Questions: A binomial experiment has the given number of trials N and the given success probability P. N=8, P=0.5 Find the variance and standard deviation. Round the variance to two decimal places and standard deviation to at least three decimal places.

Solution Steps

The mean \( \mu \) of a binomial distribution is calculated using the formula: \[ \mu = n \cdot p \] Substituting the values \( n = 8 \) and \( p = 0.5 \): \[ \mu = 8 \cdot 0.5 = 4.0 \]

The variance \( \sigma^2 \) of a binomial distribution is given by: \[ \sigma^2 = n \cdot p \cdot q \] where \( q = 1 - p \). Thus, \( q = 0.5 \). Substituting the values: \[ \sigma^2 = 8 \cdot 0.5 \cdot 0.5 = 2.0 \]

The standard deviation \( \sigma \) is the square root of the variance: \[ \sigma = \sqrt{n \cdot p \cdot q} \] Substituting the values: \[ \sigma = \sqrt{8 \cdot 0.5 \cdot 0.5} = \sqrt{2} \approx 1.414 \]

Final Answer