I need an actual example to understand this problem better, it would be great if

Question

I need an actual example to understand this problem better, it would be great if it could be a real life example or something that I can easily understand.

One can calculate the 95% confidence interval for the mean with the population standard deviation known. This will give us an upper and a lower confidence limit. What happens if we decide to calculate the 99% confidence interval? Describe how the increase in the confidence level has changed the width of the confidence interval. Do the same for the confidence interval set at 80%. Please include an example with actual numerical values for the intervals.

Explanation / Answer

say we have a sample of 100 students, and they take an average of 15 minutes with population standard deviation of 5 minutes to get to school.

For 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 =    15          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    5          
n = sample size =    100          
              
Thus,              
Margin of Error E =    0.979981992          
Lower bound =    14.02001801          
Upper bound =    15.97998199          
              
Thus, the confidence interval is              
              
(   14.02001801   ,   15.97998199   )   
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FOR 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 =    15          
z(alpha/2) = critical z for the confidence interval =    2.575829304          
s = sample standard deviation =    5          
n = sample size =    100          
              
Thus,              
Margin of Error E =    1.287914652          
Lower bound =    13.71208535          
Upper bound =    16.28791465          
              
Thus, the confidence interval is              
              
(   13.71208535   ,   16.28791465   )

*********************************************

As we can see, the 99% confidence interval is wider (larger margin of error) than the 95% confidence interval. [ANSWER]

*******************************************

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 =    0.1          
X = sample mean =    15          
z(alpha/2) = critical z for the confidence interval =    1.281551566          
s = sample standard deviation =    5          
n = sample size =    100          
              
Thus,              
Margin of Error E =    0.640775783          
Lower bound =    14.35922422          
Upper bound =    15.64077578          
              
Thus, the confidence interval is              
              
(   14.35922422   ,   15.64077578   )

***************************************************

As we can see, the 80% confidence interval is the narrowest, next is 95%, and the widest interval is 99% confidence interval.

Hence, as confidence level increases, the confidence interval becomes wider. [ANSWER]

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