Question

A certain process controller in your plant has been known to develop problems over time. This typically manifests as an increase in temperature variability. The typical process standard deviation when thecontroller works properly is 1.5 degrees. When the standard deviation exceeds this, the controller is taken offline and repaired. To avoid unnecessary downtime, you need to do a statistical test that only takes the controller down if you are 95% confident the variability is too high. Construct a hypothesis test that tests whether the variation is toohigh, then use the given data to evaluate the process. Should the controller be repaired?

n = 15

s= 1.7

Answer 1

Sample Size,   n=   15
Sample Standard Deviation,   s=   1.5
Confidence Level,   CL=   0.95
      
      
Degrees of Freedom,   DF=n-1 =    14
alpha,   α=1-CL=   0.05
alpha/2 ,   α/2=   0.025
Lower Chi-Square Value=   χ²1-α/2 =   5.6287
Upper Chi-Square Value=   χ²α/2 =   26.1189

  
upper bound= (n-1)s²/χ²1-α/2 =   14*1.5² / 5.6287=   5.596

95% confidence interval for std dev is       
  
upper bound=   √(upper bound =   2.37

if std dev exceeds 2.37, then you are 95% confident the variability is too high

........................

      

Ho :   σ =   1.5
Ha :   σ >   1.5
      
Level of Significance ,    α =    0.05
sample Std dev ,    s =    1.7
Sample Size ,   n =    15
      
Chi-Square Statistic   X² = (n-1)s²/σ² =    17.982
      
degree of freedom,   DF=n-1 =    14
      
one tail test   

   
Upper Critical Value   =   23.6848
p-Value   =   0.2076

p value >0.05
      
Do not reject the null hypothesis      

controller need not to be repaired

please revert back for doubt