De steekproefproportie is \(\hat{p} = {12 \over 100} = 0{,}12\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}12 ⋅ 0{,}88 \over 100}} = 0{,}032...\)

\(\hat{p} - 2 \sigma = 0{,}12 - 2 ⋅ 0{,}032... ≈ 0{,}055 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}12 + 2 ⋅ 0{,}032... ≈ 0{,}185 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}055 ; 0{,}185] \text{.}\)

De steekproefproportie is \(\hat{p} = 42\% = 0{,}42 \text{.}\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}42 ⋅ 0{,}58 \over 246}} = 0{,}0314...\)

\(\hat{p} - 2 \sigma = 0{,}42 - 2 ⋅ 0{,}0314... ≈ 0{,}357 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}42 + 2 ⋅ 0{,}0314... ≈ 0{,}483 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([35{,}7\% ; 48{,}3\%] \text{.}\)

\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 64{,}7 - 2 ⋅ {17{,}1 \over \sqrt{244}} ≈ 62{,}5 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 64{,}7 + 2 ⋅ {17{,}1 \over \sqrt{244}} ≈ 66{,}9 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([62{,}5 ; 66{,}9] \text{.}\)

\(\hat{p} = {0{,}314 + 0{,}466 \over 2} = 0{,}39\) en \(\text{breedte} = 0{,}466 - 0{,}314 = 0{,}152 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}39 ⋅ 0{,}61 \over n}} = 0{,}152 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}39 ⋅ 0{,}61 \over x}}\)
\(y_{2} = 0{,}152\)
Optie 'intersect' geeft \(x = 164{,}750...\)

De steekproefomvang is dus \(165 \text{.}\)

\(S = 0{,}29\) en \(\text{breedte} = 1{,}90 - 1{,}82 = 0{,}08 \text{.}\)

Los op \(4 ⋅ {0{,}29 \over \sqrt{n}} = 0{,}08 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {0{,}29 \over \sqrt{x}}\)
\(y_{2} = 0{,}08\)
Optie 'intersect' geeft \(x = 210{,}25\)

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