Unemployment rate

Over the last 16 years, the country's unemployment rate has changed according to the following frequency table:
years of unemployment: 2 5 2 3 3 1
unemployment rate: 0.5 1 1.5 2 2.5 3 in % (percent).

Determine the two-sided confidence interval for the variance DN with a confidence level of 0.95.

Correct answer:

a =  1.224 %
b =  1.9635 %

Step-by-step explanation:

$$\begin{gathered} n=2+5+2+3+3+1=16 \\ \alpha =0.95 \\ \mu =\frac{2\cdot 0.5+5\cdot 1+2\cdot 1.5+3\cdot 2+3\cdot 2.5+1\cdot 3}{n}=\frac{2\cdot 0.5+5\cdot 1+2\cdot 1.5+3\cdot 2+3\cdot 2.5+1\cdot 3}{16}=\frac{51}{32}\doteq 1.5938 \\ {E}_2=2\cdot (0.5-\mu {)}^2+5\cdot (1-\mu {)}^2+2\cdot (1.5-\mu {)}^2+3\cdot (2-\mu {)}^2+3\cdot (2.5-\mu {)}^2+1\cdot (3-\mu {)}^2=2\cdot (0.5-\frac{51}{32}{)}^2+5\cdot (1-\frac{51}{32}{)}^2+2\cdot (1.5-\frac{51}{32}{)}^2+3\cdot (2-\frac{51}{32}{)}^2+3\cdot (2.5-\frac{51}{32}{)}^2+1\cdot (3-\frac{51}{32}{)}^2=\frac{583}{64}\doteq 9.1094 \\ V=E_2/n=\frac{583}{64}/16=\frac{583}{64\cdot 16}=\frac{583}{1024}\doteq 0.5693 \\ \sigma =\sqrt{V}=\sqrt{0.5693}\doteq 0.7545 \\ \alpha =0.95 \\ {\alpha }_2=1-(1-0.95)/2=\frac{39}{40}=0.975 \\ u(0.975)=1.96 \\ u=1.96 \\ r=u\cdot \sigma /\sqrt{n}=1.96\cdot 0.7545/\sqrt{16}\doteq 0.3697 \\ a=\mu -r=1.5938-0.3697=1.224\% \end{gathered}$$

$b=\mu +r=1.5938+0.3697=1.9635\%$

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