Probability questions become easier to organize when the distribution, the region, and the calculator command are kept together. The tool below covers the main distributions used in AP Statistics.

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Six distributions, typed parameters, four region shapes, and both directions: a region gives a probability, a probability gives a cutoff. The binomial keeps P(X = k) and P(X ≤ k) side by side, because those are the two the calculator distinguishes as binompdf and binomcdf and they are the pair most often swapped. The t curve carries the standard normal behind it in outline, so raising the degrees of freedom shows one collapsing onto the other. For the two sampling distributions the center and spread are printed as expressions with the numbers substituted, and a warning appears when a normal-approximation condition fails.

Enter the parameters from a problem, choose the region, and compare the graph with the corresponding numerical probability or cutoff. The TI-84 command is shown underneath.

Forward and inverse questions

A forward probability question gives a boundary and asks for area. Examples include:

  • \[P(X<70)\]
  • \[P(2<T<3)\]
  • the probability of at most 6 successes

Calculator commands such as normalcdf, tcdf, and binomcdf work in this direction. An inverse question gives a probability and asks for the boundary. Examples include:

  • the 90th percentile
  • the critical value leaving 2.5% in the upper tail

Commands such as invNorm and invT work in the inverse direction. Before choosing a command, ask which of those two quantities the problem supplied.

Binomial probability

For a binomial random variable, binompdf gives \(P(X=k)\), the probability of exactly \(k\) successes, and binomcdf gives \(P(X\le k)\). For “at least 6,” use the complement \(P(X\ge6) = 1-P(X\le5)\). The off-by-one matters, and the complement stops at 5 because 6 belongs in the event we want to keep.

Normal and \(t\)

For a standard normal variable, \(P(-1.96<Z<1.96) \approx0.95\). A \(t\)-distribution with small degrees of freedom has heavier tails, so the same fixed interval contains less probability. As the degrees of freedom increase, the \(t\)-distribution approaches the standard normal distribution, which is why \(t\) critical values are larger when the sample is small.

Sampling distributions

The sample-proportion and sample-mean modes use normal curves as sampling models. For a sample proportion, \(\mu_{\hat p}=p\) and \(\sigma_{\hat p} = \sqrt{\tfrac{p(1-p)}{n}}\), and the usual large-count condition requires expected successes and failures to be sufficiently large.

For a sample mean, \(\mu_{\bar X}=\mu\) and \(\sigma_{\bar X} = \tfrac{\sigma}{\sqrt n}\). Normality is exact when the population is normal and approximate for sufficiently large samples under the Central Limit Theorem.

The warnings in the tool are part of the statistical reasoning. A calculator can evaluate a probability even when the model used to justify that calculation is poor.