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.

Image courtesy of inchcalculator

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.