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This is a sample problem from ISI MStat 2016 Problem 10, which tests the student's ability to write a model and then test the equality of parameters in it using appropriate statistics.

A cake weighing one kilogram is cut into two pieces, and each piece is weighed separately. Denote the measured weights of the two pieces by \( X \) and \( Y \) . Assume that the errors in obtaining \( X \) and \(Y \) are independent and normally distributed with mean zero and the same (unknown) variance. Devise a test for the hypothesis that the true weights of the two pieces are equal.

1.Testing of Hypothesis

2.Model formation

3.Idea about RSS (Residual Sum of Squares)

Let us write the two cases in the form of a model:

\( X= \mu_1 + \epsilon_1 \)

\(Y = \mu_2 + \epsilon_2 \)

where, \( \mu_1,\mu_2 \) are the true weights of the two slices and \( \epsilon_1 , \epsilon_2 \sim N(0, \sigma^2) \) (independently).

So, you get \( X \sim N(\mu_1,\sigma^2) \) and \( Y \sim N(\mu_2, \sigma^2 ) \).

Also, see that \( X,Y \) are independent.

So, we need to test \( H_0: \mu_1=\mu_2 =\frac{1}{2} \) against \(H_1: \mu_1 \neq \mu_2 \).

See that, under \( H_0 \), \( X-Y \sim N(0,2 \sigma^2) \)

So, \( \frac{X-Y}{\sqrt{2} \sigma} \sim N(0,1) \).

But have you noticed that \( \sigma \) is unknown? So this isn't a statistic after all.

Can you replace \( \sigma \) by an appropriate quantity so that you can conduct the test?

Hint: What do you know about RSS? Does it estimate something?

Okay, let's move from cakes to doughnuts!!

Yeah, I know this is off topic and nothing related to statistics but it's good for the brain to alter cuisines once a while!

This is the famous **doughnut slicing problem**:

What is the largest number of pieces you can slice a doughnut into using only 3 cuts? (Note that you can only make planar cuts and you are not allowed to rearrange the pieces between the cuts)

I would request you to try this on your own without looking up solutions directly.

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Replace sigma by s