Confidence Intervals with Wolfram Mathematica

Question

I'd be interested in calculating confidence intervals of data. For example using R:

data = c(1,0,2,1,2,0,1,0)
confidence_interv = (sd(data)/sqrt(8))*qnorm(.95)

Where 8 is the length of the data array. I'd like to do the same with Mathematica, maybe using a quicker way. And, is it possible to plot the data and the confidence interval around the single data? Thanks

Answer 1

data = {1, 0, 2, 1, 2, 0, 1, 0};
lm = LinearModelFit[data, {1, x}, x];
lm["SinglePredictionConfidenceIntervalTable", ConfidenceLevel -> 0.95]

Show[ListPlot[data], 
 Plot[{lm[x], 
   lm["SinglePredictionBands", ConfidenceLevel -> 0.95]}, {x, 0, 9}, 
  Filling -> {2 -> {1}}], PlotRange -> {Automatic, {-4, 6}}, 
 Frame -> True, ImageSize -> 300]

See also Showing the correlation of two variables using a plot

Answer 2

A little different interpretation of the question..

 Needs["HypothesisTesting`"]
 data = Table[RandomVariate[NormalDistribution[1, 1]], {2000}];
 ci = NormalCI[ Mean[data] , StandardDeviation[data], 
          ConfidenceLevel -> .99]

{-1.55307, 3.58071}

 Histogram[data,
   Epilog -> {Text[Style["99% confidence range", FontSize -> 20], {1, 100}],
              Red,Line[{{#, 0}, {#, 100}}] & /@ ci ,
              Arrowheads[{-.05, .05}], Arrow[{{ci[[1]], 70}, {ci[[2]], 70}}]}] 

Incidentally you can get the interval like this as well: (w/o loading the HypothesisTesting package)

  InverseCDF[
        NormalDistribution[Mean[data], StandardDeviation[data]],
            (1 + # {-1, 1})/2 &@.99 ]

{-1.55307, 3.58071}