\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 592 - 2 ⋅ {68 \over \sqrt{155}} ≈ 581 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 592 + 2 ⋅ {68 \over \sqrt{155}} ≈ 603 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([581 , 603] \text{.}\)

De steekproefproportie is \(\hat{p} = {78 \over 245} = 0{,}318...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}318... ⋅ 0{,}681... \over 245}} = 0{,}029...\)

\(\hat{p} - 2 \sigma = 0{,}318... - 2 ⋅ 0{,}029... ≈ 0{,}259 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}318... + 2 ⋅ 0{,}029... ≈ 0{,}378 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}259 ; 0{,}378] \text{.}\)

De steekproefproportie is \(\hat{p} = 11\% = 0{,}11 \text{.}\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}11 ⋅ 0{,}89 \over 240}} = 0{,}0201...\)

\(\hat{p} - 2 \sigma = 0{,}11 - 2 ⋅ 0{,}0201... ≈ 0{,}070 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}11 + 2 ⋅ 0{,}0201... ≈ 0{,}150 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([7{,}0\% , 15{,}0\%] \text{.}\)

\(\hat{p} = {0{,}374 + 0{,}506 \over 2} = 0{,}44\) en \(\text{breedte} = 0{,}506 - 0{,}374 = 0{,}132 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}44 ⋅ 0{,}56 \over n}} = 0{,}132 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}44 ⋅ 0{,}56 \over x}}\)
\(y_{2} = 0{,}132\)
Optie 'intersect' geeft \(x = 226{,}262...\)

De steekproefomvang is dus \(226 \text{.}\)

\(S = 0{,}28\) en \(\text{breedte} = 2{,}55 - 2{,}47 = 0{,}08 \text{.}\)

Los op \(4 ⋅ {0{,}28 \over \sqrt{n}} = 0{,}08 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {0{,}28 \over \sqrt{x}}\)
\(y_{2} = 0{,}08\)
Optie 'intersect' geeft \(x = 196\)

De steekproefomvang is dus \(196 \text{.}\)