De steekproefproportie is \(\hat{p} = {47 \over 196} = 0{,}239...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}239... ⋅ 0{,}760... \over 196}} = 0{,}030...\)

\(\hat{p} - 2 \sigma = 0{,}239... - 2 ⋅ 0{,}030... ≈ 0{,}179 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}239... + 2 ⋅ 0{,}030... ≈ 0{,}301 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}179 ; 0{,}301] \text{.}\)

De steekproefproportie is \(\hat{p} = 11\% = 0{,}11 \text{.}\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}11 ⋅ 0{,}89 \over 233}} = 0{,}0204...\)

\(\hat{p} - 2 \sigma = 0{,}11 - 2 ⋅ 0{,}0204... ≈ 0{,}069 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}11 + 2 ⋅ 0{,}0204... ≈ 0{,}151 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([6{,}9\% ; 15{,}1\%] \text{.}\)

\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 21{,}4 - 2 ⋅ {4{,}2 \over \sqrt{190}} ≈ 20{,}8 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 21{,}4 + 2 ⋅ {4{,}2 \over \sqrt{190}} ≈ 22{,}0 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([20{,}8 ; 22{,}0] \text{.}\)

\(\hat{p} = {0{,}410 + 0{,}570 \over 2} = 0{,}49\) en \(\text{breedte} = 0{,}570 - 0{,}410 = 0{,}16 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}49 ⋅ 0{,}51 \over n}} = 0{,}16 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}49 ⋅ 0{,}51 \over x}}\)
\(y_{2} = 0{,}16\)
Optie 'intersect' geeft \(x = 156{,}187...\)

De steekproefomvang is dus \(156 \text{.}\)

\(S = 12\) en \(\text{breedte} = 774 - 770 = 4 \text{.}\)

Los op \(4 ⋅ {12 \over \sqrt{n}} = 4 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {12 \over \sqrt{x}}\)
\(y_{2} = 4\)
Optie 'intersect' geeft \(x = 144\)

De steekproefomvang is dus \(144 \text{.}\)