Let X1, ⋯,Xn be a random sample from a normal distribution with known mean μ and unknown variance σ2. Three possible confidence intervals for σ2 area) (∑i=1n(Xi−X―)2a1, ∑i=1n(Xi−X―)2a2)b) (∑i=1n(Xi−μ)2b1, ∑i=1n(Xi−μ)2b2)c) (n(X―−μ)2c1, n(X―−μ)2c2)where a1, a2, b1, b2, c1, c2 are constants.Find values of these six constants which give confidence level 0.90 for each of the three intervals when n=10 and compare the expected widths of the tree intervels in this caseWith σ2=1, what value of n is required to achieve a 90% confidence interval of expected width less than 1 in cases (b) and (c) above?

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Let X1, ⋯,Xn be a random sample from a normal distribution with known mean μ and unknown variance σ2. Three possible confidence intervals for σ2 area) (∑i=1n(Xi−X―)2a1, ∑i=1n(Xi−X―)2a2)b) (∑i=1n(Xi−μ)2b1, ∑i=1n(Xi−μ)2b2)c) (n(X―−μ)2c1, n(X―−μ)2c2)where a1, a2, b1, b2, c1, c2 are constants.Find values of these six constants which give confidence level 0.90 for each of the three intervals when n=10 and compare the expected widths of the tree intervels in this caseWith σ2=1, what value of n is required to achieve a 90% confidence interval of expected width less than 1 in cases (b) and (c) above?

Leonardo Mcpherson · Accepted Answer

Step 1X1,…,Xn&amp;nbsp;are i.i.d.&amp;nbsp;N(μ,σ2)&amp;nbsp;random variables andX―=n−1∑i=1nXi&amp;nbsp;The distributions and confidence intervals are as follow.a)&amp;nbsp;1σ2∑i=1n(Xi−X―)2∼χn−12andPrl(∑i=1n(Xi−X―)2χn−1,α22≤σ2≤∑i=1n(Xi−X―)2χn−1,1−α22r)=1−α.b)&amp;nbsp;1σ2∑i=1n(Xi−μ)2∼χn2andPrl(∑i=1n(Xi−μ)2χn,α22≤σ2≤∑i=1n(Xi−μ)2χn,1−α22r)=1−α.c)&amp;nbsp;n(X―−μ)2σ2∼χ12andPrl(n(X―−μ)2χ1,α22≤σ2≤n(X―−μ)2χ1,1−α22r)=1−α.I hope this helps.

kachnaemra · Accepted Answer

Step 1I believe you have the denominators switched. Also, say c1=χ{1,1−α2}2 and c2=χ{1,α2}2 then for (c), the expected width of the interval isE[n(X―−μ)2(1c2−1c1)]=n(1c2−1c1)×E(X―−μ)2=n(1c2−1c1);;×σ2n=(1c2−1c1)σ2since nσ2(X―−μ)∼χ12and thus E(X―−μ)2=σ2nstrange the expected width does not depend on n.

Let \(\displaystyle{X}_{{{1}}},\ \cdots,{X}_{{{n}}}\) be a random sample

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