First slide

Introduction to statistics

Question

The first of two samples has 100 items with mean 15 and SD = 3. If the whole group has 250 items with mean 15.6 and $SD = \sqrt{13.44}$ , the SD of the second group is

Moderate

A

4
B

5
C

6
D

3.52

Solution

We know , $σ^{2} = \frac{n_{1} (σ_{1}^{2} + d_{1}^{2}) + n_{2} (σ_{2}^{2} + d_{2}^{2})}{n_{1} + n_{2}}$
where $d_{1} = m_{1} - a, d_{2} = m_{2} - a, a$ being the mean of the whole group.
$\begin{array}{l} ∴ 15 .6 = \frac{100 \times 15 + 150 \times m_{2}}{250} \Rightarrow m_{2} = 16 \\ Thus, 13 .44 = \frac{[\begin{matrix} (100 \times 9 + 150 \times σ^{2}) + \\ 100 \times (0 .6)^{2} + 150 \times (0 .4)^{2} \end{matrix}]}{250} \Rightarrow σ = 4 \end{array}$

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