zero.
Match the interval to the test. If a two-sided interval excludes the null value, the matching test rejects.
Read the confidence level and tail carefully. Two-sided 95% and one-sided 5% use different critical values.
Eliminate options with wrong degrees of freedom before computing; they often include a distractor built on n − 1.

Practice questions from Linear Regression

Hypothesis Testing and Confidence Intervals for Coefficients in other exams

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

Hypothesis Testing and Confidence Intervals for Coefficients: frequently asked questions

What is the difference between a p-value and a t-statistic in regression?

The t-statistic measures how many standard errors the estimate lies from the null value. The p-value converts that statistic into a probability of getting a result at least that extreme if the null is true. A larger absolute t gives a smaller p-value.

How do I construct a confidence interval for a regression slope?

Take the estimated slope and add and subtract the critical t value times the slope's standard error. Use n − k − 1 degrees of freedom to find the critical value. For large samples at 95%, the critical value is about 1.96.

What is the formula for the standard error of a regression coefficient?

In simple regression, SE(β̂₁) = s ÷ √Σ(Xᵢ − X̄)², where s = √(SSR ÷ (n − 2)). The exam usually gives you the standard error directly. Know that it falls when residual variation is smaller or when X is more spread out.

When should I use the normal critical value instead of t?

Use the t distribution when degrees of freedom are small or when the question gives a t table. With a large sample the two are close, and 1.96 is a fair approximation at 95% two-sided. Follow what the question states.