Biostadistic

Sample Answer

The 95% confidence interval for the average height of the population is 1.7108 m to 1.7892 m.

This means that we are 95% confident that the true average height of the population is between 1.7108 meters and 1.7892 meters.

Calculation Details

1. Identify the Parameters

Sample Mean (xˉ): 1.75 m

Sample Size (n): 400

Population Variance (σ2): 0.16

Population Standard Deviation (σ): 0.16​=0.4 m

Confidence Level: 95%

z-score (z): ≈1.96 (for a 95% confidence level)

Calculate the Standard Error of the Mean (SEM)

The standard error of the mean is calculated as:

$\text{SEM} = \frac{\sigma}{\sqrt{n}}$$\text{SEM} = \frac{0.4}{\sqrt{400}} = \frac{0.4}{20} = 0.02 \text{ m}$

3. Calculate the Margin of Error (ME)

The margin of error is calculated as the $z$-score multiplied by the standard error:

$\text{ME} = z \cdot \text{SEM}$$\text{ME} \approx 1.96 \cdot 0.02 = 0.0392 \text{ m}$

4. Determine the Confidence Interval

The confidence interval is calculated using the formula:

$\left( \bar{x} - \text{ME}, \bar{x} + \text{ME} \right)$

Lower Bound:

$1.75 - 0.0392 = 1.7108 \text{ m}$

Upper Bound:

$1.75 + 0.0392 = 1.7892 \text{ m}$