Questions: Sixty-four percent of adults in a certain country believe that life on other planets is plausible. You randomly select five adults and ask them whether they believe that life on other planets is plausible. The random variable represents the number of adults who believe that life on other planets is plausible. Find the mean, variance, and standard deviation of the binomial distribution for the random variable interpret the results Find the mean of the binomial distribution μ= (Round to two decimal places as needed.)

Solution Steps

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

\[ \mu = n \cdot p \]

where:

• \( n = 5 \) (the number of trials),
• \( p = 0.64 \) (the probability of success).

Substituting the values:

\[ \mu = 5 \cdot 0.64 = 3.2 \]

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

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

where:

• \( q = 1 - p = 0.36 \) (the probability of failure).

Substituting the values:

\[ \sigma^2 = 5 \cdot 0.64 \cdot 0.36 = 1.15 \]

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

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

Substituting the values:

\[ \sigma = \sqrt{5 \cdot 0.64 \cdot 0.36} \approx 1.07 \]

Final Answer