Theoretical DistributionsMTP June 2023 Series IQuestion 3991 of 230
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The speeds of bikes follow a normal distribution model with a mean of 80\displaystyle 80 km/hr. and a standard deviation of 9.4\displaystyle 9.4 km/hr. Find the probability that a bike picked at random is travelling at more than 95\displaystyle 95 km/hr.? [P(z)=P(1.60)=0.4452]\displaystyle [P(z) = P(1.60) = 0.4452]

Options

A0.0548\displaystyle 0.0548
B0.38\displaystyle 0.38
C0.49\displaystyle 0.49
D0.278\displaystyle 0.278
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Correct Answer

✅ Option a — 0.0548\displaystyle 0.0548

All Options:

  • A0.0548\displaystyle 0.0548
  • B0.38\displaystyle 0.38
  • C0.49\displaystyle 0.49
  • D0.278\displaystyle 0.278

Detailed Solution & Explanation

**Probability in Normal Distribution** Let X\displaystyle X represent the speed of a bike. We are given XsimN(mu,sigma2)\displaystyle X \\sim N(\\mu, \\sigma^2) where: - Mean (mu\displaystyle \\mu) = 80\displaystyle 80 km/hr - Standard Deviation (sigma\displaystyle \\sigma) = 9.4\displaystyle 9.4 km/hr We need to find the probability that a randomly selected bike travels at more than 95\displaystyle 95 km/hr, which is P(X>95)\displaystyle P(X > 95). **Step 1: Standardize the variable** We convert X=95\displaystyle X = 95 to standard normal variable Z\displaystyle Z: Z=fracX−musigma=frac95−809.4=frac159.4approx1.5957approx1.60Z = \\frac{X - \\mu}{\\sigma} = \\frac{95 - 80}{9.4} = \\frac{15}{9.4} \\approx 1.5957 \\approx 1.60 **Step 2: Calculate the probability** P(X>95)=P(Z>1.60)P(X > 95) = P(Z > 1.60) Since the standard normal distribution is symmetrical and the total area under one half of the curve is 0.5\displaystyle 0.5: P(Z>1.60)=0.5000−P(0<Zle1.60)P(Z > 1.60) = 0.5000 - P(0 < Z \\le 1.60) We are given the area from Z=0\displaystyle Z = 0 to 1.60\displaystyle 1.60 is 0.4452\displaystyle 0.4452. Substituting this: P(Z>1.60)=0.5000−0.4452=0.0548P(Z > 1.60) = 0.5000 - 0.4452 = 0.0548 This matches Option A. Hence, **Option A** 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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