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.
μ = 2 (round to the nearest tenth as needed.)
find the variance of the binomial distribution.
σ² = □ (round to the nearest tenth as needed.)

Explanation:

Step1: Recall binomial distribution formulas

For a binomial distribution \(X\sim B(n,p)\), the mean \(\mu = np\), the variance \(\sigma^{2}=np(1 - p)\). Here \(n = 6\) (number of trials, i.e., number of voters selected) and \(p=0.34\) (probability that a voter thinks federal government should get more involved in fighting local crime).

Step2: Calculate the mean

Using the formula \(\mu=np\), substitute \(n = 6\) and \(p = 0.34\).
\(\mu=6\times0.34 = 2.04\approx2.0\)

Step3: Calculate the variance

Using the formula \(\sigma^{2}=np(1 - p)\), substitute \(n = 6\), \(p = 0.34\), and \(1-p=0.66\).
\(\sigma^{2}=6\times0.34\times0.66=6\times0.2244 = 1.3464\approx1.3\)

Answer:

The mean \(\mu = 2.0\) and the variance \(\sigma^{2}=1.3\)