\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 13{,}0 - 2 ⋅ {3{,}9 \over \sqrt{172}} ≈ 12{,}4 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 13{,}0 + 2 ⋅ {3{,}9 \over \sqrt{172}} ≈ 13{,}6 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([12{,}4 ; 13{,}6] \text{.}\)

De steekproefproportie is \(\hat{p} = {24 \over 227} = 0{,}105...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}105... ⋅ 0{,}894... \over 227}} = 0{,}020...\)

\(\hat{p} - 2 \sigma = 0{,}105... - 2 ⋅ 0{,}020... ≈ 0{,}065 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}105... + 2 ⋅ 0{,}020... ≈ 0{,}147 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}065 ; 0{,}147] \text{.}\)

De steekproefproportie is \(\hat{p} = 29\% = 0{,}29 \text{.}\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}29 ⋅ 0{,}71 \over 238}} = 0{,}0294...\)

\(\hat{p} - 2 \sigma = 0{,}29 - 2 ⋅ 0{,}0294... ≈ 0{,}231 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}29 + 2 ⋅ 0{,}0294... ≈ 0{,}349 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([23{,}1\% ; 34{,}9\%] \text{.}\)

\(\hat{p} = {0{,}354 + 0{,}506 \over 2} = 0{,}43\) en \(\text{breedte} = 0{,}506 - 0{,}354 = 0{,}152 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}43 ⋅ 0{,}57 \over n}} = 0{,}152 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}43 ⋅ 0{,}57 \over x}}\)
\(y_{2} = 0{,}152\)
Optie 'intersect' geeft \(x = 169{,}736...\)

De steekproefomvang is dus \(170 \text{.}\)

\(S = 2{,}65\) en \(\text{breedte} = 7{,}36 - 6{,}56 = 0{,}80 \text{.}\)

Los op \(4 ⋅ {2{,}65 \over \sqrt{n}} = 0{,}80 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {2{,}65 \over \sqrt{x}}\)
\(y_{2} = 0{,}80\)
Optie 'intersect' geeft \(x = 175{,}562...\)

De steekproefomvang is dus \(176 \text{.}\)