Question

If 3% of calculators are defective, find the standard deviation of the defectives in 300 calculators?

• A. 2.954
• B. 8.73
• C. 3.9
• D. 4.3

Hint:

For binomial problems:
1. Check if \( n \times p \) is small; here \( \text{Mean} = 9 \).
2. Standard deviation is always less than the square root of the mean because \( q < 1 \).
Since \( \sqrt{9} = 3 \), the standard deviation must be slightly less than 3, pointing directly to 2.954.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem can be modeled using the Binomial distribution, which represents the number of successes (or defectives) in a fixed number of independent trials.
Key Formula or Approach:
For a Binomial distribution with parameters $n$ (number of trials) and $p$ (probability of success/defective):
- Variance:
\[ \sigma^2 = n p q \]
- Standard Deviation:
\[ \sigma = \sqrt{n p q} \]
where \( q = 1 - p \) is the probability of non-defective.

Step 2: Detailed Explanation:
Let us identify the parameters from the given problem:
- Total number of calculators, \( n = 300 \)
- Probability of a defective calculator, \( p = 3% = 0.03 \)
- Probability of a non-defective calculator, \( q = 1 - p = 1 - 0.03 = 0.97 \)
Now, calculate the variance of the defective calculators:
\[ \sigma^2 = n p q = 300 \times 0.03 \times 0.97 \]
\[ \sigma^2 = 9 \times 0.97 = 8.73 \]
Next, calculate the standard deviation by taking the square root of the variance:
\[ \sigma = \sqrt{8.73} \]
Evaluating the square root:
\[ \sigma \approx 2.95466 \]
This value corresponds to option (A).

Step 3: Final Answer:
The standard deviation of the defective calculators is approximately 2.954.