De steekproefproportie is \(\hat{p}={61 \over 186}=0{,}327...\)

\(\sigma =\sqrt{{\hat{p}(1-\hat{p}) \over n}}=\sqrt{{0{,}327...⋅0{,}672... \over 186}}=0{,}034...\)

\(\hat{p}-2\sigma =0{,}327...-2⋅0{,}034...≈0{,}259\text{.}\)

\(\hat{p}+2\sigma =0{,}327...+2⋅0{,}034...≈0{,}397\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval is \([0{,}259; 0{,}397]\text{.}\)

De steekproefproportie is \(\hat{p}=41\%=0{,}41\text{.}\)

\(\sigma =\sqrt{{\hat{p}(1-\hat{p}) \over n}}=\sqrt{{0{,}41⋅0{,}59 \over 176}}=0{,}0370...\)

\(\hat{p}-2\sigma =0{,}41-2⋅0{,}0370...≈0{,}336\text{.}\)

\(\hat{p}+2\sigma =0{,}41+2⋅0{,}0370...≈0{,}484\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval is \([33{,}6\%; 48{,}4\%]\text{.}\)

\(\bar{X}-2⋅{S \over \sqrt{n}}=8{,}51-2⋅{0{,}52 \over \sqrt{121}}≈8{,}42\text{.}\)

\(\bar{X}+2⋅{S \over \sqrt{n}}=8{,}51+2⋅{0{,}52 \over \sqrt{121}}≈8{,}60\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([8{,}42; 8{,}60]\text{.}\)

\(\hat{p}={0{,}176+0{,}284 \over 2}=0{,}23\) en \(\text{breedte}=0{,}284-0{,}176=0{,}108\text{.}\)

Los op \(4⋅\sqrt{{0{,}23⋅0{,}77 \over n}}=0{,}108\text{.}\)

Voer in
\(y_1=4⋅\sqrt{{0{,}23⋅0{,}77 \over x}}\)
\(y_2=0{,}108\)
Optie 'intersect' geeft \(x=242{,}935...\)

De steekproefomvang is dus \(243\text{.}\)

\(S=11{,}9\) en \(\text{breedte}=38{,}0-34{,}4=3{,}6\text{.}\)

Los op \(4⋅{11{,}9 \over \sqrt{n}}=3{,}6\text{.}\)

Voer in
\(y_1=4⋅{11{,}9 \over \sqrt{x}}\)
\(y_2=3{,}6\)
Optie 'intersect' geeft \(x=174{,}827...\)

De steekproefomvang is dus \(175\text{.}\)