Theoretical DistributionsMCQMTP June 24 Series IQuestion 3605 of 230
All Questions

The Interval (μ−3σ,μ+3σ)\displaystyle (\mu - 3\sigma, \mu + 3\sigma) covers

Options

A95%\displaystyle 95\% area of normal distribution
B96%\displaystyle 96\% area of normal distribution
C99%\displaystyle 99\% area of normal distribution
DAll but not 0.27%\displaystyle 0.27\% area of a normal distribution
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Correct Answer

✅ Option d — All but not 0.27%\displaystyle 0.27\% area of a normal distribution

All Options:

  • A95%\displaystyle 95\% area of normal distribution
  • B96%\displaystyle 96\% area of normal distribution
  • C99%\displaystyle 99\% area of normal distribution
  • DAll but not 0.27%\displaystyle 0.27\% area of a normal distribution

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Detailed Solution & Explanation

**Empirical Rule of a Normal Distribution**\n\nFor a normal distribution N(mu,sigma2)\displaystyle N(\\mu, \\sigma^2), the area under the curve represents the total probability (which equals 1\displaystyle 1 or 100\displaystyle 100\\%). The percentage of area covered within standard deviations from the mean is given by:\n\n1. **One Standard Deviation:**\n - Interval: (mu−sigma,mu+sigma)\displaystyle (\\mu - \\sigma, \\mu + \\sigma)\n - Area covered: approx68.27\displaystyle \\approx 68.27\\%\n - Area outside: approx31.73\displaystyle \\approx 31.73\\%\n\n2. **Two Standard Deviations:**\n - Interval: (mu−2sigma,mu+2sigma)\displaystyle (\\mu - 2\\sigma, \\mu + 2\\sigma)\n - Area covered: approx95.45\displaystyle \\approx 95.45\\%\n - Area outside: approx4.55\displaystyle \\approx 4.55\\%\n\n3. **Three Standard Deviations:**\n - Interval: (mu−3sigma,mu+3sigma)\displaystyle (\\mu - 3\\sigma, \\mu + 3\\sigma)\n - Area covered: approx99.73\displaystyle \\approx 99.73\\%\n - Area outside: 100\displaystyle 100\\% - 99.73\\% = 0.27\\%\n\n**Analyzing Option D:**\n"All but not 0.27\displaystyle 0.27\\% area of a normal distribution" means:\ntextAreaCovered=100\\text{Area Covered} = 100\\% - 0.27\\% = 99.73\\%\n\nThis matches the theoretical area covered by the three-sigma interval (mu−3sigma,mu+3sigma)\displaystyle (\\mu - 3\\sigma, \\mu + 3\\sigma) exactly.\n\nHence, **Option D** is the correct answer.

About This Chapter: Theoretical Distributions

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Binomial, Poisson, Normal Distribution

This chapter covers Binomial, Poisson, Normal Distribution and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

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For a Poisson distribution,

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