Question 1206714

a) A survey of {{{500 }}}people finds that {{{55}}}% plan to vote for Smith for governor.


margin of error = {{{z * sqrt(p * (1 - p) / n)}}}

where {{{z}}} is the z-score for the {{{95}}}% confidence level ({{{z=1.96}}}), {{{p }}}is the sample proportion ({{{p=0.55}}}), and {{{n}}} is the sample size ({{{n=500}}}).

margin of error = {{{1.96 * sqrt(0.55 * (1 - 0.55) / 500)}}}

margin of error = {{{1.96 * sqrt(0.000495)}}}

margin of error ={{{ 1.96 * 0.022248595461286987}}}

margin of error ={{{ 0.0436 }}}


confidence interval = p +- margin of error
confidence interval = {{{0.55 +- 0.0436}}}

confidence interval is:

{{{0.55 - 0.0436=0.5064 }}} or {{{50.64}}}% 
{{{0.55 + 0.0436=0.5936}}} or {{{59.36}}}%


We can be {{{95}}}% confident that the true proportion of people who plan to vote for Smith for governor is between {{{50.64}}}% and {{{59.36}}}%.



b) A survey of 1500 people finds that {{{57}}}% support stricter penalties for child abuse.


margin of error = {{{1.96 * sqrt(0.57 * (1 - 0.57) / 1500)}}}

margin of error = {{{1.96 * 0.01278280094502}}}

margin of error = {{{0.02505}}}


confidence interval ={{{ 0.57 +-0.02505}}}


confidence interval is:

{{{0.57 -0.02505=0.5450}}}

{{{0.57 +0.02505=0.5951}}}


We can be {{{95}}}% confident that the true proportion of people who support stricter penalties for child abuse is between {{{54.50}}}% and {{{59.51}}}%.