\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 14{,}6 - 2 ⋅ {4{,}2 \over \sqrt{165}} ≈ 13{,}9 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 14{,}6 + 2 ⋅ {4{,}2 \over \sqrt{165}} ≈ 15{,}3 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([13{,}9 ; 15{,}3] \text{.}\)

De steekproefproportie is \(\hat{p} = {30 \over 112} = 0{,}267...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}267... ⋅ 0{,}732... \over 112}} = 0{,}041...\)

\(\hat{p} - 2 \sigma = 0{,}267... - 2 ⋅ 0{,}041... ≈ 0{,}184 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}267... + 2 ⋅ 0{,}041... ≈ 0{,}352 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}184 ; 0{,}352] \text{.}\)

De steekproefproportie is \(\hat{p} = 43\% = 0{,}43 \text{.}\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}43 ⋅ 0{,}57 \over 110}} = 0{,}0472...\)

\(\hat{p} - 2 \sigma = 0{,}43 - 2 ⋅ 0{,}0472... ≈ 0{,}336 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}43 + 2 ⋅ 0{,}0472... ≈ 0{,}524 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([33{,}6\% ; 52{,}4\%] \text{.}\)

\(\hat{p} = {0{,}278 + 0{,}402 \over 2} = 0{,}34\) en \(\text{breedte} = 0{,}402 - 0{,}278 = 0{,}124 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}34 ⋅ 0{,}66 \over n}} = 0{,}124 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}34 ⋅ 0{,}66 \over x}}\)
\(y_{2} = 0{,}124\)
Optie 'intersect' geeft \(x = 233{,}506...\)

De steekproefomvang is dus \(234 \text{.}\)

\(S = 71\) en \(\text{breedte} = 193 - 173 = 20 \text{.}\)

Los op \(4 ⋅ {71 \over \sqrt{n}} = 20 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {71 \over \sqrt{x}}\)
\(y_{2} = 20\)
Optie 'intersect' geeft \(x = 201{,}64\)

De steekproefomvang is dus \(202 \text{.}\)