Contoh Soal 2 Distribusi Normal

$\begin{array}{l}\\ 6.& \text{Luas daerah yang diarsir di bawah}\\ & \text{kurva normal baku berikut adalah}....\\ & \text{a}.0,4861\\ & \text{b}.0,4878\\ & \text{c}.0,4881\\ & \text{d}.0,4938\\ & \text{e}.0,4946\\ \end{array}$.

$.\begin{array}{rl}& \mathbf{\text{Jawab}}:\\ & \text{Perhatikan tabel}\text{distribusi normal}\text{berikut}\\ & \begin{array}{|ccccccccccc|}\hline \text{z}& 0& 1& 2& 3& 4& 5& 6& 7& 8& 9\\ ⋮& & & & & & \downarrow & & & & \\ ⋮& & & & & & & & & & \\ 2,2& & & & & & 0,4878& & & & \\ ⋮& & & & & & & & & & \\ \hline\end{array}\\ & \text{Sehingga nilai}\text{z}=2,25\text{luasnya}=0,4878\end{array}$.

$\begin{array}{l}\\ 7.& \text{Luas daerah yang diarsir di bawah}\\ & \text{kurva normal baku berikut adalah}....\\ & \text{a}.0,1138\\ & \text{b}.0,3810\\ & \text{c}.0,3862\\ & \text{d}.0,4948\\ & \text{e}.0,5000\\ \end{array}$.

$.\begin{array}{rl}& \mathbf{\text{Jawab}}:\\ & \text{Perhatikan tabel}\text{distribusi normal}\text{berikut}\\ & \begin{array}{|ccccccccccc|}\hline \text{z}& 0& 1& 2& 3& 4& 5& 6& 7& 8& 9\\ ⋮& & & & & & & \downarrow & & \downarrow & \\ ⋮& & & & & & & & & & \\ 1,1& & & & & & & \downarrow & & 0,3810& \\ ⋮& & & & & & & & & & \\ 2,5& & & & & & & 0,4948& & & \\ ⋮& & & & & & & & & & \\ \hline\end{array}\\ & \text{Sehingga nilai}\text{z}=1,18\text{luasnya}=0,3810\\ & \text{Dan nilai}\text{z}=2,56\text{luasnya}=0,4948\\ & \text{maka luas arsiran}\\ & =P(1,18<Z<2,56)\\ & =P(0<Z<2,56)-P(0<Z<1,18)\\ & =0,4948-0,3810\\ & =0,1138\end{array}$.

$\begin{array}{l}\\ 8.& \text{Luas daerah yang diarsir pada gambar}\\ & \text{di bawah adalah 0,9332, maka nilai}\text{z}=....\end{array}$.

$.\begin{array}{l}\\ & \text{a}.1,05\\ & \text{b}.1,06\\ & \text{c}.1,16\\ & \text{d}.1,50\\ & \text{e}.1,60\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Luas arsiran adalah}\\ & =P(Z<z)=0,9332\\ & =0,5+0,4332\\ & =0,5+P(0<Z<z)\\ & \text{lihat/konfirmasi ke tabel}\\ & \text{z}=1,50\\ & \begin{array}{rl}& \text{Perhatikan tabel}\text{distribusi normal}\text{berikut}\\ & \begin{array}{|ccccccccccc|}\hline \text{z}& 0& 1& 2& 3& 4& 5& 6& 7& 8& 9\\ ⋮& \downarrow & & & & & & & & & \\ ⋮& & & & & & & & & & \\ 1,5& 0,4332& & & & & & & & & \\ ⋮& & & & & & & & & & \\ \hline\end{array}\\ & \text{Sehingga luasnya}=0,4878,\text{batas z}=1,50\end{array}\end{array}$.

$\begin{array}{l}\\ 9.& \text{Pada suatu kelas seorang guru matematika}\\ & \text{menyatakan bahwa nilai ulangan mapel }\\ & \text{yang diampunya tidak kurang dari}68\\ & \text{Untuk menentukan uji tersebut, maka guru}\\ & \text{yang bersangkutan memilih 10 anak secara}\\ & \text{acak untuk ditanyai nilai hasil ulangannya}\\ & \text{Hipotesis}{\text{H}}_0\text{dan}{\text{H}}_1\text{yang tepat dari kondisi}\\ & \text{kondisi di atas adalah}....\\ & \text{a}.\begin{array}{c}{\text{H}}_0:\mu =68\\ {\text{H}}_1:\mu \ne 68\end{array}\\ & \text{b}.\begin{array}{c}{\text{H}}_0:\mu =68\\ {\text{H}}_1:\mu >68\end{array}\\ & \text{c}.\begin{array}{c}{\text{H}}_0:\mu \ge 68\\ {\text{H}}_1:\mu <68\end{array}\\ & \text{d}.\begin{array}{c}{\text{H}}_0:\mu \le 68\\ {\text{H}}_1:\mu >q68\end{array}\\ & \text{e}.\begin{array}{c}{\text{H}}_0:\mu >68\\ {\text{H}}_1:\mu \ne 68\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{c}}\\ & \text{Cukup jelas}\\ & \text{Dan ini contoh uji satu pihak, yaitu kiri}\end{array}$.

$\begin{array}{l}\\ 10.& \text{Seorang petugas}customerservice\text{menyatakan}\\ & \text{bahwa di dealer A dapat mensevis rata-rata }\\ & \text{sebanyak}75\text{unit sepeda motor perhari}.\text{Untuk}\\ & \text{mengecek kebenaran pernyataan di atas diambil}\\ & \text{sampel beberapa hari secara random sebanyak }\\ & \text{20 hari. Dari penelitian ini diperoleh rata-rata}\\ & \text{78 unit dengan simpangan bakunya 5. Hasil}\\ & \text{perhitungan}{\text{z}}_0-\text{nya adalah}....\\ & \text{a}.2,35\\ & \text{b}.2,43\\ & \text{c}.2,55\\ & \text{d}.2,68\\ & \text{e}.2,75\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{d}}\\ & \begin{array}{rl}& \text{Diketahui data dianggap berdistribusi}\\ & \text{normal baku}N(0,1)\text{dengan}\\ & \text{Rata-rata sampel}=\bar{\text{x}}=78\text{unit sepeda motor}\\ & \text{Rata-rata populasinya yang diuji}={\mu }_0=75\\ & \text{Simpangan bakunya}=\sigma =5\text{unit}\\ & \text{dengan banyak data sampel}=n=20\text{hari}\\ & \text{Penghitungan nilainya}z-\text{nya}\\ & ={\displaystyle \frac{\bar{\text{x}}-{\mu }_0}{{\displaystyle \frac{\sigma }{\sqrt{n}}}}}\\ & ={\displaystyle \frac{78-75}{{\displaystyle \frac{5}{\sqrt{20}}}}={\displaystyle \frac{3}{{\displaystyle \frac{5}{2\sqrt{5}}}}}}\\ & ={\displaystyle \frac{6}{\sqrt{5}}=\frac{6}{2,236}=2,683}\end{array}\end{array}$.