Question

An elementary school determines that 90% of parents whose children are enrolled in the Gifted & Talented Program complete and return the program evaluation survey at the end of the school year. At the end of the 2005-2006 school year, the school mailed 45 surveys to the parents. Consider the BINOMDIST function in Excel, BINOMDIST(number_s,trials,probability_s,cumulative). What is the value of the number of successes in this problem?

Answer 1

Answer:

For this case the expected value is given by:

$E(X)= np = 45*0.9 = 40.5$

The expected number of successes are on this case 40.5

And if we want to find a probability $P(X=a)$ we can use the following Excel function:

"=BINOM.DIST(a,49,0.9,FALSE)"

And if we want the probability:

$P(X \leq a)$

We can use the following code:

"=BINOM.DIST(a,49,0.9,TRUE)"

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".

Solution to the problem

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

$X \sim Binom(n=45, p=0.9)$

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:

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

For this case the expected value is given by:

$E(X)= np = 45*0.9 = 40.5$

The expected number of successes are on this case 40.5

And if we want to find a probability $P(X=a)$ we can use the following Excel function:

"=BINOM.DIST(a,49,0.9,FALSE)"

And if we want the probability:

$P(X \leq a)$

We can use the following code:

"=BINOM.DIST(a,49,0.9,TRUE)"