Question

According to 2015 census data, 42.7 percent of Colorado residents were born in Colorado. If a sample of 250 Colorado residents is selected at random, what is the standard deviation of the number of residents in the sample who were born in Colorado? A.6.75 B.7.82 C.10.33 D.11.97 E.61.17

Answer 1

Answer:

$sd(X)=\sqrt{np(1-p)}=\sqrt{250*0.427*(1-0.427)}=7.82$

The best option is:

B.7.82

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".  

Data given

$p_C$ represent the real population proportion for residents born in Colorado  

$\hat p_C =0.427$ represent the estimated proportion for rsidents born in Colorado

$n_C=250$ is the sample size selected

Solution to the problem

Let X the random variable of interest (number of residents in the sample), on this case we now that:  

$X \sim Binom(n=250, p=0.427)$  

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:

$nCx=\frac{n!}{(n-x)! x!}$  

The expected value is given by this formula:

$E(X) = np=250*0.427=106.75$

And the standard deviation for the random variable is given by:

$sd(X)=\sqrt{np(1-p)}=\sqrt{250*0.427*(1-0.427)}=7.82$

The best option is:

B.7.82