What is the probability that 5 large loaves are heavier than 10 small loaves?

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Homework Help Overview

The discussion revolves around the probability of the total weight of five large loaves of bread being greater than that of ten small loaves. The weights of both types of loaves are modeled as normally distributed random variables.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore the properties of normally distributed variables and the implications of combining them. Questions arise regarding the correct interpretation of the inequality involving the weights of the loaves.

Discussion Status

Some participants have provided insights into the distribution of the sums of the weights and clarified the assumptions regarding independence and normality. There is an acknowledgment of differing interpretations of the problem setup, but no explicit consensus has been reached.

Contextual Notes

There is a discussion about the independence of the random variables representing the weights of the loaves and the implications of using different loaves in the calculations.

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The weight of a large loaf of bread is a normal variable with mean 420g and standard deviation 30g. The weight of a small loaf is a normal variable with mean 220g and standard deviation 10g.

1) Find the probability that 5 large loaves of bread are heavier than 10 small loaves.

My Working:
Let X be the weight of a large loaf, Y be the weight of a small loaf

X~N(420, 900) , Y~N(220,100)

5X>10Y
5X-10Y>0
X-2Y>0

E(X-2Y)=E(X) - 2E(Y)=420-440=-20
Var(X-2Y)=Var(X)+4Var(Y)=900+400=1300

P(X-2Y>0)=P(Z>20/(1300)^1/2)=P(Z>0.5547)=0.2896

But the answer given is totally different. Is there anything I miss in my working?
 
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Are [tex]X,Y[/tex] normally distributed random variables? If so, then:

[tex]Z_{X} = \frac{X-420}{30}[/tex], and [tex]Z_{Y} = \frac{Y-220}{10}[/tex]

Also [tex]var(aX+bY) = a^{2}var(X) + b^{2}var(Y) + 2abcov(X,Y)[/tex]

Since [tex]X,Y[/tex] are independent, then [tex]cov(X,Y) = 0[/tex]
 
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You are working with different loaves of bread here, with EACH of their weights being normally distributed. It would not be correct to consider the case where 5X > 10Y, because this would imply you are thinking about the scenario where 5 times the weight of ONE large loaf is greater than 10 times the weight of ONE small loaf.

Instead you should think about the distribution of [tex]X_{1} + X_{2} + X_{3} + X_{4} + X_{5}[/tex].

Observe that [tex]5X \sim N(2100,22500)[/tex] but [tex]X_{1} + X_{2} + X_{3} + X_{4} + X_{5} \sim N(2100,4500)[/tex], so these 2 distributions are indeed different.
 
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Thanks, I understand it now.