Questions: Evaluate σ=sqrt(n p(1-p)) for n=1582, p=2/5. σ= (Simplify your answer. Type an integer or decimal rounded to one decimal place as needed.)

Transcript text: Evaluate $\sigma=\sqrt{n p(1-p)}$ for $n=1582, p=\frac{2}{5}$. \[ \sigma= \] (Simplify your answer. Type an integer or decimal rounded to one decimal place as needed.)

Solution

Solution Steps

Step 1: Calculate the Mean

The mean $ \mu $ of a binomial distribution is calculated using the formula: \[ \mu = n \cdot p \] Substituting the given values $ n = 1582 $ and $ p = \frac{2}{5} $: \[ \mu = 1582 \cdot \frac{2}{5} = 632.8 \]

Step 2: Calculate the Variance

The variance $ \sigma^2 $ of a binomial distribution is given by: \[ \sigma^2 = n \cdot p \cdot q \] where $ q = 1 - p $. Thus, $ q = 1 - \frac{2}{5} = \frac{3}{5} $. Now substituting the values: \[ \sigma^2 = 1582 \cdot \frac{2}{5} \cdot \frac{3}{5} = 379.7 \]

Step 3: Calculate the Standard Deviation

The standard deviation $ \sigma $ is the square root of the variance: \[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{379.7} \approx 19.5 \]