What is the area covered by a bell-shaped curve from −6 sigma to \(+\)6 sigma in normal distribution?
- 99.73\(\%\)
- 99.99\(\%\)
- 99.9999\(\%\)
- 99.999666\(\%\)
The Correct Option is D
Solution and Explanation
The question asks about the area covered by a bell-shaped curve from \(-6\) sigma to \(+6\) sigma in a normal distribution. This can be understood using the properties of the normal distribution and the empirical rule (also known as the 68-95-99.7 rule).
A normal distribution is a symmetric, bell-shaped curve that is characterized by its mean (\(\mu\)) and standard deviation (\(\sigma\)). In a standard normal distribution:
- 68% of the data is within 1 sigma (\(\mu \pm \sigma\))
- 95% of the data is within 2 sigma (\(\mu \pm 2\sigma\))
- 99.7% of the data is within 3 sigma (\(\mu \pm 3\sigma\))
The area covered by the curve between \(-6\) sigma to \(+6\) sigma will include virtually all of the data, since this range captures the data so far from the mean that the tail ends of the distribution are considered.
For \(-6\) sigma to \(+6\), the area under the curve is approximately \(99.999666\%\). This value can be determined by understanding cumulative distribution function values or through statistical tables and software.
Thus, the area covered by a bell-shaped curve from \(-6\) sigma to \(+6\) sigma is 99.999666\%, which aligns with statistical analyses of normal distribution tails.
This comprehensively explains why the correct answer is 99.999666\%, ruling out other options which denotes different (smaller) areas.
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