stat A civil engineer is analyzing Me compressive strength o

A civil engineer is analyzing Me compressive strength of concrete. Completive Strength Is approximately normally distributed with variance 1000 mil. A random Amu\'s% of 12 specimens has a mean corn Preserves strength a --e.32,42 pal. Construct a 95% two-sided Cr on mean compressive strength. Construct a 99% two-sided Cl on mean compressive strength. And compare the width of this confidence interval with the width Al the one fOund In part al make comments.

Solution

a)

Note that  
          
Margin of Error E = t(alpha/2) * s / sqrt(n)              
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 =    3255.42          
t(alpha/2) = critical t for the confidence interval =    2.20098516          
s = sample standard deviation = sqrt(1000) =   31.6227766          
n = sample size =    12          
df = n - 1 =    11          
Thus,              
Margin of Error E =    20.09215368          
Lower bound =    3235.327846          
Upper bound =    3275.512154          
              
Thus, the confidence interval is              
              
(   3235.327846   ,   3275.512154   ) [ANSWER]

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b)

Note that              
Margin of Error E = t(alpha/2) * s / sqrt(n)              
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 =    0.005          
X = sample mean =    3255.42          
t(alpha/2) = critical t for the confidence interval =    3.105806516          
s = sample standard deviation = sqrt(1000) =   31.6227766          
n = sample size =    12          
df = n - 1 =    11          
Thus,              
Margin of Error E =    28.3520048          
Lower bound =    3227.067995          
Upper bound =    3283.772005          
              
Thus, the confidence interval is              
              
(   3227.067995   ,   3283.772005   ) [ANSWER]

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The 99% confidence interval is wider. This is becasue the critical t value is larger, making the margin of error larger as well. This makes sense because you have to expand the interval in order to be \"more confident\" that you got the true mean inside your interval.