Theoretical DistributionsMCQPYQ July 21Question 3549 of 230
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For a certain type of mobile, the length of time between charges of the battery is normally distributed with a mean of 50 hours and a standard deviation of 15 hours. A person owns one of these mobiles and want to know the probability that the length of time will be between 50 and 70 hours is (given q(1.33)=0.9082,q(0)=0.5\displaystyle q(1.33) = 0.9082, q(0) = 0.5)?

Options

A- 0.4082
B- 0.5
C0.4082
D0.5
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Correct Answer

✅ Option c — 0.4082

All Options:

  • A- 0.4082
  • B- 0.5
  • C0.4082
  • D0.5

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Detailed Solution & Explanation

**Probability Calculation for a Normally Distributed Variable** Let X\displaystyle X represent the length of time between charges (in hours). Given: - X∼N(μ,σ2)\displaystyle X \sim N(\mu, \sigma^2) where μ=50\displaystyle \mu = 50 and σ=15\displaystyle \sigma = 15 We want to find the probability: P(50≤X≤70)P(50 \le X \le 70) **Step 1: Standardize the limits to find the Z-scores** Using the formula Z=X−μσ\displaystyle Z = \frac{X - \mu}{\sigma}: - For X=50\displaystyle X = 50: Z1=50−5015=0Z_1 = \frac{50 - 50}{15} = 0 - For X=70\displaystyle X = 70: Z2=70−5015=2015≈1.33Z_2 = \frac{70 - 50}{15} = \frac{20}{15} \approx 1.33 **Step 2: Express the probability in terms of the standard normal variable Z\displaystyle Z** P(50≤X≤70)=P(0≤Z≤1.33)P(50 \le X \le 70) = P(0 \le Z \le 1.33) Using the cumulative area function Φ(z)\displaystyle \Phi(z) (given as q(z)\displaystyle q(z)): P(0≤Z≤1.33)=Φ(1.33)−Φ(0)P(0 \le Z \le 1.33) = \Phi(1.33) - \Phi(0) =q(1.33)−q(0)= q(1.33) - q(0) Substituting the given values q(1.33)=0.9082\displaystyle q(1.33) = 0.9082 and q(0)=0.5\displaystyle q(0) = 0.5: P(0≤Z≤1.33)=0.9082−0.5=0.4082P(0 \le Z \le 1.33) = 0.9082 - 0.5 = 0.4082 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.

Key Concepts to Understand

Probability

A numerical measure of the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Formula: P(A) = Number of favorable outcomes / Total number of equally likely outcomes.

Standard Deviation

A measure of the dispersion or spread of values in a data set around the mean. It is the square root of Variance. A lower standard deviation indicates data points are clustered close to the mean; a higher value indicates greater spread.

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