\(\bar{X}-2⋅{S \over \sqrt{n}}=891-2⋅{80 \over \sqrt{160}}≈878\text{.}\)

\(\bar{X}+2⋅{S \over \sqrt{n}}=891+2⋅{80 \over \sqrt{160}}≈904\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([878, 904]\text{.}\)

De steekproefproportie is \(\hat{p}={45 \over 133}=0{,}338...\)

\(\sigma =\sqrt{{\hat{p}(1-\hat{p}) \over n}}=\sqrt{{0{,}338...⋅0{,}661... \over 133}}=0{,}041...\)

\(\hat{p}-2\sigma =0{,}338...-2⋅0{,}041...≈0{,}256\text{.}\)

\(\hat{p}+2\sigma =0{,}338...+2⋅0{,}041...≈0{,}420\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval is \([0{,}256; 0{,}420]\text{.}\)

De steekproefproportie is \(\hat{p}=31\%=0{,}31\text{.}\)

\(\sigma =\sqrt{{\hat{p}(1-\hat{p}) \over n}}=\sqrt{{0{,}31⋅0{,}69 \over 196}}=0{,}0330...\)

\(\hat{p}-2\sigma =0{,}31-2⋅0{,}0330...≈0{,}244\text{.}\)

\(\hat{p}+2\sigma =0{,}31+2⋅0{,}0330...≈0{,}376\text{.}\)

Het \(95\%\text{-}\)betrouwbaarheidsinterval is \([24{,}4\%; 37{,}6\%]\text{.}\)

\(\hat{p}={0{,}176+0{,}304 \over 2}=0{,}24\) en \(\text{breedte}=0{,}304-0{,}176=0{,}128\text{.}\)

Los op \(4⋅\sqrt{{0{,}24⋅0{,}76 \over n}}=0{,}128\text{.}\)

Voer in
\(y_1=4⋅\sqrt{{0{,}24⋅0{,}76 \over x}}\)
\(y_2=0{,}128\)
Optie 'intersect' geeft \(x=178{,}125\)

De steekproefomvang is dus \(178\text{.}\)

\(S=13{,}0\) en \(\text{breedte}=73{,}4-69{,}8=3{,}6\text{.}\)

Los op \(4⋅{13{,}0 \over \sqrt{n}}=3{,}6\text{.}\)

Voer in
\(y_1=4⋅{13{,}0 \over \sqrt{x}}\)
\(y_2=3{,}6\)
Optie 'intersect' geeft \(x=208{,}641...\)

De steekproefomvang is dus \(209\text{.}\)