Questions: Sixty-six 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=3.30 (Round to two decimal places as needed.) Find the variance of the binomial distribution. σ^2=1.12 (Round to two decimal places as needed.) Find the standard deviation of the binomial distribution. σ=1.06 (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.)

Sixty-six 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=3.30 (Round to two decimal places as needed.) Find the variance of the binomial distribution. σ^2=1.12 (Round to two decimal places as needed.) Find the standard deviation of the binomial distribution. σ=1.06 (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.)
Transcript text: Sixty-six 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. $\mu=3.30$ (Round to two decimal places as needed.) Find the variance of the binomial distribution. $\sigma^{2}=1.12$ (Round to two decimal places as needed.) Find the standard deviation of the binomial distribution. $\sigma=1.06$ (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.)
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Solution

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Solution Steps

Step 1: Calculate the Mean

The mean \( \mu \) of a binomial distribution can be calculated using the formula: \[ \mu = n \cdot p \] where \( n = 5 \) (the number of trials) and \( p = 0.66 \) (the probability of success). Thus, \[ \mu = 5 \cdot 0.66 = 3.30 \]

Step 2: Calculate the Variance

The variance \( \sigma^2 \) of a binomial distribution is given by the formula: \[ \sigma^2 = n \cdot p \cdot (1 - p) \] Substituting the values, we have: \[ \sigma^2 = 5 \cdot 0.66 \cdot (1 - 0.66) = 5 \cdot 0.66 \cdot 0.34 = 1.12 \]

Step 3: Calculate the Standard Deviation

The standard deviation \( \sigma \) is the square root of the variance: \[ \sigma = \sqrt{\sigma^2} \] Thus, \[ \sigma = \sqrt{1.12} \approx 1.06 \]

Step 4: Interpret the Results

On average, \( \mu = 3.30 \) 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 \( \sigma \approx 1.06 \).

Final Answer

The mean of the binomial distribution is \( \boxed{3.30} \), the variance is \( \boxed{1.12} \), and the standard deviation is \( \boxed{1.06} \). On average, \( 3.30 \) 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 \( 1.06 \).

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