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. Find the sample variance s 2 of the sample and find a 95% confidence interval for \sigma 2 and find a 98\% confidence interval for \sigma.

UkusakazaL 2021-08-10 Answered
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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Answers (1)

question2answer
Answered 2021-08-16 Author has 155 answers

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
10261026=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
117520116964=556
5) subtract 1 from the number of items in your data set, 91=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=(n1)s2ψ2right<σ2<(n1)s2ψ2 left where,
s2= variance
ψ2 right =1confidence level2
ψ2 left =1ψ2 right
n= sample size
since α=0.05
ψ2 right =1confidence level2=10.952=0.052=0.025
ψ2 left =1ψ2 right =10.025=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 =[869.517.5345<σ2<869.52.18]
=[55617.5345<σ2<5562.1797]
[31.7089,255.081]
Step 3
(c)
to calculate confidence interval for standard deviation
ci=(n1)s2ψ2 right <σ2<(n1)s2ψ2 left
where,
s= standard deviation
ψ2 right =1confidencevel2

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