From the combined standard deviation, with the mean, and standard
deviation of the data sets, the required variance can be found.
Reasons:
The given data information are;
Mean of the first set of observation,
= 9.0
Standard deviation of the the first set, σ₁ = 4.0
Sample size of second observation, n₂ = 20
Mean of second sample,
= 10
Standard deviation of the combined set, σ₁₂ = 3.0
Sample size of combined set of observation, n₂ = 35
Solution:
The number of elements in the second sample, n₁ = 35 - 20 = 15
The standard deviation of the combined set is given by the formula;
![{\sigma_{12}} = \mathbf{\sqrt{\dfrac{n_1 \cdot \left ( \sigma^2_{1}-d_1^2 \right ) + n_2 \cdot \left ( \sigma^2_{2}-d_2^2 \right )}{n_{1}+n_{2}}}}](https://tex.z-dn.net/?f=%7B%5Csigma_%7B12%7D%7D%20%3D%20%5Cmathbf%7B%5Csqrt%7B%5Cdfrac%7Bn_1%20%5Ccdot%20%5Cleft%20%28%20%5Csigma%5E2_%7B1%7D-d_1%5E2%20%5Cright%20%29%20%2B%20n_2%20%5Ccdot%20%5Cleft%20%28%20%5Csigma%5E2_%7B2%7D-d_2%5E2%20%5Cright%20%29%7D%7Bn_%7B1%7D%2Bn_%7B2%7D%7D%7D%7D)
Where;
![\overline x_{12} =\mathbf{\dfrac{n_1 \cdot \overline x_1 +n_2 \cdot \overline x_2 }{n_1 + n_2}}](https://tex.z-dn.net/?f=%5Coverline%20x_%7B12%7D%20%3D%5Cmathbf%7B%5Cdfrac%7Bn_1%20%5Ccdot%20%5Coverline%20x_1%20%2Bn_2%20%5Ccdot%20%5Coverline%20x_2%20%20%7D%7Bn_1%20%2B%20n_2%7D%7D)
Therefore;
![\overline x_{12} =\dfrac{15 \times 9.0 +20 \times10.0 }{15 + 20} = \dfrac{67}{7}](https://tex.z-dn.net/?f=%5Coverline%20x_%7B12%7D%20%3D%5Cdfrac%7B15%20%5Ctimes%209.0%20%2B20%20%5Ctimes10.0%20%7D%7B15%20%2B%2020%7D%20%3D%20%5Cdfrac%7B67%7D%7B7%7D)
d₁ =
- ![\mathbf{\overline x_1}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Coverline%20x_1%7D)
![d_1 = \dfrac{67}{7} - 9.0 = \dfrac{4}{7}](https://tex.z-dn.net/?f=d_1%20%3D%20%5Cdfrac%7B67%7D%7B7%7D%20-%209.0%20%3D%20%5Cdfrac%7B4%7D%7B7%7D)
d₂ =
-
![d_2 = \dfrac{67}{7} - 10.0 = -\dfrac{3}{7}](https://tex.z-dn.net/?f=d_2%20%3D%20%5Cdfrac%7B67%7D%7B7%7D%20-%2010.0%20%3D%20-%5Cdfrac%7B3%7D%7B7%7D)
Therefore;
![3=\sqrt{\dfrac{15 \times \left ( 4.0^2-\left(\dfrac{4}{7}\right) ^2 \right ) + 20 \cdot \left ( \sigma^2_{2}-\left(-\dfrac{3}{7}\right) ^2 \right )}{15+20}}](https://tex.z-dn.net/?f=3%3D%5Csqrt%7B%5Cdfrac%7B15%20%5Ctimes%20%5Cleft%20%28%204.0%5E2-%5Cleft%28%5Cdfrac%7B4%7D%7B7%7D%5Cright%29%20%5E2%20%5Cright%20%29%20%2B%2020%20%5Ccdot%20%5Cleft%20%28%20%5Csigma%5E2_%7B2%7D-%5Cleft%28-%5Cdfrac%7B3%7D%7B7%7D%5Cright%29%20%5E2%20%5Cright%20%29%7D%7B15%2B20%7D%7D)
Solving gives;
![Variance, \ \sigma^2_2 = \mathbf{\dfrac{{117} }{28}}](https://tex.z-dn.net/?f=Variance%2C%20%5C%20%5Csigma%5E2_2%20%3D%20%5Cmathbf%7B%5Cdfrac%7B%7B117%7D%20%7D%7B28%7D%7D)
![\mathrm{The \ variance \ of \ the \ second \ set, \ \sigma_2^2} = \dfrac{{117} }{28} = 4\dfrac{5}{28}](https://tex.z-dn.net/?f=%5Cmathrm%7BThe%20%5C%20variance%20%5C%20%20of%20%5C%20%20the%20%5C%20%20second%20%5C%20%20set%2C%20%5C%20%20%5Csigma_2%5E2%7D%20%3D%20%20%5Cdfrac%7B%7B117%7D%20%7D%7B28%7D%20%3D%204%5Cdfrac%7B5%7D%7B28%7D)
Learn more here:
brainly.com/question/16869429