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

• Harmony
In summary, the weight of a large loaf of bread is normally distributed with a mean of 420g and a standard deviation of 30g, while the weight of a small loaf is normally distributed with a mean of 220g and a standard deviation of 10g. To find the probability that 5 large loaves are heavier than 10 small loaves, we need to consider the distribution of the sum of the weights of the 5 large loaves, which is different from simply multiplying the weight of one large loaf by 5.
Harmony
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?

Are $$X,Y$$ normally distributed random variables? If so, then:

$$Z_{X} = \frac{X-420}{30}$$, and $$Z_{Y} = \frac{Y-220}{10}$$

Also $$var(aX+bY) = a^{2}var(X) + b^{2}var(Y) + 2abcov(X,Y)$$

Since $$X,Y$$ are independent, then $$cov(X,Y) = 0$$

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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 $$X_{1} + X_{2} + X_{3} + X_{4} + X_{5}$$.

Observe that $$5X \sim N(2100,22500)$$ but $$X_{1} + X_{2} + X_{3} + X_{4} + X_{5} \sim N(2100,4500)$$, so these 2 distributions are indeed different.

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Thanks, I understand it now.

## What is Normal Distribution?

Normal distribution, also known as Gaussian distribution, is a probability distribution that is symmetric around the mean value. It is commonly used in statistics to model continuous random variables such as heights, weights, and test scores.

## What does the shape of a Normal Distribution curve look like?

The shape of a Normal Distribution curve is bell-shaped, with the highest point at the mean value and the curve symmetrical on both sides. The curve approaches zero as it extends to positive and negative infinity.

## What is the Central Limit Theorem and how does it relate to Normal Distribution?

The Central Limit Theorem states that when independent random variables are added, their sum tends towards a normal distribution regardless of the shape of the original distribution. This means that many real-world data sets can be approximated by a normal distribution, making it a valuable tool in statistical analysis.

## How is Normal Distribution used in hypothesis testing?

In hypothesis testing, Normal Distribution is used to determine the probability of obtaining a sample mean that is significantly different from the population mean. The z-test and t-test, which are commonly used in hypothesis testing, assume that the data follows a normal distribution.

## Can a data set have a perfect Normal Distribution?

In theory, a data set can have a perfect Normal Distribution, but in practice, it is very rare. Many factors, such as sample size and outliers, can affect the shape of the distribution. It is more common for data sets to have a close approximation to a normal distribution rather than a perfect one.