\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 95{,}1 - 2 ⋅ {16{,}3 \over \sqrt{133}} ≈ 92{,}3 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 95{,}1 + 2 ⋅ {16{,}3 \over \sqrt{133}} ≈ 97{,}9 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([92{,}3 ; 97{,}9] \text{.}\)

De steekproefproportie is \(\hat{p} = {20 \over 174} = 0{,}114...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}114... ⋅ 0{,}885... \over 174}} = 0{,}024...\)

\(\hat{p} - 2 \sigma = 0{,}114... - 2 ⋅ 0{,}024... ≈ 0{,}067 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}114... + 2 ⋅ 0{,}024... ≈ 0{,}163 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}067 ; 0{,}163] \text{.}\)

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

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}42 ⋅ 0{,}58 \over 217}} = 0{,}0335...\)

\(\hat{p} - 2 \sigma = 0{,}42 - 2 ⋅ 0{,}0335... ≈ 0{,}353 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}42 + 2 ⋅ 0{,}0335... ≈ 0{,}487 \text{.}\)

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

\(\hat{p} = {0{,}202 + 0{,}378 \over 2} = 0{,}29\) en \(\text{breedte} = 0{,}378 - 0{,}202 = 0{,}176 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}29 ⋅ 0{,}71 \over n}} = 0{,}176 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}29 ⋅ 0{,}71 \over x}}\)
\(y_{2} = 0{,}176\)
Optie 'intersect' geeft \(x = 106{,}353...\)

De steekproefomvang is dus \(106 \text{.}\)

\(S = 59\) en \(\text{breedte} = 684 - 664 = 20 \text{.}\)

Los op \(4 ⋅ {59 \over \sqrt{n}} = 20 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {59 \over \sqrt{x}}\)
\(y_{2} = 20\)
Optie 'intersect' geeft \(x = 139{,}24\)

De steekproefomvang is dus \(139 \text{.}\)