Meil on antud juhuslik valim \(n=75\), \(\bar{x}=54.5\) ja standardhälve \(s=10.2\)
standardh<-10.2
valimisuurus<-75
valimikesk<-54.5Kuna \((1-\alpha)=0.5\), siis \(\alpha=1-0.5=0.5\). Sellest saame, et \(\frac{\alpha}{2}=0.25\). Meil tarvis leida \(\bar{z}_{\frac{\alpha}{2}}\) ehk antud juhul \(\bar{z}_{0.25}\). Me teame, et kehtib valem \(\bar{z}_{\frac{\alpha}{2}} = z_{1-\frac{\alpha}{2}}\). Seega peame leidma hoopis \(z_{1-\frac{\alpha}{2}}\), mis on antud juhul \(z_{1-0.25} = z_{0.75}\).
kvantiil_1 <- qnorm(0.75)
kvantiil_1## [1] 0.6744898Leiame usaldusintervalli laiuse valemiga \(2\cdot \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}\)
2*kvantiil_1*10.2/sqrt(75)## [1] 1.588821Leiame usaldusintervalli, kui see avaldub kujul \((\bar{x}- \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}; \bar{x}+ \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}})\)
alumine_1<-valimikesk-kvantiil_1*standardh/sqrt(valimisuurus)
ylemine_1<-valimikesk+kvantiil_1*standardh/sqrt(valimisuurus)
intervall_1<-c(alumine_1, ylemine_1)
intervall_1## [1] 53.70559 55.29441Kuna \((1-\alpha)=0.9\), siis \(\alpha=1-0.9=0.1\). Sellest saame, et \(\frac{\alpha}{2}=0.05\). Meil tarvis leida \(\bar{z}_{\frac{\alpha}{2}}\) ehk antud juhul \(\bar{z}_{0.05}\). Me teame, et kehtib valem \(\bar{z}_{\frac{\alpha}{2}} = z_{1-\frac{\alpha}{2}}\). Seega peame leidma hoopis \(z_{1-\frac{\alpha}{2}}\), mis on antud juhul \(z_{1-0.05} = z_{0.95}\).
kvantiil_2 <- qnorm(0.95)
kvantiil_2## [1] 1.644854Leiame usaldusintervalli laiuse valemiga \(2\cdot \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}\)
2*kvantiil_2*standardh/sqrt(valimisuurus)## [1] 3.874599Leiame usaldusintervalli, kui see avaldub kujul \((\bar{x}- \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}; \bar{x}+ \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}})\)
alumine_2<-valimikesk-kvantiil_2*standardh/sqrt(valimisuurus)
ylemine_2<-valimikesk+kvantiil_2*standardh/sqrt(valimisuurus)
intervall_2<-c(alumine_2, ylemine_2)
intervall_2## [1] 52.5627 56.4373Kuna \((1-\alpha)=0.95\), siis \(\alpha=1-0.95=0.05\). Sellest saame, et \(\frac{\alpha}{2}=0.025\). Meil tarvis leida \(\bar{z}_{\frac{\alpha}{2}}\) ehk antud juhul \(\bar{z}_{0.025}\). Me teame, et kehtib valem \(\bar{z}_{\frac{\alpha}{2}} = z_{1-\frac{\alpha}{2}}\). Seega peame leidma hoopis \(z_{1-\frac{\alpha}{2}}\), mis on antud juhul \(z_{1-0.025} = z_{0.975}\).
kvantiil_3 <- qnorm(0.975)
kvantiil_3## [1] 1.959964Leiame usaldusintervalli laiuse valemiga \(2\cdot \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}\)
2*kvantiil_3*standardh/sqrt(valimisuurus)## [1] 4.61687Leiame usaldusintervalli, kui see avaldub kujul \((\bar{x}- \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}; \bar{x}+ \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}})\)
alumine_3<-valimikesk-kvantiil_3*standardh/sqrt(valimisuurus)
ylemine_3<-valimikesk+kvantiil_3*standardh/sqrt(valimisuurus)
intervall_3<-c(alumine_3, ylemine_3)
intervall_3## [1] 52.19157 56.80843Kuna \((1-\alpha)=0.99\), siis \(\alpha=1-0.99=0.01\). Sellest saame, et \(\frac{\alpha}{2}=0.005\). Meil tarvis leida \(\bar{z}_{\frac{\alpha}{2}}\) ehk antud juhul \(\bar{z}_{0.005}\). Me teame, et kehtib valem \(\bar{z}_{\frac{\alpha}{2}} = z_{1-\frac{\alpha}{2}}\). Seega peame leidma hoopis \(z_{1-\frac{\alpha}{2}}\), mis on antud juhul \(z_{1-0.005} = z_{0.995}\).
kvantiil_4 <- qnorm(0.995)
kvantiil_4## [1] 2.575829Leiame usaldusintervalli laiuse valemiga \(2\cdot \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}\)
2*kvantiil_4*standardh/sqrt(valimisuurus)## [1] 6.067595Leiame usaldusintervalli, kui see avaldub kujul \((\bar{x}- \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}; \bar{x}+ \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}})\)
alumine_4<-valimikesk-kvantiil_4*standardh/sqrt(valimisuurus)
ylemine_4<-valimikesk+kvantiil_4*standardh/sqrt(valimisuurus)
intervall_4<-c(alumine_4, ylemine_4)
intervall_4## [1] 51.4662 57.5338Kui suurt valimit vajaksime, et usaldusintervalli laius ei ületaks \(2,0\)?
Valem on: \(2.0=2\cdot \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}\)
Avaldame \(n\): \[ \begin{align*} 2.0&=2\cdot \bar{z}_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}} \Leftrightarrow \\ 2\cdot \sqrt{n}&=2\cdot \bar{z}_{\frac{\alpha}{2}}\cdot s\Leftrightarrow \\ \sqrt{n}&= \bar{z}_{\frac{\alpha}{2}}\cdot s\Leftrightarrow \\ n&= (\bar{z}_{\frac{\alpha}{2}}\cdot s)^2 \end{align*} \] ### 1) Usaldusnivoo \((1-\alpha) = 0.5\) ehk \(\bar{z}_{\frac{\alpha}{2}}=0.6744898\): Siis \(n\) väärtus on:
(kvantiil_1*standardh)^2## [1] 47.33159(0.67*standardh)^2## [1] 46.70356Siis \(n\) väärtus on:
(kvantiil_2*standardh)^2## [1] 281.4847(1.64*standardh)^2## [1] 279.826Siis \(n\) väärtus on:
(kvantiil_3*standardh)^2## [1] 399.6654(1.96*standardh)^2## [1] 399.6801Siis \(n\) väärtus on:
(kvantiil_4*standardh)^2## [1] 690.2946(2.58*standardh)^2## [1] 692.5319