Questions: 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=2.0 (Round to the nearest tenth as needed.) Find the variance of the binomial distribution. σ^2= (Round to the nearest tenth as needed.)

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=2.0 (Round to the nearest tenth as needed.)
Find the variance of the binomial distribution.
σ^2= (Round to the nearest tenth as needed.)
Transcript text: $34 \%$ 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. $\mu=2.0$ (Round to the nearest tenth as needed.) Find the variance of the binomial distribution. $\sigma^{2}=$ $\square$ (Round to the nearest tenth as needed.)
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Solution

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Solution Steps

Step 1: Calculate the Mean

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

\[ \mu = n \cdot p \]

where:

  • \( n = 6 \) (the number of trials),
  • \( p = 0.34 \) (the probability of success).

Substituting the values:

\[ \mu = 6 \cdot 0.34 = 2.0 \]

Step 2: Calculate the Variance

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

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

where:

  • \( q = 1 - p = 0.66 \) (the probability of failure).

Substituting the values:

\[ \sigma^2 = 6 \cdot 0.34 \cdot 0.66 \approx 1.3 \]

Step 3: Summary of Results

From the calculations, we have:

  • Mean \( \mu = 2.0 \)
  • Variance \( \sigma^2 \approx 1.3 \)

Final Answer

\(\boxed{\mu = 2.0}\)
\(\boxed{\sigma^2 \approx 1.3}\)

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