Question

Suppose that x has a binomial distribution with n = 50 and p = .6, so that μ = np = 30 and σ = np(1 − p) = 3.4641. Calculate the following probabilities using the normal approximation with the continuity correction. (Hint: 25 < x < 38 is the same as 26 ≤ x ≤ 37. Round your answers to four decimal places.)

Answer 1

Answer:

$P(25<X<38)$

And using the continuity correction factor we got:

$P(25<X<38) = P(25.5 < X< 37.5)$

And we can use the z score formula given by:

$z = \frac{X -\mu}{\sigma}$

And if we find the z score for the two values we got:

$z= \frac{25.5 -30}{3.4641}=-1.299$

$z= \frac{37.5 -30}{3.4641}=2.165$

$P(25<X<38) = P(25.5 < X< 37.5) =P(Z<2.165)-P(Z<-1.299)$And using the normal standard table or excel we got:

$P(25<X<38) = P(25.5 < X< 37.5) =P(Z<2.165)-P(Z<-1.299)= 0.985-0.097= 0.888$

Step-by-step explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Let X the random variable of interest, on this case we now that:

$X \sim Binom(n=50, p=0.6)$

The probability mass function for the Binomial distribution is given as:

$P(X)=(nCx)(p)^x (1-p)^{n-x}$

Where (nCx) means combinatory and it's given by this formula:

$nCx=\frac{n!}{(n-x)! x!}$

We need to check the conditions in order to use the normal approximation.

$np=50*0.6=30 \geq 10$

$n(1-p)=50*(1-0.6)=20 \geq 10$

So we see that we satisfy the conditions and then we can apply the approximation.

If we appply the approximation the new mean and standard deviation are:

$E(X)=np=50*0.6=30$

$\sigma=\sqrt{np(1-p)}=\sqrt{50*0.6(1-0.6)}=3.4641$

Solution to the problem

For this case we want this probability:

$P(25<X<38)$

And using the continuity correction factor we got:

$P(25<X<38) = P(25.5 < X< 37.5)$

And we can use the z score formula given by:

$z = \frac{X -\mu}{\sigma}$

And if we find the z score for the two values we got:

$z= \frac{25.5 -30}{3.4641}=-1.299$

$z= \frac{37.5 -30}{3.4641}=2.165$

$P(25<X<38) = P(25.5 < X< 37.5) =P(Z<2.165)-P(Z<-1.299)$

And using the normal standard table or excel we got:

$P(25<X<38) = P(25.5 < X< 37.5) =P(Z<2.165)-P(Z<-1.299)= 0.985-0.097= 0.888$