Questions: 32% 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=1.9 (Round to the nearest tenth as needed.) Find the variance of the binomial distribution. σ^2=1.3 (Round to the nearest tenth as needed.) Find the standard deviation of the binomial distribution. σ= (Round to the nearest tenth as needed.)

Solution Steps

The mean \( \mu \) of a binomial distribution can be calculated using the formula:

\[ \mu = n \cdot p \]

where:

• \( n = 6 \) (the number of trials),
• \( p = 0.32 \) (the probability of success).

Substituting the values:

\[ \mu = 6 \cdot 0.32 = 1.92 \]

Rounding to the nearest tenth, we have:

\[ \mu \approx 1.9 \]

The variance \( \sigma^2 \) of a binomial distribution is given by the formula:

\[ \sigma^2 = n \cdot p \cdot q \]

where:

• \( q = 1 - p = 0.68 \) (the probability of failure).

Substituting the values:

\[ \sigma^2 = 6 \cdot 0.32 \cdot 0.68 = 1.3056 \]

Rounding to the nearest tenth, we have:

\[ \sigma^2 \approx 1.3 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{n \cdot p \cdot q} \]

Substituting the values:

\[ \sigma = \sqrt{6 \cdot 0.32 \cdot 0.68} = \sqrt{1.3056} \approx 1.1447 \]

Rounding to the nearest tenth, we have:

\[ \sigma \approx 1.1 \]

Final Answer