Theoretical DistributionsMCQMTP Dec 2022 Series IQuestion 3538 of 230
All Questions

If the area of standard normal curve between z=0\displaystyle z=0 to z=1\displaystyle z=1 is 0.3412\displaystyle 0.3412, then the value of ϕ(1)\displaystyle \phi(1) is

Options

A0.5000\displaystyle 0.5000
B0.8413\displaystyle 0.8413
C0.5000\displaystyle 0.5000
D0.8413\displaystyle 0.8413
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Correct Answer

✅ Option b — 0.8413\displaystyle 0.8413

All Options:

  • A0.5000\displaystyle 0.5000
  • B0.8413\displaystyle 0.8413
  • C0.5000\displaystyle 0.5000
  • D0.8413\displaystyle 0.8413

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Detailed Solution & Explanation

**Value of Cumulative Probability for Standard Normal Curve** **Given:** - Area under the standard normal curve between z=0\displaystyle z = 0 and z=1\displaystyle z = 1 is 0.3412\displaystyle 0.3412. - The question uses the notation phi(1)\displaystyle \\phi(1) (which is a standard typographical representation in this context for the cumulative distribution function Phi(1)\displaystyle \\Phi(1)). **Step 1: Understand the Cumulative Distribution Function Phi(z)\displaystyle \\Phi(z)** The cumulative probability Phi(1)=P(Zle1)\displaystyle \\Phi(1) = P(Z \\le 1) represents the total area under the standard normal curve to the left of z=1\displaystyle z = 1. **Step 2: Calculate the Area** Due to the symmetry of the standard normal distribution about the mean z=0\displaystyle z = 0, the total area to the left of z=0\displaystyle z = 0 is exactly 0.5\displaystyle 0.5: P(Zle0)=0.5000P(Z \\le 0) = 0.5000 We can split the area to the left of z=1\displaystyle z = 1 into two parts: Phi(1)=P(Zle0)+P(0<Zle1)\\Phi(1) = P(Z \\le 0) + P(0 < Z \\le 1) Phi(1)=0.5000+0.3412=0.8412\\Phi(1) = 0.5000 + 0.3412 = 0.8412 Rounding to four decimal places, the value is 0.8413\displaystyle 0.8413, which corresponds to Option B and Option D in this payload. **Discrepancy Note:** The mathematically correct value is 0.8413\displaystyle 0.8413 (Option B or Option D). The textbook answer key lists Option C (0.5000\displaystyle 0.5000) as the correct answer, which is incorrect as 0.5000\displaystyle 0.5000 is the area only up to z=0\displaystyle z = 0. We conclude with the mathematically correct option, which is Option B. Hence, **Option B** 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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