The times in minutes X1,X2, and X3 to grade the 3 questions on a quiz are independent and normally distributed with means μ1= 6, μ2= 4, and μ3= 7, and standard deviations σ1= 1, σ2= 0.5, and σ3= 1.5. Let Y be the total time it takes to grade one such quiz. a) Find μY and σYb) Find the probability that Y is between 15 and 20 minutes

Answered: The times in minutes X1,X2, and X3 to…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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The times in minutes X₁,X₂, and X₃to grade the 3 questions on a quiz are independent and normally distributed with means μ₁= 6, μ₂= 4, and μ₃= 7, and standard deviations σ₁= 1, σ₂= 0.5, and σ₃= 1.5. Let Y be the total time it takes to grade one such quiz.

a) Find μ_Y and σ_Y

b) Find the probability that Y is between 15 and 20 minutes

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Expert Solution

Step 1

It is known that, X, Y and Z are three independent normal variables, such that, X ~ N(µ₁, σ₁²), Y ~ N(µ₂, σ₂²), and Z ~ N(µ₃, σ₃²), then X + Y + Z ~ N(µ₁ + µ₂ + µ₃, σ₁² + σ₂² + σ₃²),

Step 2

In this case, X₁ ~ N(6, 1²), X₂ ~ N(4, 0.5²), and X₃ ~ N(7, 1.5²),.

Thus,

Y = X₁ + X₂ + X₃ ~ N(6 + 4 + 7, 1² + 0.5² + 1.5²) ~ N(17, 3.5).

Thus, µ_Y = 17 and σ_Y = 1.8708 (= √3.5).

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