De steekproefproportie is \(\hat{p} = {26 \over 115} = 0{,}226...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}226... ⋅ 0{,}773... \over 115}} = 0{,}039...\)

\(\hat{p} - 2 \sigma = 0{,}226... - 2 ⋅ 0{,}039... ≈ 0{,}148 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}226... + 2 ⋅ 0{,}039... ≈ 0{,}304 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}148 ; 0{,}304] \text{.}\)

De steekproefproportie is \(\hat{p} = 35\% = 0{,}35 \text{.}\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}35 ⋅ 0{,}65 \over 200}} = 0{,}0337...\)

\(\hat{p} - 2 \sigma = 0{,}35 - 2 ⋅ 0{,}0337... ≈ 0{,}283 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}35 + 2 ⋅ 0{,}0337... ≈ 0{,}417 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([28{,}3\% ; 41{,}7\%] \text{.}\)

\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 71{,}0 - 2 ⋅ {5{,}8 \over \sqrt{111}} ≈ 69{,}9 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 71{,}0 + 2 ⋅ {5{,}8 \over \sqrt{111}} ≈ 72{,}1 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([69{,}9 ; 72{,}1] \text{.}\)

\(\hat{p} = {0{,}054 + 0{,}166 \over 2} = 0{,}11\) en \(\text{breedte} = 0{,}166 - 0{,}054 = 0{,}112 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}11 ⋅ 0{,}89 \over n}} = 0{,}112 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}11 ⋅ 0{,}89 \over x}}\)
\(y_{2} = 0{,}112\)
Optie 'intersect' geeft \(x = 124{,}872...\)

De steekproefomvang is dus \(125 \text{.}\)

\(S = 7{,}8\) en \(\text{breedte} = 17{,}8 - 15{,}8 = 2{,}0 \text{.}\)

Los op \(4 ⋅ {7{,}8 \over \sqrt{n}} = 2{,}0 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {7{,}8 \over \sqrt{x}}\)
\(y_{2} = 2{,}0\)
Optie 'intersect' geeft \(x = 243{,}36\)

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