Essentials of Mathematical Probability and Statistics

As with the confidence intervals above for proportions, the Central Limit Theorem also allows you to create an interval centered on a sample mean for estimating the population mean $\mu \text{.}$

Enjoy this interactive calculator for confidence intervals for the mean.

    

        
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# Confidence Interval for an unknown mean
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print("Calculator for Confidence Interval for an unknown mean")
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@interact
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def _(conf=input_box(95,width=10,label='confidence level (in %) = '),
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     samplemean =input_box(default=55,width=10,label='sample mean'),
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     stdev = input_box(default=8,width=10,label='standard deviation'),
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     n = input_box(default=100,width=10,label='sample size')):
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    T = RealDistribution('gaussian', 1)   # use built-in
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    alpha = 1-conf/100
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    z0 = T.cum_distribution_function_inv(conf/100+alpha/2).n(digits=5)   # presuming a two-sided confidence interval  
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#  Need to consider here whether to use z or t distribution based upon n and stdev
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    E = z0*stdev/sqrt(n)
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    low = samplemean-E
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    high = samplemean+E
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    pretty_print("z-score =",z0," and std dev =",stdev," gives ",[low, high])
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    G = points([(samplemean,0)],size=50,color='red')
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    G += points([(low,0),(high,0)],size=50,color='blue')
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    G += line([(low,0),(samplemean,0)],color='green',thickness=3)
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    G += line([(high,0),(samplemean,0)],color='green',thickness=3)  
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    show(G,figsize=(5,1),xmin=samplemean-stdev/2,xmax=samplemean+stdev/2)

    
    
    Evaluate (Sage)
    

        

            
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        Messages
        


    

It should be noted that the use of the Central Limit Theorem makes the use of InvNorm an approximation. It can be shown that so long as n is larger than 30 then generally this approximation is reasonable. If not, then use replace the z-score with a corresponding value from the t-distribution.

Further, note that the confidence intervals for the mean before presumed that $\sigma$ was known. However, one might not have this value and therefore as estimate could be optained by using a sample standard deviation. In this case, using a t-statistic rather than a z-statistic is also justified.

Here is an interactive cell illustrating confidence intervals for the mean using the t-score.

    

        
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# Confidence Interval for an unknown mean using student t
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print("Calculator for Confidence Interval for an unknown mean if n small or using sample standard deviation")
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@interact
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def _(conf=input_box(95,width=10,label='confidence level (in %) = '),
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     samplemean =input_box(default=55,width=10,label='sample mean'),
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     stdev = input_box(default=8,width=10,label='standard deviation'),
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     n = input_box(default=100,width=10,label='sample size')):
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    T = RealDistribution('t', n)   # use built-in student t distribution
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    alpha = 1-conf/100
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    t0 = T.cum_distribution_function_inv(conf/100+alpha/2).n(digits=5)   # presuming a two-sided confidence interval  
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#  Need to consider here whether to use z or t distribution based upon n and stdev
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    E = t0*stdev/sqrt(n)
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    low = samplemean-E
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    high = samplemean+E
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    pretty_print("t-score =",t0," and std dev =",stdev," gives ",[low, high])
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    G = points([(samplemean,0)],size=50,color='red')
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    G += points([(low,0),(high,0)],size=50,color='blue')
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    G += line([(low,0),(samplemean,0)],color='green',thickness=3)
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    G += line([(high,0),(samplemean,0)],color='green',thickness=3)  
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    show(G,figsize=(5,1),xmin=samplemean-stdev,xmax=samplemean+stdev)

    
    
    Evaluate (Sage)
    

        

            
                Language:
                
                    Sage
                    Gap
                    GP
                    HTML
                    Macaulay2
                    Maxima
                    Octave
                    Python
                    R
                    Singular
                
            
        
    
    




    

    

        

        Messages
        


    

Additionally, this derivation assumes that $\mu$ is not known...indeed the goal is to approximate that mean using $\bar{x}\text{...}$but that $\sigma$ is known. This is often not the case. It can however be shown that if n is larger than 30, replacing $\sigma$ with the sample standard deviation s gives an acceptable confidence interval.

Notice, in practice you might want to take n to be a little larger than the absolute minimum value prescribed above since you are dealing with approximations (Central Limit Theorem and the use of an estimate for s rather than the actual $\sigma \text{.}$)