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. $mu=square$ (round to two decimal places as needed.)

Explanation:

Step1: Identify binomial parameters

Let $n = 5$ (number of trials, i.e., number of adults selected) and $p=0.68$ (probability of success, i.e., probability an adult believes life on other planets is plausible).

Step2: Calculate the mean

The formula for the mean of a binomial distribution is $\mu = np$. Substitute $n = 5$ and $p = 0.68$ into the formula: $\mu=5\times0.68 = 3.40$.

Step3: Calculate the variance

The formula for the variance of a binomial distribution is $\sigma^{2}=np(1 - p)$. Substitute $n = 5$ and $p = 0.68$: $\sigma^{2}=5\times0.68\times(1 - 0.68)=5\times0.68\times0.32 = 1.088$.

Step4: Calculate the standard deviation

The formula for the standard deviation is $\sigma=\sqrt{\sigma^{2}}$. So $\sigma=\sqrt{1.088}\approx1.04$.

Step5: Interpret the results

The mean of 3.40 means that, on average, out of 5 randomly - selected adults, we expect 3.40 adults to believe that life on other planets is plausible. The variance of 1.088 measures the spread of the number of adults who believe in life on other planets around the mean. The standard deviation of about 1.04 gives us a measure of the typical deviation from the mean number of adults who believe in life on other planets.

Answer:

The mean $\mu = 3.40$, the variance $\sigma^{2}=1.09$ (rounded to two decimal places), and the standard deviation $\sigma\approx1.04$ (rounded to two decimal places).