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.

## ISI MStat 2016 Problem 10:

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.

## Prerequisites:

1.Testing of Hypothesis

2.Model formation

3.Idea about RSS (Residual Sum of Squares)

## Solution:

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?

## Food For Thought:

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.

Replace sigma by s