Question

In normal distribution with mean \(\mu\) and variance \(\sigma^2\), the area under the normal curve between \(\mu \pm \sigma\) is:

• A. 0.9544
• B. 0.6826
• C. 0.5000
• D. 0.9973

Hint:

Exam Tip:
Remember the empirical rule:

• \(68\%\) (\(0.6826\)) within \(\mu \pm \sigma\).
• \(95\%\) (\(0.9544\)) within \(\mu \pm 2\sigma\).
• \(99.7\%\) (\(0.9973\)) within \(\mu \pm 3\sigma\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The area under the normal curve between \(\mu - \sigma\) and \(\mu + \sigma\) represents the probability that a normally distributed random variable falls within one standard deviation of the mean.

Step 2: Key Formula or Approach:
From the empirical rule (68-95-99.7 rule):
• Within \(1\sigma\) of the mean: 68.26% of the data.
• Within \(2\sigma\) of the mean: 95.44% of the data.
• Within \(3\sigma\) of the mean: 99.73% of the data.

Step 3: Detailed Explanation:
The area between \(\mu - \sigma\) and \(\mu + \sigma\) is approximately 0.6826.
This is a standard result for the normal distribution.

Step 4: Final Answer:
Therefore, option (B) is correct.