A sample of 225 elements from a population with a standard d

A sample of 225 elements from a population with a standard deviation of 75 is selected. Sample mean is 180. Find 90%, 95% and 99% confidence interval for mu. What is observed about confidence interval as you increase confidence level?

Solution

a)

90% confidence:

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.05          
X = sample mean =    180          
z(alpha/2) = critical z for the confidence interval =    1.644853627          
s = sample standard deviation =    75          
n = sample size =    225          
              
Thus,              
Margin of Error E =    8.224268135          
Lower bound =    171.7757319          
Upper bound =    188.2242681          
              
Thus, the confidence interval is              
              
(   171.7757319   ,   188.2242681   ) [ANSWER]

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b)

95% confidence:

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    180          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    75          
n = sample size =    225          
              
Thus,              
Margin of Error E =    9.799819923          
Lower bound =    170.2001801          
Upper bound =    189.7998199          
              
Thus, the confidence interval is              
              
(   170.2001801   ,   189.7998199   ) [ANSWER]

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c)

99% confidence:

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.005          
X = sample mean =    180          
z(alpha/2) = critical z for the confidence interval =    2.575829304          
s = sample standard deviation =    75          
n = sample size =    225          
              
Thus,              
Margin of Error E =    12.87914652          
Lower bound =    167.1208535          
Upper bound =    192.8791465          
              
Thus, the confidence interval is              
              
(   167.1208535   ,   192.8791465   ) [ANSWER]

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The confidence interval becomes wider as confidence level increases.