Question

34% of likely U.S. voters think that the federal government should get more involved in fighting local crime. You randomly select six likely U.S. voters and ask them whether they think that the federal government should get more involved in fighting local crime. The random variable represents the number of likely U.S. voters who think that the federal government should get more involved in fighting local crime. Find the mean of the binomial distribution. muequals nothing ​(Round to the nearest tenth as​needed.) Find the variance of the binomial distribution. sigmasquaredequals nothing ​(Round to the nearest tenth as​needed.) Find the standard deviation of the binomial distribution. sigmaequals nothing ​(Round to the nearest tenth as​ needed.) Interpret the results in the context of the​ real-life situation. In most samples of 6 U.S.​ voters, the number of voters who think that the federal government should get more involved in local crime would differ from the mean by no more than nothing. ​(Type an integer or decimal rounded to the nearest tenth as​ needed.)

Answer 1

Answer:

Find the mean of the binomial distribution

$E(X) =\mu = np = 6*0.34=2.04 \approx 2.0$

Find the variance of the binomial distribution

$Var(X) = np(1-p) = 6*0.34*(1-0.34) = 1.346 \approx 1.4$

Find the standard deviation of the binomial distribution.

$Sd(X) =\sigma= \sqrt{np(1-p)} =\sqrt{ 6*0.34*(1-0.34)} = 1.16 \approx 1.2$

For this case the deviation would be 1.2 so we expect that the number of voters who think that the federal government should get more involved in local crime would differ from the mean by no more 1.2. We expect that at least 68% of the data would be within 1 deviation from the mean and within 2*1.2 =2.4 if we want at least 95% of the data.

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 "number of likely U.S. voters who think that the federal government should get more involved in fighting local crime", on this case we now that:

$X \sim Binom(n=6, p=0.34)$

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:

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

Find the mean of the binomial distribution

$E(X) =\mu = np = 6*0.34=2.04 \approx 2.0$

Find the variance of the binomial distribution

$Var(X) =\sigma^2= np(1-p) = 6*0.34*(1-0.34) = 1.346 \approx 1.4$

Find the standard deviation of the binomial distribution.

$Sd(X) =\sigma= \sqrt{np(1-p)} =\sqrt{ 6*0.34*(1-0.34)} = 1.16 \approx 1.2$

Interpret the results in the context of the​ real-life situation. In most samples of 6 U.S.​ voters, the number of voters who think that the federal government should get more involved in local crime would differ from the mean by no more than nothing. ​(Type an integer or decimal rounded to the nearest tenth as​ needed.)

For this case the deviation would be 1.2 so we expect that the number of voters who think that the federal government should get more involved in local crime would differ from the mean by no more 1.2. We expect that at least 68% of the data would be within 1 deviation from the mean and within 2*1.2 =2.4 if we want at least 95% of the data.