De steekproefproportie is \(\hat{p} = {77 \over 216} = 0{,}356...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}356... ⋅ 0{,}643... \over 216}} = 0{,}032...\)

\(\hat{p} - 2 \sigma = 0{,}356... - 2 ⋅ 0{,}032... ≈ 0{,}291 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}356... + 2 ⋅ 0{,}032... ≈ 0{,}422 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}291 ; 0{,}422] \text{.}\)

De steekproefproportie is \(\hat{p} = 18\% = 0{,}18 \text{.}\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}18 ⋅ 0{,}82 \over 216}} = 0{,}0261...\)

\(\hat{p} - 2 \sigma = 0{,}18 - 2 ⋅ 0{,}0261... ≈ 0{,}128 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}18 + 2 ⋅ 0{,}0261... ≈ 0{,}232 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([12{,}8\% ; 23{,}2\%] \text{.}\)

\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 8{,}06 - 2 ⋅ {2{,}08 \over \sqrt{191}} ≈ 7{,}76 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 8{,}06 + 2 ⋅ {2{,}08 \over \sqrt{191}} ≈ 8{,}36 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([7{,}76 ; 8{,}36] \text{.}\)

\(\hat{p} = {0{,}202 + 0{,}318 \over 2} = 0{,}26\) en \(\text{breedte} = 0{,}318 - 0{,}202 = 0{,}116 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}26 ⋅ 0{,}74 \over n}} = 0{,}116 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}26 ⋅ 0{,}74 \over x}}\)
\(y_{2} = 0{,}116\)
Optie 'intersect' geeft \(x = 228{,}775...\)

De steekproefomvang is dus \(229 \text{.}\)

\(S = 23{,}7\) en \(\text{breedte} = 78{,}6 - 72{,}2 = 6{,}4 \text{.}\)

Los op \(4 ⋅ {23{,}7 \over \sqrt{n}} = 6{,}4 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {23{,}7 \over \sqrt{x}}\)
\(y_{2} = 6{,}4\)
Optie 'intersect' geeft \(x = 219{,}410...\)

De steekproefomvang is dus \(219 \text{.}\)