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}=1.01 \\) (round to two decimal places as needed.)
find the standard deviation of the binomial distribution.
\\( \sigma=1 \\) (round to two decimal places as needed.)
interpret the results.
on average, \\( \square \\) 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 \\( \square \\).
(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 \(\mu=3.6\). Since \(\mu=np\), we can find \(p=\frac{\mu}{n}=\frac{3.6}{5}=0.72\).

Step2: Calculate the variance

Using the formula \(\sigma^{2}=np(1 - p)\), substitute \(n = 5\) and \(p = 0.72\). Then \(\sigma^{2}=5\times0.72\times(1 - 0.72)=5\times0.72\times0.28 = 1.008\approx1.01\).

Step3: Calculate the standard deviation

Using the formula \(\sigma=\sqrt{\sigma^{2}}\), since \(\sigma^{2}=1.01\), then \(\sigma=\sqrt{1.01}\approx1.00\).

Step4: Interpret the results for the last blank

The formula for the range of values around the mean in a binomial distribution (using the empirical rule for a roughly bell - shaped distribution, approximated by the binomial when \(n\) is not too small and \(p\) is not too close to 0 or 1) is \(\mu\pm2\sigma\). The value of \(2\sigma\approx2\times1 = 2\).

Answer:

The variance \(\sigma^{2}=1.01\), the standard deviation \(\sigma = 1.00\), and on average, \(3\) out of every \(5\) adults in the country believe that life on other planets is plausible. The number of adults who think life on other planets is plausible would differ from the mean by no more than \(2\).