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

Answer 2

The standard deviation of the number of residents in the sample who were born in Colorado

B.7.82

hence option B is correct

Given

42.7 % of Colorado residents were born in Colorado.

If a sample of 250 Colorado residents is selected at random

We have to find out  the standard deviation of the number of residents in the sample who were born in Colorado

The options are given below  

A.6.75

B.7.82

C.10.33

D.11.97

E.61.17

This problem is given problem of binomial distribution  is a discrete probability distribution of two independent events for the given parameters.

Number of Colorado students taken in sample = n =  250

Let, the percentage of Colorado residents that were born in Colorado be p

Let, the percentage of Colorado residents that were not  born in Colorado be q

The percentage of Colorado residents that were born in Colorado = 42.7% = p = 0.427

$\rm \m for \; a \; binomeal\; distribution \to p = 1-q ......(1)\\Equation \ (1)\; holds\; good$

So The percentage of Colorado residents that were not born in Colorado =   q = (100-42.7) =  57.3 %

The standard deviation for binomial distribution is given by the formula as formulated in the equation number (1)

$\rm \sigma = \sqrt{ n\times p\times q } .......(1)$

So we can write

Standard deviation of binomial distribution

$\rm \sigma = \sqrt{ 250 \times \0.427 \times 0.573 } = 7.82$

The standard deviation of the number of residents in the sample who were born in Colorado

B.7.82

hence option B is correct

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https://brainly.com/question/15363285