Question

Define the steps to read a cumulative binomial probability table.

Answer 1

A binomial probability refers to the probability of getting EXACTLY r successes in an exceedingly specific number of trials. Cumulative binomial probability refers to the probability that the worth of a binomial random variable falls within a specified range. Cumulative binomial probability tables are used to find $P(X \leqslant x)$for the distribution$X-B(n,p)$

Complete step by step solution:
We know that cumulative binomial probability tables are used to find $P(X \leqslant x)$for the distribution$X-B(n,p)$.
Basic rules that are used to find probabilities in a binomial distribution table:
$P(X < x) = P(X \leqslant x - 1)$…….(1)
$P(X \geqslant x) = 1 - P(X \leqslant x - 1)$…….(2)
$P(X > x) = 1 - P(X \leqslant x)$………(3)
$P(A < X \leqslant B) = P(X \leqslant B) - P(X \leqslant A)$……..(4)
There is a separate table for every sample size .First step is to find the proper table of
n (where n is equal to your sample size).
Then find the column on the same table with the probability of your distribution. The number in the row $x = a$
For example, to find $P(4 \leqslant X \leqslant 9)$for the distribution $X-B(14,0.55)$, head to the table for $n = 14$.
Then find the column $p = 0.55$. Seek for the row $x = 9$in this column, which
supplies 0.8328, then searches for$x = 4$ in this column, which provides 0.0114. (Here we have assumed the values of the probabilities)
Thus using the formula (4) from the above list of formulas we’ll get, $P(4 \leqslant X \leqslant 9) = P(X \leqslant 9) - P(X \leqslant 4) = 0.8328 - 0.0114 = 0.8214$

Note: The binomial cumulative distribution function enables you to obtain the probability of observing less or equal to x successes in n trials, with the probability p of success on one trial. The binomial cumulative distribution function for a given value x and a given pair of parameters n and p is y = F(x|n,p) = \sum\limits_{i = 0}^x {\left( {\begin{array}{*{20}{c}} n \\\ i \end{array}} \right){p^i}{{(1 - p)}^{(n - i)}}{I_{(0,1,...,n)}}(i)}