De steekproefproportie is \(\hat{p} = {58 \over 242} = 0{,}239...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}239... ⋅ 0{,}760... \over 242}} = 0{,}027...\)

\(\hat{p} - 2 \sigma = 0{,}239... - 2 ⋅ 0{,}027... ≈ 0{,}185 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}239... + 2 ⋅ 0{,}027... ≈ 0{,}295 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}185 ; 0{,}295] \text{.}\)

De steekproefproportie is \(\hat{p} = 48\% = 0{,}48 \text{.}\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}48 ⋅ 0{,}52 \over 183}} = 0{,}0369...\)

\(\hat{p} - 2 \sigma = 0{,}48 - 2 ⋅ 0{,}0369... ≈ 0{,}406 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}48 + 2 ⋅ 0{,}0369... ≈ 0{,}554 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([40{,}6\% ; 55{,}4\%] \text{.}\)

\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 376 - 2 ⋅ {77 \over \sqrt{110}} ≈ 361 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 376 + 2 ⋅ {77 \over \sqrt{110}} ≈ 391 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([361 , 391] \text{.}\)

\(\hat{p} = {0{,}286 + 0{,}474 \over 2} = 0{,}38\) en \(\text{breedte} = 0{,}474 - 0{,}286 = 0{,}188 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}38 ⋅ 0{,}62 \over n}} = 0{,}188 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}38 ⋅ 0{,}62 \over x}}\)
\(y_{2} = 0{,}188\)
Optie 'intersect' geeft \(x = 106{,}654...\)

De steekproefomvang is dus \(107 \text{.}\)

\(S = 5{,}6\) en \(\text{breedte} = 42{,}1 - 40{,}5 = 1{,}6 \text{.}\)

Los op \(4 ⋅ {5{,}6 \over \sqrt{n}} = 1{,}6 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {5{,}6 \over \sqrt{x}}\)
\(y_{2} = 1{,}6\)
Optie 'intersect' geeft \(x = 196\)

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