To monitor complex chemical processes, chemical engineers will consider key process indicators, which may be just yield but most often depend on several quantities. Before trying to improve a process, 9 measurements were made on a key performance indicator. 123, 106, 114, 128, 113, 109, 120, 102, 111. Assume that the key performance indicator has a normal distribution. (a) Find the sample variance s 2 of the sample. (b) Find a 95% confidence interval for σ2. (c) Find a 98% confidence interval for σ.

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To monitor complex chemical processes, chemical engineers will consider key process indicators, which may be just yield but most often depend on several quantities. Before trying to improve a process, 9 measurements were made on a key performance indicator. 123, 106, 114, 128, 113, 109, 120, 102, 111. Assume that the key performance indicator has a normal distribution. (a) Find the sample variance s 2 of the sample. (b)  Find a 95% confidence interval for σ2. (c) Find a 98% confidence interval for σ.

Ian Adams · Accepted Answer

Step 1 (a) to find variance value 1) add them up 123+106+114+128+113+109+120+102+111=1026 2) square your answer 1026⋅1026=1052676 ...and divide by the number of items. we have 9 items , 10526769=116964 set this number aside for a moment. 3) take your set of original numbers from step 1, and square them individually this time 1232+1062+1142+1282+1132+1092+1202+1022+1112=117520 4) subtract the amount in step 2 from the amount in step 3 117520−116964=556 5) subtract 1 from the number of items in your data set, 9−1=8 6) divide the number in step 4 by the number in step 5. this gives you the variance 5568=69.5 ---------------------------------------------------------------------------------------- to find standard deviation value take the square root of your answer from step 6. this gives you the standard deviation 8.3367 Step 2 (b) CONFIDENCE INTERVAL FOR VARIANCE ci=(n−1)s2ψ2right&amp;lt;σ2&amp;lt;(n−1)s2ψ2 left where, s2= variance ψ2 right =1−confidence level2 ψ2 left =1−ψ2 right n= sample size since α=0.05 ψ2 right =1−confidence level2=1−0.952=0.052=0.025 ψ2 left =1−ψ2 right =1−0.025=0.975 the two critical values ψ2 left, ψ2 right at 8 df are 17.5345 , 2.18 variance (s2)=69.5 sample size (n)=9 confidence interval =[8⋅69.517.5345&amp;lt;σ2&amp;lt;8⋅69.52.18] =[55617.5345&amp;lt;σ2&amp;lt;5562.1797] [31.7089,255.081] Step 3 (c) to calculate confidence interval for standard deviation ci=(n−1)s2ψ2 right &amp;lt;σ2&amp;lt;(n−1)s2ψ2 left where, s= standard deviation ψ2 right =1−confidence≤vel2

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