Statistics Question 7

Question 7 - 31 January - Shift 1

If the variance of the frequency distribution

$\mathbf{x} _{\mathbf{i}}$ 2 3 4 5 6 7 8
Frequency $f_i$ 3 6 16 $\alpha$ 9 5 6

is 3. Then $\alpha$ is equal to _____.

Show Answer

Answer: 5

Solution:

Formula: Probability distribution

$\mathbf{f} _{\mathbf{i}}$ $d_i=$ $\mathbf{x} _i-5$ $\mathbf{f} _{\mathbf{i}} \mathbf{d} _{\mathbf{i}}^{\mathbf{2}}$ $\mathbf{f} _{\mathbf{i}} \mathbf{d} _{\mathbf{i}}$
2 3 -3 27 -9
3 6 -2 24 -12
4 16 -1 16 -16
5 $\alpha$ 0 0 0
6 9 1 9 9
7 5 2 20 10
8 6 3 54 18

$\sigma_x^{2}=\sigma_d^{2}=\frac{\sum f_i d_i{ }^{2}}{\sum f_i}-(\frac{\sum f_i d_i}{\sum f_i})^{2}$

$=\frac{150}{45+\alpha}-0=3$

$\Rightarrow 150=135+3 \alpha$

$\Rightarrow 3 \alpha=15 \Rightarrow \alpha=5$