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.