Questions: 4. Eighty-five percent of Americans favor spending government money to develop alternative sources of fuel for automobiles. For a random sample of 120 Americans, find the mean, variance, and standard deviation for the number who favor government spending for alternative fuels. Round answers to the nearest hundredth.

Solution Steps

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

\[ \mu = n \cdot p \]

Substituting the values \( n = 120 \) and \( p = 0.85 \):

\[ \mu = 120 \cdot 0.85 = 102.0 \]

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

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

where \( q = 1 - p \). Thus, \( q = 1 - 0.85 = 0.15 \). Now substituting the values:

\[ \sigma^2 = 120 \cdot 0.85 \cdot 0.15 = 15.3 \]

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

\[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{120 \cdot 0.85 \cdot 0.15} \approx 3.91 \]

Final Answer