Explanation
Algebra / Rational Expressions / Percent / Proportions and Percents
From the statement, we know that:
• a sample of 726 suspected criminals is drawn,
,
• and from these people, 232 were captured.
Part 1
We have to estimate the proportion of people who were caught. We compute the quotient between 232 and 726, and we get:
[tex]p=\frac{232}{726}\cong0.3195592\cong0.320.[/tex]
Part 2
We must construct the 98% confidence interval for the population proportion p.
1) First, we compute the coefficient α of the confidence interval:
[tex]\begin{gathered} α=1-98\%=1-0.98=0.02, \\ α/2=0.02/2=0.01. \end{gathered}[/tex]
2) From a table of z-scores, the z-score for α/2 is:
[tex]Z_{α/2}=Z_{0.01}=2.326.[/tex]
3) The margin of error ε of the confidence interval p - ε < p < p + ε, is given by:
[tex]ε=Z_{α/2}\cdot\sqrt{\frac{p\cdot(1-p)}{n}}[/tex]
Where n = 726 is the size of the sample.
Replacing the data obtained in the previous steps, we get:
[tex]ε=2.326\cdot\sqrt{\frac{\frac{232}{726}\cdot(1-\frac{232}{726})}{726}}\cong0.040254\cong0.04.[/tex]
The confidence interval is:
[tex]\begin{gathered} p-ε
The lower limit of the 98% confidence interval is 0.316.
Answer
Part 1
• As a fraction: ,232/726
,
• In decimal form: ,0.320
Part 2
• Lower endpoint: ,0.316
,
• Upper endpoint: ,0.324
