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 local crime.
find the mean of the binomial distribution.
μ = 2 (round to the nearest tenth as needed.)
find the variance of the binomial distribution.
σ² = 1.3 (round to the nearest tenth as needed.)
find the standard deviation of the binomial distribution.
σ = 1.1 (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 □.
(type an integer or decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Calculate the variance

The formula for the variance of a binomial distribution is $\sigma^{2}=np(1 - p)$. Here, $n = 6$ (number of trials, i.e., number of voters selected) and $p=0.34$ (probability of success, i.e., probability that a voter thinks federal government should get more involved in fighting local crime).
So, $\sigma^{2}=6\times0.34\times(1 - 0.34)=6\times0.34\times0.66 = 1.3464\approx1.3$

Step2: Calculate the standard deviation

The formula for the standard deviation of a binomial distribution is $\sigma=\sqrt{np(1 - p)}$. Since we know that $np(1 - p)=\sigma^{2}=1.3464$, then $\sigma=\sqrt{1.3464}\approx1.1$

Step3: Interpretation

The mean $\mu = 2$ means that, on average, out of 6 randomly - selected likely U.S. voters, 2 voters think that the federal government should get more involved in fighting local crime. The standard deviation $\sigma\approx1.1$ means that the number of voters in most samples of 6 likely U.S. voters who think the federal government should get more involved in local crime would differ from the mean ($\mu = 2$) by no more than $1.1$.

Answer:

The variance $\sigma^{2}=1.3$, the standard deviation $\sigma = 1.1$, and the interpretation is that on average, out of 6 randomly - selected likely U.S. voters, 2 voters think that the federal government should get more involved in fighting local crime and the number of voters in most samples of 6 likely U.S. voters who think the federal government should get more involved in local crime would differ from the mean ($\mu = 2$) by no more than $1.1$.