Question

find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p. n = 80, p = 0.6 the mean, μ, is (round to the nearest tenth as needed.)

Explanation:

Step1: Calculate the mean

The formula for the mean of a binomial distribution is $\mu = np$. Substitute $n = 80$ and $p=0.6$ into the formula.
$\mu=80\times0.6$
$\mu = 48.0$

Step2: Calculate the variance

The formula for the variance of a binomial distribution is $\sigma^{2}=np(1 - p)$. Substitute $n = 80$ and $p = 0.6$ into the formula. First, find $1-p=1 - 0.6=0.4$. Then $\sigma^{2}=80\times0.6\times0.4$
$\sigma^{2}=19.2$

Step3: Calculate the standard - deviation

The formula for the standard deviation of a binomial distribution is $\sigma=\sqrt{\sigma^{2}}$. Since $\sigma^{2}=19.2$, then $\sigma=\sqrt{19.2}\approx4.4$

Answer:

The mean, $\mu$, is $48.0$. The variance, $\sigma^{2}$, is $19.2$. The standard deviation, $\sigma$, is approximately $4.4$.