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Question : 3 of 16

Marks: +1, -0

The mean and variance of the observations

x_1, x_2, x_3, \ldots, x_{15}

are respectively 2 and 4. If the mean and variance of the observations

y_1, y_2, \ldots, y_{10}

are respectively 2 and 5 , then the variance of the observations

x_1, x_2, \ldots, x_{15}, y_1, y_2, \ldots, y_{10}

[AP EAPCET 23 May 2025 S1]

6.5
5.3
3.4
4.4

Solution:

Use formula for combined variance

Let

n_1=15, \mu_1=2, \sigma_1^2=4

n_2=10, \mu_2=2, \sigma_2^2=5

\; \;\text{ Combined mean }\; \; = \mu = \; \frac{n_1 \mu_1 + n_2 \mu_2}{n_1+n_2}

\; = \frac{15 \times 2 + 10 \times 2}{25} = 2

Combined variance

= \sigma^2

= \; \frac{n_1 \sigma_1^2 + n_2 \sigma_2^2 + n_1 (\mu_1-\mu)^2 + n_2 (\mu_2-\mu)^2}{n_1+n_2}

Since,

\mu_1 = \mu_2 =

the last two terms becomes zero.

So,

\sigma^2 = \; \frac{15 \times 4 + 10 \times 5}{25} = \; \frac{60+50}{25} = \; \frac{110}{25}

\; \text{ = }\; 4.4

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