Understanding confidence intervals is integral to the field of statistics, particularly when it comes to estimating population means. These intervals are not just a measure of where a mean might lie but are a cornerstone of statistical inference, providing insights into the reliability of an estimate.

## Confidence Intervals

Confidence intervals (CI) provide a range where we expect the true population parameter, like the mean, to be, given a certain confidence level. This is a way to measure how certain we are about an estimate.

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### Key Points:

**Confidence Interval:**A range expected to contain the true population parameter.**Components:**Lower limit, upper limit, confidence level (e.g., 95%).**Use:**Helps quantify uncertainty in estimates.

### When Population Variance is Known:

**Formula:**$CI = \bar{x} \pm z \times \frac{\sigma}{\sqrt{n}}$**Variables:**$\bar{x}$ = sample mean, $z$= z-score for confidence level, $\sigma$ = population standard deviation, $n$ = sample size.

## Examples

### Problem 1: Known Population Standard Deviation

**Given:**

- Sample Mean $(\bar{x}): 68$
- Population Standard Deviation $(\sigma): 15$
- Sample Size $(n): 50$
- Confidence Level: 95% (with $z$≈ 1.96)

**Objective:**

Find the 95% confidence interval for the average.

#### Solution:

1. **Calculate Standard Error (SE):**

- Formula: $SE = \frac{\sigma}{\sqrt{n}}$
- Calculation: $SE = \frac{15}{\sqrt{50}} \approx 2.12$

2. **Determine Margin of Error (ME):**

- Formula: $ME = z \times SE$
- Calculation: $ME = 1.96 \times 2.12 \approx 4.16$

3. **Construct the Confidence Interval (CI):**

- Lower Limit: $\bar{x} - ME = 68 - 4.16 = 63.84$
- Upper Limit: $\bar{x} + ME = 68 + 4.16 = 72.16$

Thus, the 95% confidence interval for the population mean ranges from approximately $63.84$ to $72.16$.

### Problem 2: Unknown Population Variance, Large Sample Size

**Given Data:**

- Sample Mean $(\bar{x}): 170 cm$
- Sample Standard Deviation $(s): 20 cm$
- Sample Size $(n): 100$
- Confidence Level: 95% (Z-score ≈ 1.96)

**Objective: **

Determine the 95% confidence interval for the population mean height.

#### Solution:

1. **Calculate the Standard Error (SE):** $SE = \frac{s}{\sqrt{n}} = \frac{20}{\sqrt{100}} = 2 \text{ cm}$

2. **Calculate the Margin of Error (ME):** $ME = Z \times SE = 1.96 \times 2 = 3.92 \text{ cm}$

3. **Construct the Confidence Interval (CI):** $CI = \bar{x} \pm ME = 170 \pm 3.92$

$\therefore CI = [166.08 \text{ cm}, 173.92 \text{ cm}]$

Thus, the 95% confident that the true population mean height is between 166.08 cm and 173.92 cm.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.