De steekproefproportie is \(\hat{p} = {25 \over 144} = 0{,}173...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}173... ⋅ 0{,}826... \over 144}} = 0{,}031...\)

\(\hat{p} - 2 \sigma = 0{,}173... - 2 ⋅ 0{,}031... ≈ 0{,}110 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}173... + 2 ⋅ 0{,}031... ≈ 0{,}237 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}110 ; 0{,}237] \text{.}\)

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

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}42 ⋅ 0{,}58 \over 176}} = 0{,}0372...\)

\(\hat{p} - 2 \sigma = 0{,}42 - 2 ⋅ 0{,}0372... ≈ 0{,}346 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}42 + 2 ⋅ 0{,}0372... ≈ 0{,}494 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([34{,}6\% ; 49{,}4\%] \text{.}\)

\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 348 - 2 ⋅ {94 \over \sqrt{224}} ≈ 335 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 348 + 2 ⋅ {94 \over \sqrt{224}} ≈ 361 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([335 , 361] \text{.}\)

\(\hat{p} = {0{,}248 + 0{,}412 \over 2} = 0{,}33\) en \(\text{breedte} = 0{,}412 - 0{,}248 = 0{,}164 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}33 ⋅ 0{,}67 \over n}} = 0{,}164 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}33 ⋅ 0{,}67 \over x}}\)
\(y_{2} = 0{,}164\)
Optie 'intersect' geeft \(x = 131{,}528...\)

De steekproefomvang is dus \(132 \text{.}\)

\(S = 15{,}2\) en \(\text{breedte} = 79{,}0 - 73{,}8 = 5{,}2 \text{.}\)

Los op \(4 ⋅ {15{,}2 \over \sqrt{n}} = 5{,}2 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {15{,}2 \over \sqrt{x}}\)
\(y_{2} = 5{,}2\)
Optie 'intersect' geeft \(x = 136{,}710...\)

De steekproefomvang is dus \(137 \text{.}\)