One can calculate the 95 confidence interval for the mean wi

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 explain this to me and include an example with actual numerical values for the intervals.

Solution

Say we have a sample mean of 100, standard deviation of 30, and sample size of 50.

For 95% confidence:

Note that              
              
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 =    100          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    30          
n = sample size =    50          
              
Thus,              
              
Lower bound =    91.68457705          
Upper bound =    108.3154229          
              
Thus, the confidence interval is              
              
(   91.68457705   ,   108.3154229   ) [ANSWER]

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For a 99% confidence:

Note that              
              
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 =    100          
z(alpha/2) = critical z for the confidence interval =    2.575829304          
s = sample standard deviation =    30          
n = sample size =    50          
              
Thus,              
              
Lower bound =    89.07168179          
Upper bound =    110.9283182          
              
Thus, the confidence interval is              
              
(   89.07168179   ,   110.9283182   ) [ANSWER]

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As we can see, the 99% confidence interval is WIDER than that of 95%.

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For 80% confidence:

Note that              
              
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 =    100          
z(alpha/2) = critical z for the confidence interval =    1.281551566          
s = sample standard deviation =    30          
n = sample size =    50          
              
Thus,              
              
Lower bound =    94.56283719          
Upper bound =    105.4371628          
              
Thus, the confidence interval is              
              
(   94.56283719   ,   105.4371628   ) [ANSWER]

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As you can see, this is NARROWER than the 95% and 99% confidence intervals.

Thus, we see that as confidence level increases, the confidence interval WIDENS.