\(\bar{X}-2⋅{S \over \sqrt{n}}=251-2⋅{13 \over \sqrt{152}}≈249\text{.}\)

\(\bar{X}+2⋅{S \over \sqrt{n}}=251+2⋅{13 \over \sqrt{152}}≈253\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([249, 253]\text{.}\)

De steekproefproportie is \(\hat{p}={39 \over 109}=0{,}357...\)

\(\sigma =\sqrt{{\hat{p}(1-\hat{p}) \over n}}=\sqrt{{0{,}357...⋅0{,}642... \over 109}}=0{,}045...\)

\(\hat{p}-2\sigma =0{,}357...-2⋅0{,}045...≈0{,}266\text{.}\)

\(\hat{p}+2\sigma =0{,}357...+2⋅0{,}045...≈0{,}450\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval is \([0{,}266; 0{,}450]\text{.}\)

De steekproefproportie is \(\hat{p}=18\%=0{,}18\text{.}\)

\(\sigma =\sqrt{{\hat{p}(1-\hat{p}) \over n}}=\sqrt{{0{,}18⋅0{,}82 \over 194}}=0{,}0275...\)

\(\hat{p}-2\sigma =0{,}18-2⋅0{,}0275...≈0{,}125\text{.}\)

\(\hat{p}+2\sigma =0{,}18+2⋅0{,}0275...≈0{,}235\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval is \([12{,}5\%; 23{,}5\%]\text{.}\)

\(\hat{p}={0{,}094+0{,}186 \over 2}=0{,}14\) en \(\text{breedte}=0{,}186-0{,}094=0{,}092\text{.}\)

Los op \(4⋅\sqrt{{0{,}14⋅0{,}86 \over n}}=0{,}092\text{.}\)

Voer in
\(y_1=4⋅\sqrt{{0{,}14⋅0{,}86 \over x}}\)
\(y_2=0{,}092\)
Optie 'intersect' geeft \(x=227{,}599...\)

De steekproefomvang is dus \(228\text{.}\)

\(S=0{,}14\) en \(\text{breedte}=5{,}25-5{,}21=0{,}04\text{.}\)

Los op \(4⋅{0{,}14 \over \sqrt{n}}=0{,}04\text{.}\)

Voer in
\(y_1=4⋅{0{,}14 \over \sqrt{x}}\)
\(y_2=0{,}04\)
Optie 'intersect' geeft \(x=196\)

De steekproefomvang is dus \(196\text{.}\)