Theoretical DistributionsMTP Oct 21Question 3985 of 230
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For a normal distribution with mean as 500 and SD as 120, what is the value of k so that the interval [500, k] covers 40.32 per cent area of the normal curve? (Given ϕ(1.30)=0.9032\displaystyle \phi(1.30) = 0.9032)

Options

A740
B750
C656
D800
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Correct Answer

✅ Option c — 656

All Options:

  • A740
  • B750
  • C656
  • D800

Detailed Solution & Explanation

**Finding Interval Parameter for a Normal Curve Area** Given: - Mean (mu\displaystyle \\mu) = 500\displaystyle 500 - Standard Deviation (sigma\displaystyle \\sigma) = 120\displaystyle 120 - The area of the interval [500,k]\displaystyle [500, k] under the curve is 40.32\displaystyle 40.32\\% = 0.4032. We express the probability as: P(500leXlek)=0.4032P(500 \\le X \\le k) = 0.4032 Convert this to the standard normal variable Z=fracX−musigma=fracX−500120\displaystyle Z = \\frac{X - \\mu}{\\sigma} = \\frac{X - 500}{120}: - For X=500\displaystyle X = 500, Z=0\displaystyle Z = 0 - For X=k\displaystyle X = k, Z=zk=frack−500120\displaystyle Z = z_k = \\frac{k - 500}{120} So: P(0leZlezk)=0.4032P(0 \\le Z \\le z_k) = 0.4032 Phi(zk)−Phi(0)=0.4032\\Phi(z_k) - \\Phi(0) = 0.4032 Since Phi(0)=0.5\displaystyle \\Phi(0) = 0.5: Phi(zk)−0.5=0.4032impliesPhi(zk)=0.9032\\Phi(z_k) - 0.5 = 0.4032 \\implies \\Phi(z_k) = 0.9032 We are given that Phi(1.30)=0.9032\displaystyle \\Phi(1.30) = 0.9032. Comparing the terms, we get: zk=1.30z_k = 1.30 Now, solve for k\displaystyle k: frack−500120=1.30\\frac{k - 500}{120} = 1.30 k−500=1.30times120=156k - 500 = 1.30 \\times 120 = 156 k=500+156=656k = 500 + 156 = 656 This corresponds to Option C. Hence, **Option C** 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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