\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 17{,}4 - 2 ⋅ {1{,}5 \over \sqrt{194}} ≈ 17{,}2 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 17{,}4 + 2 ⋅ {1{,}5 \over \sqrt{194}} ≈ 17{,}6 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([17{,}2 ; 17{,}6] \text{.}\)

De steekproefproportie is \(\hat{p} = {78 \over 217} = 0{,}359...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}359... ⋅ 0{,}640... \over 217}} = 0{,}032...\)

\(\hat{p} - 2 \sigma = 0{,}359... - 2 ⋅ 0{,}032... ≈ 0{,}294 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}359... + 2 ⋅ 0{,}032... ≈ 0{,}425 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}294 ; 0{,}425] \text{.}\)

De steekproefproportie is \(\hat{p} = 36\% = 0{,}36 \text{.}\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}36 ⋅ 0{,}64 \over 179}} = 0{,}0358...\)

\(\hat{p} - 2 \sigma = 0{,}36 - 2 ⋅ 0{,}0358... ≈ 0{,}288 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}36 + 2 ⋅ 0{,}0358... ≈ 0{,}432 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([28{,}8\% ; 43{,}2\%] \text{.}\)

\(\hat{p} = {0{,}198 + 0{,}362 \over 2} = 0{,}28\) en \(\text{breedte} = 0{,}362 - 0{,}198 = 0{,}164 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}28 ⋅ 0{,}72 \over n}} = 0{,}164 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}28 ⋅ 0{,}72 \over x}}\)
\(y_{2} = 0{,}164\)
Optie 'intersect' geeft \(x = 119{,}928...\)

De steekproefomvang is dus \(120 \text{.}\)

\(S = 1{,}69\) en \(\text{breedte} = 4{,}86 - 4{,}34 = 0{,}52 \text{.}\)

Los op \(4 ⋅ {1{,}69 \over \sqrt{n}} = 0{,}52 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {1{,}69 \over \sqrt{x}}\)
\(y_{2} = 0{,}52\)
Optie 'intersect' geeft \(x = 169\)

De steekproefomvang is dus \(169 \text{.}\)