Soalan 6:
Jisim suatu kumpulan yang terdiri daripada lapan orang murid mempunyai min 45 kg dan varians 2.5 kg2. Hitung
(a) hasil tambah jisim bagi lapan orang murid ini.
(b) hasil tambah kuasa dua jisim murid ini.
Penyelesaian:
(a)
$$ \begin{aligned} \text { Min } & =\frac{\sum x}{N} \\ 45 & =\frac{\sum x}{8} \\ \sum x & =360 \end{aligned} $$
(b)
$$ \begin{aligned} \text { Varians } & =\frac{\sum x^2}{N}-(\bar{x})^2 \\ 2.5 & =\frac{\sum x^2}{8}-45^2 \\ \frac{\sum x^2}{8} & =2.5+45^2 \\ \frac{\sum x^2}{8} & =2027.5 \\ \sum x^2 & =16220 \end{aligned} $$
Jisim suatu kumpulan yang terdiri daripada lapan orang murid mempunyai min 45 kg dan varians 2.5 kg2. Hitung
(a) hasil tambah jisim bagi lapan orang murid ini.
(b) hasil tambah kuasa dua jisim murid ini.
Penyelesaian:
(a)
$$ \begin{aligned} \text { Min } & =\frac{\sum x}{N} \\ 45 & =\frac{\sum x}{8} \\ \sum x & =360 \end{aligned} $$
(b)
$$ \begin{aligned} \text { Varians } & =\frac{\sum x^2}{N}-(\bar{x})^2 \\ 2.5 & =\frac{\sum x^2}{8}-45^2 \\ \frac{\sum x^2}{8} & =2.5+45^2 \\ \frac{\sum x^2}{8} & =2027.5 \\ \sum x^2 & =16220 \end{aligned} $$
Soalan 7:
Min bagi suatu set nombor (m – 4), m, (m + 2), 2m, (2m + 3) ialah 10.
(a) Hitung,
(i) nilai m
(ii) sisihan piawai
(b) Setiap nombor dalam set tersebut didarab dengan 3 dan kemudian ditambah dengan 2. Hitung varians bagi set data yang baharu.
Penyelesaian:
(a)(i)
$$ \begin{aligned} & \operatorname{Min}=10 \\ & \begin{array}{l} \frac{(m-4)+m+(m+2)+2 m+(2 m+3)}{5}=10 \\ 7 m+1=50 \\ \quad 7 m=49 \\ \quad m=7 \end{array} \end{aligned} $$
(a)(ii)
$$ \begin{aligned} & m=7 \\ & \begin{aligned} \text { Maka set nombor } & =(7-4), 7,(7+2), 2(7),[2(7)+3] \\ & =3,7,9,14,17 \end{aligned} \end{aligned} $$
$$ \begin{aligned} &\text { Sisihan piawai, } \sigma=\sqrt{\frac{\sum x^2}{N}-(\bar{x})^2}\\ &\begin{aligned} & =\sqrt{\frac{3^2+7^2+9^2+14^2+17^2}{5}-10^2} \\ & =\sqrt{\frac{624}{5}-10^2} \\ & =\sqrt{24.8} \\ & =4.980 \end{aligned} \end{aligned} $$
(b)
$$ \begin{aligned} & \text { Set nombor baharu } \\ & =(3 \times 3+2),(7 \times 3+2),(9 \times 3+2),(14 \times 3+2),(17 \times 3+2) \\ & =11,23,29,44,53 \end{aligned} $$
$$ \begin{aligned} & \text { Varians baharu }=\frac{\sum x^2}{N}-(\bar{x})^2 \\ & =\frac{11^2+23^2+29^2+44^2+53^2}{5}-\left(\frac{11+23+29+44+53}{5}\right)^2 \\ & =1247.2-1024 \\ & =223.2 \end{aligned} $$
Min bagi suatu set nombor (m – 4), m, (m + 2), 2m, (2m + 3) ialah 10.
(a) Hitung,
(i) nilai m
(ii) sisihan piawai
(b) Setiap nombor dalam set tersebut didarab dengan 3 dan kemudian ditambah dengan 2. Hitung varians bagi set data yang baharu.
