Question

You randomly sample a variable from a population. Your sampled values are 1, 3, 4, 4 You compute the sample variance 82. What is s, rounded to the nearest O.x.

Answer 1

Here, we're given the sample- 1,3,4,4
We're asked to find sample variance and sample s.d.-

To, obtain the sample variance, first we've to obtain the sample mean:

C ΙΣ 2 n 1=1

where,
$x_{i}:$ ith sample value, for all i=1(1)4
n: sample size
$\therefore \overline{x}=\frac{1+3+4+4}{4}=\frac{12}{4}=3$
Now,
We're to obtain the Sample Variance(s² )-

Formula for Sample Variance is given by-

$s^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}$

  
1 Σα; – 3)? 4 - 1 1=1

  $=\frac{1}{4-1}\left \{ (1-3)^{2}+(3-3)^{2}+(4-3)^{2}+(4-3)^{2} \right \}$
$=\frac{1}{4-1}\left \{ 6 \right \}$
  $=\frac{6}{3}$
  $=2$

$\therefore$ Sample Variance is=2

As we know if we take the square root of the Sample variance we'll get the Sample Standard Deviation(s).

Formula for Sample Standard Deviation is given by-

$\therefore s=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}$
$\Rightarrow s=\sqrt{s^{2}}$
$\Rightarrow s=\sqrt{2}$
$\Rightarrow s=1.4$

$\therefore$ Sample Standard Deviation=1.4

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