# Confidence Intervals and Estimation

Mathematics I · Data Analysis and Probability · https://tryals.app/learn/mathematics-i/confidence-intervals-and-estimation

## Point to Interval Estimation

A single sample number (**point estimate**, like $\bar{x}$) almost never hits the true population parameter exactly. A **confidence interval** gives a range of plausible values paired with a **confidence level**, honestly capturing our uncertainty.

The general formula is:
$$\bar{x} \pm (\text{critical value}) \times \text{standard error}$$

This equals your point estimate plus or minus a **margin of error**. The margin multiplies a **critical value** (set by your confidence level, like $1.96$ for 95%) by the **standard error** $\frac{\sigma}{\sqrt{n}}$.

| Lever | Change | Effect on Interval | Why |
| --- | --- | --- | --- |
| **Confidence** | Higher (99% vs 95%) | **Wider** | Bigger net catches more |
| **Sample Size** | Larger ($n$) | **Narrower** | More data adds precision |

## Decoding Confidence

**Common Pitfall:** Thinking a 95% interval means a "95% probability the true value is in this specific interval." The true parameter is fixed, and your calculated interval either contains it (1) or doesn't (0). Probability doesn't apply after the interval is built.

Instead, the **confidence level** describes the **procedure**: if you draw repeated samples and build an interval each time, about 95% of those intervals will contain the true parameter. Confidence belongs to the method, not the single interval.

$$\bar{x} \pm z \cdot \frac{\sigma}{\sqrt{n}}$$

## Practice questions

5 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Complete the correct reading of a confidence level.

**Answer:** A confidence level of 95% is a property of **the procedure that generates the intervals**.

**Why:** The parameter is fixed; the interval is what changes from sample to sample. So 95% describes the long-run capture rate of the method, not a probability attached to the one interval you happen to have computed.

Page: https://tryals.app/practice/mathematics-i/confidence-intervals-and-estimation/complete-the-correct-reading-of-a-confidence-level

### 2. A 95% confidence interval is described as having a 95% chance of containing the true value. Why does a statistician object, and what is the correct reading?

A. Because the probability changes as more data is collected
B. Because confidence intervals describe the sample mean rather than the population mean
C. Because the parameter is fixed; the 95% describes the procedure
D. Because the correct figure is 90% once estimation error is accounted for

**Answer:** C. Because the parameter is fixed; the 95% describes the procedure

**Why:** Nothing is random once the interval exists, the sampling was random, so the guarantee attaches to the method rather than to this interval. The reading people want is available from a Bayesian credible interval, at the price of specifying a prior, which is exactly what the frequentist approach declines to do.

Page: https://tryals.app/practice/mathematics-i/confidence-intervals-and-estimation/a-95-confidence-interval-is-described-as-having-a-95-chance-of

### 3. A 95% confidence interval for a mean is often confused with a range containing 95% of the data. What is the difference?

A. The confidence interval is always the wider of the two
B. The interval estimates where the MEAN lies, and shrinks as n grows
C. They are the same interval computed by two different formulas
D. The prediction interval applies only to normal data

**Answer:** B. The interval estimates where the MEAN lies, and shrinks as n grows

**Why:** Averaging pins down a mean and does nothing to reduce individual spread. A confidence interval can be a millimetre wide while individual heights still vary by 30 cm.

Page: https://tryals.app/practice/mathematics-i/confidence-intervals-and-estimation/a-95-confidence-interval-for-a-mean-is-often-confused-with-a-range

### 4. Holding the confidence level fixed, by what factor must the sample size be multiplied to halve the margin of error?

**Answer:** 4

**Why:** Halving the margin needs root n doubled, so n must be quadrupled. This square-root law is why polls quote margins of about 3 points at n = 1000: getting to 1.5 points would demand 4000 respondents.

Page: https://tryals.app/practice/mathematics-i/confidence-intervals-and-estimation/holding-the-confidence-level-fixed-by-what-factor-must-the-sample

### 5. Sort each reading of a 95% confidence interval for the mean.

**Answer:**

- Correct interpretation: Repeat the study many times and 95 intervals in 100 would hold the true mean, This interval either contains the true mean or it does not
- Incorrect interpretation: There is a 95% chance the true mean is inside this interval, 95% of the data values lie inside the interval

**Why:** The confidence level describes the method's long-run success rate. Item 3 confuses an interval for the mean with an interval for the data, since the interval is only a few standard errors wide, it typically contains far less than 95% of the individual values.

Page: https://tryals.app/practice/mathematics-i/confidence-intervals-and-estimation/sort-each-reading-of-a-95-confidence-interval-for-the-mean