Connection Between Binomial and Normal Distributions: Apply It 2

  • Use normal probability distribution to calculate binomial probabilities
  • Check the conditions for applying a normal distribution to approximate a binomial distribution

Continuity Correction

In addition to verifying that both the number of successes and the number of failures exceed [latex]10[/latex], we need to adjust the discrete whole numbers used in a binomial distribution.

continuity correction

In probability theory, a continuity correction is an adjustment that is made when a discrete distribution is approximated by a continuous distribution. We do this by adjusting the discrete whole numbers used in a binomial distribution so that any individual value, [latex]X[/latex], is represented in the normal distribution by the interval from [latex]X - 0.5 \mbox{ to } X + 0.5[/latex] or [latex]X ±0.5[/latex].

Why is this necessary? Consider the following example.

The probability that George successfully makes exactly [latex]270[/latex] free throws is [latex]P(X= 270) = 0.0766[/latex]. So, there is a [latex]7.66\%[/latex] chance that he will make exactly [latex]270[/latex] free throws out of [latex]300[/latex]. The [latex]P(X= 270.2)[/latex] free throws is [latex]0[/latex] because he cannot make [latex]0.2[/latex] of a free throw. The number of free throws made is a discrete random variable.Now suppose we assume that the number of free throws made has a normal distribution. In other words, suppose that the random variable [latex]X[/latex] is a continuous random variable. The probabilities for a continuous random variable are defined as the area under the curve, and we talk about the area under the curve over intervals using probability notation:

[latex]-P(X < a)[/latex], [latex]P(X>a)[/latex] or [latex]P(a< X< b)[/latex]

For a continuous random variable, [latex]P(X = a) = 0[/latex] because there are an infinite number of possible numbers on any interval. Also, [latex]P( X \leq a) = P(X< a)[/latex] for a continuous random variable.

Thus, we will use [latex]P(269.5 < X < 270.5)[/latex] to approximate [latex]P(X = 270)[/latex].

Thus, provided that certain conditions ([latex]np \ge 10[/latex] and [latex]n(1-p) \ge 10[/latex]) are met, we can then use the continuity of correction to make the adjustment to the discrete random variable and use the normal distribution to approximate the binomial probabilities.