Construct the indicated confidence intervals for (a) the population variance $$\sigma ^ { 2 }$$ and (b) the population standard deviation $$\sigma$$.

Wotzdorfg 2021-05-16 Answered

Construct the indicated confidence intervals for (a) the population variance \(\sigma^{2}\) and (b) the population standard deviation \(\sigma\). Assume the sample is from a normally distributed population. c = 0.90, s = 35, n = 18

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comentezq
Answered 2021-05-17 Author has 19351 answers
Determine the critical values usinh table 6 with df=n-1=18-1=17:
\(\displaystyle{\chi_{{{1}-{0.05}}}^{{2}}}={\chi_{{{0.95}}}^{{2}}}={8.672}\)
\(\displaystyle{\chi_{{{0.05}}}^{{2}}}={27.587}\)
The boundaries of the confidence internal for the standart deviation are then:
\(\displaystyle\sqrt{{\frac{{{n}-{1}}}{{{\chi_{{\frac{\alpha}{{2}}}}^{{2}}}}}}}{s}=\sqrt{{\frac{{{18}-{1}}}{{{27.587}}}}}{35}\approx{27.475}\)
\(\displaystyle\sqrt{{\frac{{{n}-{1}}}{{{\chi_{{\frac{\alpha}{{2}}}}^{{2}}}}}}}{s}=\sqrt{{\frac{{{18}-{1}}}{{{8.762}}}}}{35}\approx{49.004}\) The boundaries of the confidence intervals for the stadart deviation:
\(\displaystyle{\left({27.475}^{{{2}}},{49.004}^{{{2}}}\right)}={\left({754.876},{2401.382}\right)}\)
Variance:(754.876,2401.392)
Standart deviation: (27.475,49.004)
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