Question

Define the steps to read a cumulative binomial probability table.

Answer 1

Hint: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 $$P(X⩽x)$$for the distribution$$X-B(n,p)$$

Complete step by step solution:
We know that cumulative binomial probability tables are used to find $$P(X⩽x)$$for the distribution$$X-B(n,p)$$.
Basic rules that are used to find probabilities in a binomial distribution table:
$$P(X<x)=P(X⩽x-1)$$…….(1)
$$P(X⩾x)=1-P(X⩽x-1)$$…….(2)
$$P(X>x)=1-P(X⩽x)$$………(3)
$$P(A<X⩽B)=P(X⩽B)-P(X⩽A)$$……..(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 $$P(4⩽X⩽9)$$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, $$P(4⩽X⩽9)=P(X⩽9)-P(X⩽4)=0.8328-0.0114=0.8214$$

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 $$y=F(x|n,p)=\sum _{i=0}^x\left(\begin{array}{c}n\\ i\end{array}\right)p^i{(1-p)}^{(n-i)}{I}_{(0,1,...,n)}(i)$$