De steekproefproportie is \(\hat{p} = {76 \over 170} = 0{,}447...\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}447... ⋅ 0{,}552... \over 170}} = 0{,}038...\)

\(\hat{p} - 2 \sigma = 0{,}447... - 2 ⋅ 0{,}038... ≈ 0{,}371 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}447... + 2 ⋅ 0{,}038... ≈ 0{,}523 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([0{,}371 ; 0{,}523] \text{.}\)

De steekproefproportie is \(\hat{p} = 46\% = 0{,}46 \text{.}\)

\(\sigma = \sqrt{{\hat{p} (1 - \hat{p}) \over n}} = \sqrt{{0{,}46 ⋅ 0{,}54 \over 240}} = 0{,}0321...\)

\(\hat{p} - 2 \sigma = 0{,}46 - 2 ⋅ 0{,}0321... ≈ 0{,}396 \text{.}\)

\(\hat{p} + 2 \sigma = 0{,}46 + 2 ⋅ 0{,}0321... ≈ 0{,}524 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval is \([39{,}6\% ; 52{,}4\%] \text{.}\)

\(\bar{X} - 2 ⋅ {S \over \sqrt{n}} = 740 - 2 ⋅ {63 \over \sqrt{102}} ≈ 728 \text{.}\)

\(\bar{X} + 2 ⋅ {S \over \sqrt{n}} = 740 + 2 ⋅ {63 \over \sqrt{102}} ≈ 752 \text{.}\)

Het \(95\% \text{-}\)betrouwbaarheidsinterval voor het populatiegemiddelde is \([728 , 752] \text{.}\)

\(\hat{p} = {0{,}172 + 0{,}288 \over 2} = 0{,}23\) en \(\text{breedte} = 0{,}288 - 0{,}172 = 0{,}116 \text{.}\)

Los op \(4 ⋅ \sqrt{{0{,}23 ⋅ 0{,}77 \over n}} = 0{,}116 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ \sqrt{{0{,}23 ⋅ 0{,}77 \over x}}\)
\(y_{2} = 0{,}116\)
Optie 'intersect' geeft \(x = 210{,}582...\)

De steekproefomvang is dus \(211 \text{.}\)

\(S = 53\) en \(\text{breedte} = 260 - 240 = 20 \text{.}\)

Los op \(4 ⋅ {53 \over \sqrt{n}} = 20 \text{.}\)

Voer in
\(y_{1} = 4 ⋅ {53 \over \sqrt{x}}\)
\(y_{2} = 20\)
Optie 'intersect' geeft \(x = 112{,}36\)

De steekproefomvang is dus \(112 \text{.}\)