Listed below are measured amounts of lead (in micrograms per cubic meter, or mug

Question

Listed below are measured amounts of lead (in micrograms per cubic meter, or mug / m^3 ) in the air. An organization has established an air quality standard for lead of 1.5 mug / m^3 . The measurements shown below were recorded at a destroyed building on different days immediately following its destruction. Assume that this sample is a simple random sample obtained from a population with a normal distribution. Use the given values to construct a 90% confidence interval estimate of the standard deviation of the amounts of lead in the air. Is there anything about this data set that suggests that the confidence interval might not be very good? The confidence interval estimate is

Explanation / Answer

Using technology, the sample standard deviation is

s = 1.568665038

As              
              
df = n - 1 =    5          
alpha = (1 - confidence level)/2 =    0.05          
              
Then the critical values for chi^2 are              
              
chi^2(alpha/2) =    11.07049769          
chi^2(alpha/2) =    1.145476226          
              
Thus, as              
              
lower bound = (n - 1) s^2 / chi^2(alpha/2) =    1.111381832          
upper bound = (n - 1) s^2 / chi^2(1 - alpha/2) =    10.74099115          
              
Thus, the confidence interval for the variance is              
              
(   1.111381832   ,   10.74099115   )
              
Also, for the standard deviation, getting the square root of the bounds,              
              
(   1.05422096   ,   3.277345138   ) [ANSWER]

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