De steekproefproportie is \(\hat{p}={76 \over 171}=0{,}444...\)

\(\sigma =\sqrt{{\hat{p}(1-\hat{p}) \over n}}=\sqrt{{0{,}444...⋅0{,}555... \over 171}}=0{,}037...\)

\(\hat{p}-2\sigma =0{,}444...-2⋅0{,}037...≈0{,}368\text{.}\)

\(\hat{p}+2\sigma =0{,}444...+2⋅0{,}037...≈0{,}520\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval is \([0{,}368; 0{,}520]\text{.}\)

De steekproefproportie is \(\hat{p}=15\%=0{,}15\text{.}\)

\(\sigma =\sqrt{{\hat{p}(1-\hat{p}) \over n}}=\sqrt{{0{,}15⋅0{,}85 \over 225}}=0{,}0238...\)

\(\hat{p}-2\sigma =0{,}15-2⋅0{,}0238...≈0{,}102\text{.}\)

\(\hat{p}+2\sigma =0{,}15+2⋅0{,}0238...≈0{,}198\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval is \([10{,}2\%; 19{,}8\%]\text{.}\)

\(\bar{X}-2⋅{S \over \sqrt{n}}=2{,}00-2⋅{0{,}19 \over \sqrt{103}}≈1{,}96\text{.}\)

\(\bar{X}+2⋅{S \over \sqrt{n}}=2{,}00+2⋅{0{,}19 \over \sqrt{103}}≈2{,}04\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([1{,}96; 2{,}04]\text{.}\)

\(\hat{p}={0{,}068+0{,}152 \over 2}=0{,}11\) en \(\text{breedte}=0{,}152-0{,}068=0{,}084\text{.}\)

Los op \(4⋅\sqrt{{0{,}11⋅0{,}89 \over n}}=0{,}084\text{.}\)

Voer in
\(y_1=4⋅\sqrt{{0{,}11⋅0{,}89 \over x}}\)
\(y_2=0{,}084\)
Optie 'intersect' geeft \(x=221{,}995...\)

De steekproefomvang is dus \(222\text{.}\)

\(S=314\) en \(\text{breedte}=802-722=80\text{.}\)

Los op \(4⋅{314 \over \sqrt{n}}=80\text{.}\)

Voer in
\(y_1=4⋅{314 \over \sqrt{x}}\)
\(y_2=80\)
Optie 'intersect' geeft \(x=246{,}49\)

De steekproefomvang is dus \(246\text{.}\)