De steekproefproportie is \(\hat{p}={31 \over 151}=0{,}205...\)

\(\sigma =\sqrt{{\hat{p}(1-\hat{p}) \over n}}=\sqrt{{0{,}205...⋅0{,}794... \over 151}}=0{,}032...\)

\(\hat{p}-2\sigma =0{,}205...-2⋅0{,}032...≈0{,}140\text{.}\)

\(\hat{p}+2\sigma =0{,}205...+2⋅0{,}032...≈0{,}271\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval is \([0{,}140; 0{,}271]\text{.}\)

De steekproefproportie is \(\hat{p}=16\%=0{,}16\text{.}\)

\(\sigma =\sqrt{{\hat{p}(1-\hat{p}) \over n}}=\sqrt{{0{,}16⋅0{,}84 \over 249}}=0{,}0232...\)

\(\hat{p}-2\sigma =0{,}16-2⋅0{,}0232...≈0{,}114\text{.}\)

\(\hat{p}+2\sigma =0{,}16+2⋅0{,}0232...≈0{,}206\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval is \([11{,}4\%; 20{,}6\%]\text{.}\)

\(\bar{X}-2⋅{S \over \sqrt{n}}=45{,}7-2⋅{8{,}5 \over \sqrt{216}}≈44{,}5\text{.}\)

\(\bar{X}+2⋅{S \over \sqrt{n}}=45{,}7+2⋅{8{,}5 \over \sqrt{216}}≈46{,}9\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([44{,}5; 46{,}9]\text{.}\)

\(\hat{p}={0{,}398+0{,}582 \over 2}=0{,}49\) en \(\text{breedte}=0{,}582-0{,}398=0{,}184\text{.}\)

Los op \(4⋅\sqrt{{0{,}49⋅0{,}51 \over n}}=0{,}184\text{.}\)

Voer in
\(y_1=4⋅\sqrt{{0{,}49⋅0{,}51 \over x}}\)
\(y_2=0{,}184\)
Optie 'intersect' geeft \(x=118{,}100...\)

De steekproefomvang is dus \(118\text{.}\)

\(S=284\) en \(\text{breedte}=937-849=88\text{.}\)

Los op \(4⋅{284 \over \sqrt{n}}=88\text{.}\)

Voer in
\(y_1=4⋅{284 \over \sqrt{x}}\)
\(y_2=88\)
Optie 'intersect' geeft \(x=166{,}644...\)

De steekproefomvang is dus \(167\text{.}\)