Question

write the binomial probability and the normal probability for the shaded region of the graph. find the value of each probability and compare the results. (round to four decimal places as needed.) a. $p(5 < x < 9)=p(5.5)+p(6.5)+p(7.5)+p(8.5)=$ b. $p(5leq xleq9)=p(5)+p(6)+p(7)+p(8)+p(9)=$ c. $p(6 < x < 8)=p(7)=$ d. $p(6leq xleq8)=p(6)+p(7)+p(8)=0.2611$

Explanation:

Step1: Identify the binomial - probability concept

For a binomial - distribution, when finding \(P(a\lt X\lt b)\) where \(X\) is a discrete random variable, and we are approximating it with a normal distribution (continuity correction), for \(P(6\lt x\lt8)\) with a discrete variable, the probability is just the probability of the single value in the middle (since there is no probability mass between consecutive integer values in a discrete distribution). Here, the single value in the range \(6\lt x\lt8\) is \(x = 7\).

Step2: Assume we have the binomial - probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\). However, since we are not given \(n\) (number of trials) and \(p\) (probability of success in a single trial), if we assume we are reading from a probability - mass function or have already calculated the probabilities for each \(x\) value, we just need to find the value of \(P(7)\). But without further information about the binomial distribution parameters, if we assume we are using a pre - calculated probability table or a given probability function, we just state the principle. In the case of a normal approximation to the binomial, for a discrete variable, when \(a\lt x\lt b\) with \(b=a + 2\) (consecutive non - inclusive integers), the probability is the probability of the middle value.

Answer:

We need more information about the binomial distribution (such as \(n\) and \(p\)) to calculate a numerical value for \(P(7)\). But conceptually, for \(P(6\lt x\lt8)\) in a binomial distribution (or normal approximation with continuity correction for a discrete variable), the probability is equal to \(P(7)\).