\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 32{,}0 - 2 ⋅ {1{,}7 \over \sqrt{231}} ≈ 31{,}8 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 32{,}0 + 2 ⋅ {1{,}7 \over \sqrt{231}} ≈ 32{,}2 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([31{,}8 ; 32{,}2] \text{.}\)

De steekproefproportie is \(\hat{p} = {49 \over 172} = 0{,}284...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}284... ⋅ 0{,}715... \over 172}} = 0{,}034...\)

\(\hat{p} - 2 \sigma = 0{,}284... - 2 ⋅ 0{,}034... ≈ 0{,}216 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}284... + 2 ⋅ 0{,}034... ≈ 0{,}354 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}216 ; 0{,}354] \text{.}\)

De steekproefproportie is \(\hat{p} = 45\% = 0{,}45 \text{.}\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}45 ⋅ 0{,}55 \over 117}} = 0{,}0459...\)

\(\hat{p} - 2 \sigma = 0{,}45 - 2 ⋅ 0{,}0459... ≈ 0{,}358 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}45 + 2 ⋅ 0{,}0459... ≈ 0{,}542 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([35{,}8\% ; 54{,}2\%] \text{.}\)

\(\hat{p} = {0{,}244 + 0{,}376 \over 2} = 0{,}31\) en \(\text{breedte} = 0{,}376 - 0{,}244 = 0{,}132 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}31 ⋅ 0{,}69 \over n}} = 0{,}132 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}31 ⋅ 0{,}69 \over x}}\)
\(y_{2} = 0{,}132\)
Optie 'intersect' geeft \(x = 196{,}418...\)

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

\(S = 2{,}9\) en \(\text{breedte} = 84{,}3 - 83{,}5 = 0{,}8 \text{.}\)

Los op \(4 ⋅ {2{,}9 \over \sqrt{n}} = 0{,}8 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {2{,}9 \over \sqrt{x}}\)
\(y_{2} = 0{,}8\)
Optie 'intersect' geeft \(x = 210{,}25\)

De steekproefomvang is dus \(210 \text{.}\)