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Estimasi Interval Selisih Mean Dua Populasi Sembarang
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Jika dipunyai [latexpage]$X_{11}, X_{12}, \cdots , X_{1n_{1}}$ dan [latexpage]$X_{21}, X_{22}, \cdots , X_{1n_{2}}$ merupakan dua sampel random yang independen satu sama lain yang diambil dari populasi yang mempunyai mean [latexpage]$\mu _{1}$ dan [latexpage]$\mu _{2}$ serta variansi [latexpage]$\sigma _{1}^{2}$ dan [latexpage]$\sigma_{2}^{2}$, maka untuk [latexpage]$n_{1}$ dan [latexpage]$n_{2}$ besar, variabel random
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1. [latexpage] $Z=\frac{(\bar{X_{1}}-\bar{X_{2}})-(\mu _{1}-\mu _{2})}{\sqrt{\frac{\sigma _{1}^{2}}{n_{1}}+\frac{\sigma _{2}^{2}}{n_{2}}}}$
berdistribusi Normal Standar, dengan
[latexpage]$\bar{X_{1}}=\sum_{i=1}^{n_{1}}{\frac{X_{1i}}{n_{1}}}, \bar{X_{2}}=\sum_{i=1}^{n_{2}}{\frac{X_{2i}}{n_{2}}}$
2. [latexpage] Jika $\sigma _{1}^{2}$ dan $\sigma _{2}^{2}$ tidak diketahui, dan diasumsikan $\sigma _{1}^{2}\ne \sigma _{2}^{2}$
[latexpage] $Z=\frac{(\bar{X_{1}}-\bar{X_{2}})-(\mu _{1}-\mu _{2})}{\sqrt{\frac{S_{1}^{2}}{n_{1}}+\frac{S_{2}^{2}}{n_{2}}}}$
berdistribusi Normal Standar, dengan [latexpage]$S_{1}^{2}$ dan $S_{2}^{2}$ merupakan variansi sampel.
3. [latexpage] Jika $\sigma _{1}^{2}$ dan $\sigma _{2}^{2}$ tidak diketahui, dan diasumsikan $\sigma _{1}^{2}=\sigma _{2}^{2}$
[latexpage] $Z=\frac{(\bar{X_{1}}-\bar{X_{2}})-(\mu _{1}-\mu _{2})}{\sqrt{S_{p}^{2}(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}$
berdistribusi Normal Standar, dengan
[latexpage]$S_{p}^{2}=\frac{(n_{1}-1)S_{1}^{2}+(n_{2}-1)S_{2}^{2}}{n_{1}+n_{2}-2}$
yang disebut sebagai pooled variance.
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Dari ketiga persamaan tersebut, diperoleh interval konfidensi (1-α)100% untuk [latexpage]$(\mu _{1}-\mu _{2})$ adalah $B\le \mu _{1}-\mu_{2}\le A$ dengan ketentuan :
a. Jika [latexpage] $\sigma _{1}^{2}$ dan [latexpage]$\sigma _{2}^{2}$ diketahui,
[latexpage] $B=(\bar{X_{1}}-\bar{X_{2}})-Z_{\frac{\alpha }{2}}\sqrt{\frac{\sigma _{1}^{2}}{n_{1}}+\frac{\sigma _{2}^{2}}{n_{2}}}$
[latexpage] $A=(\bar{X_{1}}-\bar{X_{2}})+Z_{\frac{\alpha }{2}}\sqrt{\frac{\sigma _{1}^{2}}{n_{1}}+\frac{\sigma _{2}^{2}}{n_{2}}}$
b. Jika [latexpage]$\sigma _{1}^{2}$ dan [latexpage]$\sigma _{2}^{2}$ tidak diketahui, dan diasumsikan [latexpage]$\sigma _{1}^{2}\ne \sigma _{2}^{2}$
[latexpage]$B=(\bar{X_{1}}-\bar{X_{2}})-Z_{\frac{\alpha}{2}}\sqrt{\frac{S_{1}^{2}}{n_{1}}+\frac{S_{2}^{2}}{n_{2}}}$
[latexpage]$A=(\bar{X_{1}}-\bar{X_{2}})+Z_{\frac{\alpha}{2}}\sqrt{\frac{S_{1}^{2}}{n_{1}}+\frac{S_{2}^{2}}{n_{2}}}$
dengan [latexpage]$S_{1}^{2}$ dan [latexpage]$S_{2}^{2}$ merupakan variansi sampel.
c. Jika [latexpage]$\sigma _{1}^{2}$ dan [latexpage]$\sigma _{2}^{2}$ tidak diketahui, dan diasumsikan [latexpage]$\sigma _{1}^{2}=\sigma _{2}^{2}$
[latexpage]$B=(\bar{X_{1}}-\bar{X_{2}})-Z_{\frac{\alpha}{2}}\sqrt{S_{p}^{2}(\frac{1}{n_{1}}+\frac{1}{n_{2}})}$
[latexpage]$A=(\bar{X_{1}}-\bar{X_{2}})+Z_{\frac{\alpha}{2}}\sqrt{S_{p}^{2}(\frac{1}{n_{1}}+\frac{1}{n_{2}})}$
dengan [latexpage]$S_{p}^{2}$ merupakan variansi gabungan (pooled variance).
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