Question

5. Suppose that a random sample of 25 mining caps was tested from a population of 300 mining caps and 20 exploded properly. Construct a 95% confidence interval for the proportion of mining caps that will explode properly.

Answer 1

$\text{The confidence interval for the }\\ \text{proportion can be calculated as follows:}\\ \hat p ±Z_{\frac{α}{2}}(\sqrt {\frac{\hat p (1-\hat p )}{n}}),\\ \hat p =\frac{20}{25}=0.8,\\ Z_{\frac{α}{2}}=Z_{\frac{0.05}{2}}=Z_{0.025}=1.96,\\ \text{the lower limit }= 0.8-1.96(\sqrt {\frac{0.8 (1-0.8 )}{25}})\\ ≈0.6432,\\ \text{the upper limit }= 0.8+1.96(\sqrt {\frac{0.8 (1-0.8 )}{25}})\\ ≈0.9568,\\ \text{so the confidence interval is}\\ 0.6432\leq \ p \leq 0.9568$