Calculate confidence intervals and prediction intervals

CS1 – Actuarial Statistics · IFoA (Institute and Faculty of Actuaries) · 5 min read

Welcome to the World of Confidence!

In our previous studies, we looked at point estimates—giving a single "best guess" for a value (like saying, "I think the average claim size is $500"). But in the real world, especially in actuarial work, we know that samples are never perfect. We need a "safety net" to account for uncertainty. That is exactly what Confidence Intervals (CI) and Prediction Intervals (PI) provide.

In this chapter, we will learn how to build these intervals so we can say things like, "We are 95% confident the true average claim lies between $450 and $550." Let’s dive in!

1. What exactly is a Confidence Interval?

Imagine you are trying to guess the weight of a cake. A point estimate is a single guess (e.g., 1kg). A confidence interval is a range (e.g., 0.8kg to 1.2kg).

Key Definition: A Confidence Interval is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter (like the mean $\mu$ or variance $\sigma^2$).

The Confidence Level, denoted as $1 - \alpha$, tells us how "sure" we are. Usually, we use 95% ($\alpha = 0.05$) or 99% ($\alpha = 0.01$).

The "Pivotal Quantity" Method

Don’t let the name scare you! A pivotal quantity is just a formula that links our sample data to the population parameter we want to find, but its own distribution doesn't depend on that parameter. We use this "pivot" to build our interval.

Quick Review:
- If we want a 95% interval, we leave 2.5% in the "left tail" and 2.5% in the "right tail" of our distribution.
- The value $\alpha$ is the "error probability" (the chance we are wrong).

2. Confidence Interval for the Mean ($\mu$)

This is the most common task in CS1. We want to estimate the average of a population. There are two main scenarios based on what we know about the variance ($\sigma^2$).

Scenario A: Variance ($\sigma^2$) is Known

If we know the population variance, we use the Standard Normal Distribution ($Z$).

The formula for the $1 - \alpha$ confidence interval is:
$\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

Where:
- $\bar{x}$ is the sample mean.
- $z_{\alpha/2}$ is the value from the Normal tables (e.g., 1.96 for 95%).
- $\frac{\sigma}{\sqrt{n}}$ is the Standard Error.

Scenario B: Variance ($\sigma^2$) is Unknown

This is more realistic. If we don't know $\sigma^2$, we estimate it using the sample variance $s^2$. Because we are estimating the variance, we introduce more uncertainty, so we use the t-distribution instead of the $Z$ distribution.

The formula is:
$\bar{x} \pm t_{n-1, \alpha/2} \frac{s}{\sqrt{n}}$

Important Tip: Always remember that the degrees of freedom for the t-distribution here is $n - 1$. If you forget to subtract 1, your interval will be slightly wrong!

Did you know?
As the sample size $n$ gets very large (usually $n > 100$), the t-distribution starts looking exactly like the Normal distribution. This is the Central Limit Theorem in action!

3. Confidence Interval for a Proportion ($p$)

Actuaries often estimate proportions, such as "What percentage of policyholders will make a claim?"

For large samples, we use the Normal approximation. If $\hat{p}$ is our sample proportion:
$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Common Mistake to Avoid: Ensure you use the sample proportion $\hat{p}$ in the square root (the standard error part), not the theoretical $p$, because you don't know the theoretical $p$ yet!

4. Confidence Interval for the Variance ($\sigma^2$)

Sometimes we care more about the risk (spread) than the average. To estimate variance, we use the Chi-squared ($\chi^2$) distribution.

The pivot we use is: $\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}$

The interval formula looks a bit different because the Chi-squared distribution is not symmetric:
$\left[ \frac{(n-1)s^2}{\chi^2_{n-1, \alpha/2}}, \frac{(n-1)s^2}{\chi^2_{n-1, 1-\alpha/2}} \right]$

Memory Aid: Notice the "flip"! The larger Chi-squared value goes in the denominator of the lower limit to make the total value smaller.

5. Difference Between Two Populations

In CS1, you might be asked to compare two groups (e.g., "Is the mean claim size for smokers higher than for non-smokers?").

