A model of heat transfer from a cylinder immersed in a liquid predicts that the

Question

A model of heat transfer from a cylinder immersed in a liquid predicts that the heat transfer coefficient for the cylinder will become constant at very low flow rates of the fluid. A sample of 10 measurement is taken. The result, in W/(m^2 K), are: 14.5 12.3 13.5 14.4 13.8 14.5 15.0 12.3 12.7 11.9 Assume that the variability is known to be about 1.21. Find a 95 percent confidence interval for the heat transfer coefficient. How large of a sample would be required to find an interval that is correct to within 0.3 with probability 0.98?

Explanation / Answer

A)

Note that              
              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    13.49          
t(alpha/2) = critical t for the confidence interval =    2.262157163          
s = sample standard deviation =    1.116990003          
n = sample size =    10          
df = n - 1 =    9          
Thus,              
              
Lower bound =    12.69095349          
Upper bound =    14.28904651          
              
Thus, the confidence interval is              
              
(   12.69095349   ,   14.28904651   ) [ANSWER]

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b)

Note that      
      
n = z(alpha/2)^2 s^2 / E^2      
      
where      
      
alpha/2 = (1 - confidence level)/2 =    0.01  
      
Using a table/technology,      
      
z(alpha/2) =    2.326347874  
      
Also,      
      
s = sample standard deviation =    1.116990003  
E = margin of error =    0.3  
      
Thus,      
      
n =    75.02489207  
      
Rounding up,      
      
n =    76   [ANSWER]

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