Penyelesaian:
(a)(i)
$$ \begin{aligned} & \operatorname{Min}=10 \\ & \begin{array}{l} \frac{(m-4)+m+(m+2)+2 m+(2 m+3)}{5}=10 \\ 7 m+1=50 \\ \quad 7 m=49 \\ \quad m=7 \end{array} \end{aligned} $$
(a)(ii)
$$ \begin{aligned} & m=7 \\ & \begin{aligned} \text { Maka set nombor } & =(7-4), 7,(7+2), 2(7),[2(7)+3] \\ & =3,7,9,14,17 \end{aligned} \end{aligned} $$
$$ \begin{aligned} &\text { Sisihan piawai, } \sigma=\sqrt{\frac{\sum x^2}{N}-(\bar{x})^2}\\ &\begin{aligned} & =\sqrt{\frac{3^2+7^2+9^2+14^2+17^2}{5}-10^2} \\ & =\sqrt{\frac{624}{5}-10^2} \\ & =\sqrt{24.8} \\ & =4.980 \end{aligned} \end{aligned} $$
(b)
$$ \begin{aligned} & \text { Set nombor baharu } \\ & =(3 \times 3+2),(7 \times 3+2),(9 \times 3+2),(14 \times 3+2),(17 \times 3+2) \\ & =11,23,29,44,53 \end{aligned} $$
$$ \begin{aligned} & \text { Varians baharu }=\frac{\sum x^2}{N}-(\bar{x})^2 \\ & =\frac{11^2+23^2+29^2+44^2+53^2}{5}-\left(\frac{11+23+29+44+53}{5}\right)^2 \\ & =1247.2-1024 \\ & =223.2 \end{aligned} $$
Soalan 8:
Jadual di bawah menunjukkan maklumat bagi nilai n, Σx dan Σx2 bagi suatu set data.
(a) Hitung varians.
(b) Satu nombor p ditambah kepada set data ini dan didapati min bertambah sebanyak 0.5. Hitung,
(i) nilai p
(ii) sisihan piawai bagi set data yang baharu.
Penyelesaian:
(a)
$$ \begin{aligned} \sigma^2 & =\frac{\sum x^2}{N}-\bar{x}^2 \\ & =\frac{1452}{12}-\left(\frac{66}{12}\right)^2 \\ & =121-5.5^2 \\ & =90.75 \end{aligned} $$
(b)(i)
$$ \begin{gathered} \min \text { baharu }=5.5+0.5 \\ \begin{array}{c} \frac{66+p}{12+1}=6 \\ 66+p=78 \\ p=12 \end{array} \end{gathered} $$
(b)(ii)
$$ \begin{aligned} &\text { Sisihan piawai baru, }\\ &\begin{aligned} \sigma & =\sqrt{\frac{\sum x^2}{N}-\bar{x}^2} \\ & =\sqrt{\frac{1452+12^2}{13}-6^2} \\ & =\sqrt{122.77-36} \\ & =\sqrt{86.77} \\ & =9.315 \end{aligned} \end{aligned} $$
Jadual di bawah menunjukkan maklumat bagi nilai n, Σx dan Σx2 bagi suatu set data.
(a) Hitung varians.
(b) Satu nombor p ditambah kepada set data ini dan didapati min bertambah sebanyak 0.5. Hitung,
(i) nilai p
(ii) sisihan piawai bagi set data yang baharu.
Penyelesaian:
(a)
$$ \begin{aligned} \sigma^2 & =\frac{\sum x^2}{N}-\bar{x}^2 \\ & =\frac{1452}{12}-\left(\frac{66}{12}\right)^2 \\ & =121-5.5^2 \\ & =90.75 \end{aligned} $$
(b)(i)
$$ \begin{gathered} \min \text { baharu }=5.5+0.5 \\ \begin{array}{c} \frac{66+p}{12+1}=6 \\ 66+p=78 \\ p=12 \end{array} \end{gathered} $$
(b)(ii)
$$ \begin{aligned} &\text { Sisihan piawai baru, }\\ &\begin{aligned} \sigma & =\sqrt{\frac{\sum x^2}{N}-\bar{x}^2} \\ & =\sqrt{\frac{1452+12^2}{13}-6^2} \\ & =\sqrt{122.77-36} \\ & =\sqrt{86.77} \\ & =9.315 \end{aligned} \end{aligned} $$