Difference in Means ($\mu_1 - \mu_2$)

If variances are unknown but assumed equal, we use a pooled estimate of variance ($s_p^2$):
$(\bar{x}_1 - \bar{x}_2) \pm t_{n_1+n_2-2, \alpha/2} \cdot s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$

Don't worry if this seems complex! Just think of it as taking the difference of the means and adjusting for the combined "noise" (variance) of both samples.

6. Prediction Intervals: Predicting the Future

This is a common point of confusion.
- A Confidence Interval is for a parameter (like the average $\mu$).
- A Prediction Interval is for a single future observation (let's call it $X_{new}$).

Analogy:
Think of a classroom of students.
1. Estimating the average height of all future students is easier (Confidence Interval).
2. Estimating the height of the very next student who walks through the door is much harder (Prediction Interval).

Because predicting one specific individual is harder than predicting an average, the Prediction Interval is always wider than the Confidence Interval.

The Formula (Normal case, $\sigma^2$ known):
$\bar{x} \pm z_{\alpha/2} \cdot \sigma \sqrt{1 + \frac{1}{n}}$

Notice the "$1 +$" inside the square root? That extra "1" represents the individual variability of that one future person. That’s what makes the interval wider!

7. Summary and Key Takeaways

Key Point Review:
- Use Z-distribution when variance is known or for large-sample proportions.
- Use t-distribution when variance is unknown (don't forget $n-1$ degrees of freedom).
- Use Chi-squared for intervals involving variance ($\sigma^2$).
- Prediction Intervals are for individual future values and are wider than Confidence Intervals.
- If the confidence level increases (e.g., from 95% to 99%), the interval gets wider (you need a bigger net to be more sure).

Common Exam Trap: If an exam question asks you to interpret the interval, never say "There is a 95% probability that $\mu$ is in this specific interval." Instead, say "95% of such intervals calculated from repeated samples will contain the true value of $\mu$." It’s a subtle but important distinction in Frequentist statistics!

You've got this! Practice identifying which distribution to use based on the information given in the question, and the calculations will become second nature.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the definition of a Confidence Interval (CI)?

A Confidence Interval is a range of values, calculated from sample data, that is likely to contain the value of an unknown population parameter.

What is a "pivotal quantity" in the context of constructing intervals?

A pivotal quantity is a function of the sample data and the unknown parameter whose probability distribution does not depend on that parameter.

What is the formula for a confidence interval for the mean when the population variance ($\sigma^2$) is known?

$\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

Which distribution is used for the confidence interval of the mean when the population variance is unknown?

The Student's $t$-distribution with $n - 1$ degrees of freedom.

How is a confidence interval for a population proportion ($p$) calculated for large samples?

$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

What distribution is used to find the confidence interval for the population variance ($\sigma^2$)?

The Chi-squared ($\chi^2$) distribution with $n - 1$ degrees of freedom.

What is the degrees of freedom for a pooled-variance t-interval for the difference between two means ($\mu_1 - \mu_2$)?

The degrees of freedom is $n_1 + n_2 - 2$.

What is the primary purpose of a Prediction Interval (PI)?

A Prediction Interval is used to estimate the range within which a single future observation ($X_{new}$) is expected to fall.

Why does the Prediction Interval formula contain an extra "1" inside the standard error term?

The "1" represents the inherent individual variability of the single future observation, which exists even if the population mean were perfectly known.

What is the correct frequentist interpretation of a 95% confidence interval?

95% of all intervals calculated using this method from repeated samples will contain the true population parameter.

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Calculate confidence intervals and prediction intervals

Welcome to the World of Confidence!

1. What exactly is a Confidence Interval?

The "Pivotal Quantity" Method

2. Confidence Interval for the Mean (\(\mu\))

Scenario A: Variance (\(\sigma^2\)) is Known

Scenario B: Variance (\(\sigma^2\)) is Unknown

3. Confidence Interval for a Proportion (\(p\))

4. Confidence Interval for the Variance (\(\sigma^2\))