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Actuarial Statistics · Confidence intervals and prediction intervals

Confidence Interval Basics and Pivotal Quantities Explained

Updated 11 October 2026 · Fact-checked

A confidence interval is a random interval, built from sample data, that covers the true parameter with a stated long-run probability, such as 95%. A pivotal quantity is a function of the data and the parameter whose distribution does not depend on the parameter. Invert a probability statement about it to get the interval.

Understand Confidence Interval Basics and Pivotal Quantities

A point estimate such as the sample mean gives one number. It tells you nothing about how far it may be from the true value. A confidence interval gives a range, and the range carries a measure of reliability.

The key idea is that the interval is random and the parameter is not. Before you collect data, the endpoints L(X) and U(X) are random variables. The coverage probability is P(L(X) < θ < U(X)). A 95% confidence interval is built so that this probability is 0.95 for every possible value of θ. Once you have data and compute numbers, the interval is fixed. The parameter is also fixed. So the probability that this particular interval contains θ is either 0 or 1, and you do not know which.

The correct reading is: if you repeated the sampling many times and built an interval each time by the same method, about 95% of those intervals would contain θ. Do not say there is a 95% probability that θ lies in your calculated interval. (A Bayesian credible interval does allow that statement, but it rests on a prior.)

A pivotal quantity is how you build an interval. It is a function g(X, θ) of the sample and the parameter, and its distribution is completely known and free of θ. For a normal sample with known σ, Z = (X̄ − μ) ÷ (σ/√n) ~ N(0,1) is pivotal. For unknown σ, T = (X̄ − μ) ÷ (S/√n) ~ t with n − 1 degrees of freedom is pivotal. For the variance, (n − 1)S² ÷ σ² ~ χ² with n − 1 degrees of freedom is pivotal.

Because its distribution is known, you can find constants a and b with P(a < g(X, θ) < b) = 1 − α. Then you rearrange the inequality to isolate θ. The result is the interval. A related idea is an approximate pivotal quantity, such as a maximum likelihood estimator divided by its standard error for large samples, which is roughly N(0,1).

Key rules to remember

Definition of a 100(1 − α)% confidence interval
P(L(X) < θ < U(X)) = 1 − α
Holds for all θ. The probability is over the random sample, not over θ.
Pivotal quantity
g(X, θ) has a distribution that does not depend on θ
It must contain θ and must be computable once θ is substituted. A statistic with no θ in it is not a pivot.
Probability statement from a pivot
P(a < g(X, θ) < b) = 1 − α
Choose a and b from the known distribution, usually leaving α/2 in each tail.
Normal mean, σ known
X̄ ± z(1 − α/2) × σ ÷ √n
Pivot: (X̄ − μ) ÷ (σ/√n) ~ N(0,1). For 95%, z = 1.96.
Normal mean, σ unknown
X̄ ± t(n−1, 1 − α/2) × S ÷ √n
Pivot: (X̄ − μ) ÷ (S/√n) ~ t(n−1). Needs normal data.
Normal variance
((n − 1)S² ÷ χ²(upper), (n − 1)S² ÷ χ²(lower))
Pivot: (n − 1)S² ÷ σ² ~ χ²(n−1). Upper percentile gives the lower limit. The interval is not symmetric.
Large-sample interval from MLE
θ̂ ± z(1 − α/2) × se(θ̂)
Approximate. Relies on asymptotic normality of the MLE.

How to solve Confidence Interval Basics and Pivotal Quantities questions

Use this method for any question that asks you to construct or interpret a confidence interval.

  1. 1State the model and assumptions: for example, X1,...,Xn are independent N(μ, σ²). Say whether σ is known.
  2. 2Pick a pivotal quantity containing the parameter and with a known distribution, such as Z, T or (n − 1)S² ÷ σ².
  3. 3Write the probability statement P(a < pivot < b) = 1 − α, using α/2 in each tail unless told otherwise.
  4. 4Read a and b from tables for the correct distribution and degrees of freedom.
  5. 5Rearrange the inequalities to isolate the parameter. Remember that inverting a reciprocal reverses the inequality.
  6. 6Substitute the sample values and compute the numerical limits.
  7. 7State the interval and give the correct interpretation in words: the method covers the parameter in 100(1 − α)% of repeated samples.

Quickest way: Match the parameter to the pivot

When to use it: Use in multiple-choice questions and when time is short on a written question with a standard setting.

  1. Identify the parameter: mean or variance, and whether σ is known.
  2. Pick the template: z interval, t interval or chi-square interval.
  3. Find the table value for 1 − α/2 (or both chi-square percentiles) with n − 1 degrees of freedom.
  4. Compute the estimate ± margin, or the two chi-square ratios.
  5. For interpretation options, choose the one that talks about repeated samples, not the probability that θ lies in the one interval.

Common mistakes in Confidence Interval Basics and Pivotal Quantities

  • Saying there is a 95% probability that the true parameter lies in the calculated interval.

    It sounds natural, and the interval looks like a probability statement about θ.

    Fix: Treat θ as fixed. The 95% describes the method over repeated samples. Say: the procedure captures θ in 95% of samples.

