Question

eighty - four percent of adults in a certain country 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. you randomly select five adults and ask them whether they 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 = 4.2 (round to two decimal places as needed.)
find the variance of the binomial distribution.
σ² = 0.67 (round to two decimal places as needed.)
find the standard deviation of the binomial distribution.
σ = 0.82 (round to two decimal places as needed.)
interpret the results.
on average, □ out of every 5 adults in the country believe that life on other planets is plausible. in most samples of 5 adults, the number of adults who think life on other planets is plausible would differ from the mean by no more than □.
(type integers or decimals rounded to two decimal places as needed.)

Explanation:

Step1: Recall binomial distribution formulas

For a binomial distribution \(X\sim B(n,p)\), the mean \(\mu = np\), the variance \(\sigma^{2}=np(1 - p)\), and the standard deviation \(\sigma=\sqrt{np(1 - p)}\). Here \(n = 5\) (number of trials, i.e., number of adults selected) and \(p=0.84\) (probability that an adult believes life on other planets is plausible).

Step2: Calculate the variance

We know \(\sigma^{2}=np(1 - p)\). Substitute \(n = 5\) and \(p = 0.84\) into the formula:

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Step3: Calculate the standard deviation

Since \(\sigma=\sqrt{\sigma^{2}}\), and \(\sigma^{2}=0.672\) (from step 2), then \(\sigma=\sqrt{0.672}\approx0.82\)

Step4: Interpret the results for the mean - deviation relationship

The mean \(\mu = 4.2\). The standard deviation \(\sigma\approx0.82\). By the empirical rule (for a binomial distribution approximated well by a normal distribution, which is the case when \(np\geq5\) and \(n(1 - p)\geq5\), here \(np=4.2\) and \(n(1 - p)=0.8\), but we still use the formula - based interpretation), on average, the number of adults who believe life on other planets is plausible in a sample of \(n = 5\) adults differs from the mean by no more than \(\sigma\approx0.82\)

Answer:

The variance \(\sigma^{2}=0.67\), the standard deviation \(\sigma = 0.82\), and on average, the number of adults who believe life on other planets is plausible in a sample of \(5\) adults differs from the mean by no more than \(0.82\)