Question

seventy - two 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.
\\( \mu=3.6 \\) (round to two decimal places as needed.)
find the variance of the binomial distribution.
\\( \sigma^{2}=\square \\) (round to two decimal places as needed.)

Explanation:

Step1: Identify the values of \(n\) and \(p\)

Given \(n = 5\) (number of trials, i.e., number of adults selected) and \(p=0.72\) (probability of success, i.e., probability that an adult believes life on other planets is plausible)

Step2: Use the formula for variance of a binomial distribution

The formula for the variance \(\sigma^{2}\) of a binomial distribution is \(\sigma^{2}=n\times p\times(1 - p)\)
Substitute \(n = 5\), \(p = 0.72\), and \(1-p=1 - 0.72=0.28\) into the formula:

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Step3: Round the result

Round \(1.008\) to two decimal places. Using the rounding rule (if the third decimal digit is 5 or more, round up the second decimal digit), we get \(\sigma^{2}\approx1.01\)

Answer:

\(1.01\)