The amount of weight a high school student carries in his or her bookbag follows a normal distribution with a standard deviation of σ = 2 pounds. Suppose a random sample of 20 bookbags produced a mean of 13 pounds. Construct a 95% confidence interval to estimate the mean bookbag weight for all high school students.

Answer:
Given:

Sample Mean $ (x\bar)=13 $

Population Standard Deviation $ (\sigma)=2 $

Sample Size $ (n) = 20 $

Confidence Interval Level $(CI) = 95% $

Solution:

The level of significance $(\alpha):$

$ \alpha = 1- 0.95 = 0.05 $

The critical value $ (Z_c): $

$ Z_c = Z_{\alpha/2} = Z_{0.05/2} = 1.96 $

The confidence interval $(CI): $

$\text{CI} = \bar{x} \pm Z_c \cdot \frac{\sigma}{\sqrt{n}} $

$ = 13 \pm 1.96 \cdot \frac{2}{\sqrt{20}} $

$ = (12.118, 13.882) $

Final Answer:

The 95% confidence interval $ = (12.118,13.882) $