\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 24{,}6 - 2 ⋅ {3{,}1 \over \sqrt{191}} ≈ 24{,}2 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 24{,}6 + 2 ⋅ {3{,}1 \over \sqrt{191}} ≈ 25{,}0 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([24{,}2 ; 25{,}0] \text{.}\)

De steekproefproportie is \(\hat{p} = {85 \over 198} = 0{,}429...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}429... ⋅ 0{,}570... \over 198}} = 0{,}035...\)

\(\hat{p} - 2 \sigma = 0{,}429... - 2 ⋅ 0{,}035... ≈ 0{,}359 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}429... + 2 ⋅ 0{,}035... ≈ 0{,}500 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}359 ; 0{,}500] \text{.}\)

De steekproefproportie is \(\hat{p} = 42\% = 0{,}42 \text{.}\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}42 ⋅ 0{,}58 \over 101}} = 0{,}0491...\)

\(\hat{p} - 2 \sigma = 0{,}42 - 2 ⋅ 0{,}0491... ≈ 0{,}322 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}42 + 2 ⋅ 0{,}0491... ≈ 0{,}518 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([32{,}2\% ; 51{,}8\%] \text{.}\)

\(\hat{p} = {0{,}282 + 0{,}438 \over 2} = 0{,}36\) en \(\text{breedte} = 0{,}438 - 0{,}282 = 0{,}156 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}36 ⋅ 0{,}64 \over n}} = 0{,}156 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}36 ⋅ 0{,}64 \over x}}\)
\(y_{2} = 0{,}156\)
Optie 'intersect' geeft \(x = 151{,}479...\)

De steekproefomvang is dus \(151 \text{.}\)

\(S = 1{,}43\) en \(\text{breedte} = 8{,}76 - 8{,}28 = 0{,}48 \text{.}\)

Los op \(4 ⋅ {1{,}43 \over \sqrt{n}} = 0{,}48 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {1{,}43 \over \sqrt{x}}\)
\(y_{2} = 0{,}48\)
Optie 'intersect' geeft \(x = 142{,}006...\)

De steekproefomvang is dus \(142 \text{.}\)