Question

If $p = \frac{1}{8}; n = 640; q = \frac{7}{8}$, then variance Binomial Distribution

• A. 0.07
• B. 0.7
• C. 7.0
• D. 70.0

Hint:

The variance of a binomial distribution is \( n p q \); ensure \( q = 1 - p \), and compute carefully to avoid decimal errors.

Answer 1

The Correct Option is D

Step 1: Recall the formula for the variance of a binomial distribution. 
A binomial distribution with parameters \( n \) (number of trials) and \( p \) (probability of success) has variance given by: \[ \text{Variance} = n p q, \] where \( q = 1 - p \) is the probability of failure. 
Step 2: Identify the given values. 
\( p = \frac{1}{8} \),
\( q = \frac{7}{8} \) (which satisfies \( q = 1 - p \), since \( 1 - \frac{1}{8} = \frac{7}{8} \)),
\( n = 640 \).
Step 3: Compute the variance. 
\[ \text{Variance} = n p q = 640 \cdot \frac{1}{8} \cdot \frac{7}{8}. \] First, calculate: \[ \frac{1}{8} \cdot \frac{7}{8} = \frac{7}{64}, \] \[ \text{Variance} = 640 \cdot \frac{7}{64} = \frac{640 \cdot 7}{64} = \frac{4480}{64} = 70. \] 
Step 4: Evaluate the options. 
(1) 0.07: Incorrect, as the variance is 70, not 0.07. Incorrect.
(2) 0.7: Incorrect, as the variance is 70, not 0.7. Incorrect.
(3) 7.0: Incorrect, as the variance is 70, not 7.0. Incorrect.
(4) 70.0: Correct, as the variance is 70. Correct.
Step 5: Select the correct answer. 
The variance of the binomial distribution is 70.0, matching option (4).