Confidence intreval for population proportion is written as
sample proportion ± margin of error
The formula for calculating margin of error for a population proportion is expressed as
![\text{margin of error = z}*\sqrt[]{(pq)/(n)}](https://img.qammunity.org/2023/formulas/mathematics/college/an0nkacjavcxu12xtvut7jre8bysv9ux31.png)
where
z is the z score of the confidence level
p = population proportion
q = 1 = p
From the information given,
p = 0.98
q = 1 - 0.92 = 0.08
n = 250
a) For a 90% confidence interval, z = 1.645
By substituting these values into the formula,
![\begin{gathered} \text{margin of error = 1.645}*\sqrt[]{(0.92*0.08)/(250)} \\ \text{margin of error = 0.02}8 \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/o3kzpvbtln63tojuuir8inb11cz7adjyi6.png)
Thus,
The 90% confidence interval for p is
0.92 ± 0.028
it is from
0.892 to 0.948
b)For a 95% confidence interval, z = 1.96
By substituting these values into the formula
![\begin{gathered} \text{margin of error = 1.96}*\sqrt[]{(0.92*0.08)/(250)} \\ \text{margin of error = 0.034} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/wfm3s7brd9fm1ijz962hvyesqkrilts6ww.png)
Thus,
The 95% confidence interval for p is
0.92 ± 0.034
It is from
0.886 to 0.954
c) For a 99% confidence interval, z = 2.576
By substituting these values into the formula
![\begin{gathered} \text{margin of error = 2.576}*\sqrt[]{(0.92*0.08)/(250)} \\ \text{margin of error = 0.044} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/wde3gp1tjxh5l04xpfoxb7ko7jd09gtsvz.png)
The 99% confidence interval for p is
0.92 ± 0.044
it is from
0.876 to 0.964