Plotting confidence intervals

Question

The documentation for PDF has an example for plotting a confidence interval of a standard normal distribution:

pdf = PDF[NormalDistribution[], x]
Show[Plot[pdf, {x, -1.7, -1}, Filling -> Axis, FillingStyle -> Green],
  Plot[pdf, {x, -1, 1}] , 
 Plot[pdf, {x, 1, 1.7}, Filling -> Axis, FillingStyle -> Green], 
 PlotRange -> {0, 0.4}, AxesOrigin -> {0, 0.1}, 
 Ticks -> {Automatic, None}]

The plot range is set such that this code would not work in a generalized case for NormalDistribution[u,s].

What is the appropriate format for {x, xmin, xmax} in the Plot function above for the general case of a normal distribution with mean = u and standard deviation = s?

Answer 1

This code does it

pdf = PDF[NormalDistribution[2, 1], x]
With[{mu = 2, s = 1},
 Show[
  Plot[pdf, {x, mu - 2 s, mu - s}, Filling -> Axis, 
   FillingStyle -> Green],
  Plot[pdf, {x, mu - s, mu + s}],
  Plot[pdf, {x, mu + s, mu + 2 s}, Filling -> Axis, 
   FillingStyle -> Green], PlotRange -> {0, 0.4}, 
  AxesOrigin -> {0, 0.1}, Ticks -> {Automatic, None}
  ]
 ]

Essentially, you are plotting 3 intervals $x \in [\mu - \sigma, \mu + \sigma]$, $x \in [\mu - 2 \sigma, \mu - \sigma]$, and $x \in [\mu + \sigma, \mu + 2 \sigma]$ where the last two intervals are also shaded green.

Answer 2

The limits of a 100c% confidence interval for a normal variable with mean u and standard deviation s are u ± s*InverseErf[c]*Sqrt[2]. The example doesn't say how the 1.7 cutoff was chosen, but Erf[1.7/Sqrt[2]] = .910869, so the entire plot is only a 91% confidence interval for a standard normal, not the almost-100% that is usual. To give a correct visual impression, the green should have been extended to ±3 or so with the x-axis at y=0. The white area, whose limits area ±1, is a 68% interval for a standard normal. This is how they should have done it to show a 68% interval:

Show[
  Plot[pdf, {x, -3.3, -1}, Filling -> Axis, FillingStyle -> Green],
  Plot[pdf, {x, -1, 1}, Filling -> None],
  Plot[pdf, {x, 1, 3.3}, Filling -> Axis, FillingStyle -> Green],
  PlotRange -> {0, .4}, Axes -> {True, False}, Ticks -> {Automatic, None}] 

Answer 3

I am not sure whether this will be useful. If the intention is to display the standard symmetric confidence interval of an arbitrary normal distribution, then the following will do this. Noting:

1. the confidence interval is 'highlighted' by the complement of the filled regions. If the intention is to highlight the interval in a special colour just reverse the RegionFunction inequality.
2. I have used Filling to 0. as Filling to Axis has some issues which I do not fully understand.

---

ciplot[m_, s_, ci_, opts : OptionsPattern[]] :=
  Module[{pdf, q, ns, p1, p2},
   pdf = NormalDistribution[m, s];
   q = Quantile[pdf, (1 + ci)/2];
   ns = (q - m)/s;
   p1 = Plot[PDF[pdf, x], {x, m - 1.5 ns s, m + 1.5 ns s}]; 
   p2 = Plot[PDF[pdf, x], {x, m - 1.5 ns s, m + 1.5 ns s}, 
     Filling -> 0., Evaluate@FilterRules[{opts}, Options[Plot]], 
     RegionFunction -> Function[x, Abs[x - m] > ns s]];
   Show[p2, p1, AxesOrigin -> {0, 0}, PlotRange -> All]];

I generate the animated gif for illustration purposes and manipulate could be used varying mean and standard deviation.