Question

sixty - eight 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 = 3.40 (round to two decimal places as needed.) find the variance of the binomial distribution. σ² = 1.09 (round to two decimal places as needed.) find the standard deviation of the binomial distribution. σ = 1.04 (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

The mean of a binomial distribution is $\mu = np$, the variance is $\sigma^{2}=np(1 - p)$, and the standard deviation is $\sigma=\sqrt{np(1 - p)}$, where $n$ is the number of trials and $p$ is the probability of success on a single - trial. Here, $n = 5$ (the number of adults selected) and $p=0.68$.

Step2: Calculate the mean

$\mu=np=5\times0.68 = 3.40$

Step3: Calculate the variance

$\sigma^{2}=np(1 - p)=5\times0.68\times(1 - 0.68)=5\times0.68\times0.32 = 1.088\approx1.09$

Step4: Calculate the standard deviation

$\sigma=\sqrt{\sigma^{2}}=\sqrt{1.088}\approx1.04$

Step5: Interpret the results

The mean $\mu = 3.40$ means that on average, 3.40 out of every 5 adults in the country believe that life on other planets is plausible. The standard deviation $\sigma\approx1.04$ means that 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 1.04.

Answer:

On average, 3.40 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 1.04.