  • Using a z value when σ is unknown and the sample is small.

    Students remember 1.96 and skip the check for known σ.

    Fix: If you use S, use the t distribution with n − 1 degrees of freedom. The t value is larger, so the interval is wider.

  • Swapping the chi-square percentiles for a variance interval.

    The pivot has σ² in the denominator, so the order flips.

    Fix: Lower limit = (n − 1)S² ÷ upper percentile. Upper limit = (n − 1)S² ÷ lower percentile. Check that the lower limit is smaller.

  • Calling a statistic a pivot when its distribution depends on the parameter.

    Students confuse a pivot with a sampling distribution in general. X̄ alone is N(μ, σ²/n), which depends on μ.

    Fix: Standardise so the parameter-free distribution appears. Ask: does the distribution of this quantity change with θ?

  • Thinking a wider confidence level gives a narrower interval, or that more data does not matter.

    Mixing up confidence with precision.

    Fix: Higher confidence means a larger critical value and a wider interval. Larger n shrinks the width roughly in proportion to 1/√n.

  • Taking the square root of the variance interval incorrectly or forgetting to do it for the standard deviation.

    Working in σ² and then answering for σ.

    Fix: Take square roots of both limits of the σ² interval to get an interval for σ. Do not take the square root of the estimate and then use the variance formula.

Worked examples

Example 1

A random sample of 25 claim amounts from a normal distribution has mean 4,800 and sample standard deviation 600 (in rupees). Construct a 95% confidence interval for the mean claim. Use t(24, 0.975) = 2.064.

Show the solution
  1. Model: X1,...,X25 are independent N(μ, σ²), σ unknown.
  2. Pivot: T = (X̄ − μ) ÷ (S/√n) ~ t(24).
  3. P(−2.064 < T < 2.064) = 0.95.
  4. Rearranging gives X̄ ± 2.064 × S ÷ √n.
  5. S ÷ √n = 600 ÷ 5 = 120.
  6. Margin = 2.064 × 120 = 247.68.
  7. Limits: 4,800 − 247.68 = 4,552.32 and 4,800 + 247.68 = 5,047.68.

Answer: The 95% confidence interval for μ is (₹4,552.32, ₹5,047.68). If we repeated the sampling many times, about 95% of intervals built this way would contain the true mean.

Example 2

Using the same sample (n = 25, S = 600), find a 95% confidence interval for σ². Use χ²(24) percentiles: 2.5th = 12.40 and 97.5th = 39.36. Then explain what '95%' means.

Show the solution
  1. Pivot: (n − 1)S² ÷ σ² ~ χ²(24).
  2. P(12.40 < 24S² ÷ σ² < 39.36) = 0.95.
  3. Rearranging: 24S² ÷ 39.36 < σ² < 24S² ÷ 12.40.
  4. 24S² = 24 × 360,000 = 8,640,000.
  5. Lower limit = 8,640,000 ÷ 39.36 = 219,512 (to the nearest whole number).
  6. Upper limit = 8,640,000 ÷ 12.40 = 696,774 (to the nearest whole number).
  7. Meaning: the 95% is the coverage probability of the procedure. It is not the probability that σ² lies between these two numbers.

Answer: The 95% confidence interval for σ² is approximately (219,512, 696,774). Taking square roots gives roughly (468.5, 834.7) for σ. The interval is not symmetric about S² because the chi-square distribution is skewed.

Exam tips

  • Write the pivot and its distribution first. Examiners award marks for naming it and for showing it does not depend on the parameter.
  • In interpretation questions, mention repeated sampling and say that the interval is random before the data are observed.
  • State assumptions such as normality and independence. Marks are often lost when they are missing.
  • For variance intervals, check the lower limit is below the upper limit and that both sit around, not on, S² in a skewed way.
  • In the computer-based paper, show the formula and the quantile function you used, such as qt or qchisq, with the degrees of freedom, before the result.

Practice questions from Confidence intervals and prediction intervals

Confidence Interval Basics and Pivotal Quantities in other exams

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

Confidence Interval Basics and Pivotal Quantities: frequently asked questions

What is the difference between confidence level and the probability that the parameter is in the interval?

The confidence level is the long-run proportion of intervals that cover the parameter, over repeated samples. For one calculated interval, the parameter is fixed, so it is either inside or not. In classical statistics you cannot assign it a probability of 95%.

What is a pivotal quantity?

It is a function of the sample and the unknown parameter whose distribution is fully known and does not depend on the parameter. Examples are (X̄ − μ) ÷ (σ/√n) and (n − 1)S² ÷ σ² for normal samples. You use it to write a probability statement and then rearrange it into an interval.

Why is the t interval wider than the z interval?

Using S instead of σ adds extra uncertainty. The t distribution has heavier tails than the normal, so its critical value is larger. As n grows, the t value approaches the z value.

Does a 95% interval contain 95% of the data?

No. A confidence interval is about a parameter such as the mean. It is much narrower than the range of individual observations. An interval for a future observation is a prediction interval, which is wider.