Problem set 9: Hypothesis testing in Gaussian Linear Regression

Problem 1

A new batch of coins has been produced. Before putting them into circulation, it was decided to test whether the coins have the same mass. Two coins were weighed using a scale whose measurement error follows the distribution \(\mathcal N(0,\sigma^2)\). The first coin was weighed \(n\) times and the second coin \(m\) times. All measurements were performed independently.

Construct an \(F\)-test for testing the hypothesis that the masses of the two coins are equal.

Problem 2

The measured quantity \(x\) is known to depend on the temperature \(t\) according to \[ x(t)=\beta_1+\beta_2t+\beta_3t^2. \]

A series of independent experiments was performed at the temperatures \[ t_1=-1,\qquad t_2=0,\qquad t_3=1,\qquad t_4=2,\qquad t_5=3. \]

The corresponding results were \[ X_1=1,\qquad X_2=6,\qquad X_3=10,\qquad X_4=14,\qquad X_5=19. \]

Assume that the measurement errors are independent and distributed according to \(\mathcal N(0,1)\). In addition, the optimal estimator \[ \widehat{\sigma}^2=0.4 \] of the error variance \(\sigma^2\) has been obtained.

Using an \(F\)-test, test the hypothesis \[ H_0:\quad \beta_1=\beta_2,\qquad \beta_3=0 \] at significance level \(\alpha=0.1\).

Problem 3

Let \(X_1,\ldots,X_n\) be a sample from \(\mathcal N(a_1,\sigma_1^2)\) and \(Y_1,\ldots,Y_m\) be a sample from \(\mathcal N(a_2,\sigma_2^2)\), where the two samples are independent.

Construct a test for the hypothesis \[ H_0:\quad \sigma_1^2=\sigma_2^2. \]

Problem 4

Let \(X_1,\ldots,X_n\) be a sample from \(\mathcal N(a_1,\sigma^2)\) and \(Y_1,\ldots,Y_m\) be a sample from \(\mathcal N(a_2,\sigma^2)\), where the samples are dependent. More precisely, \[ \{(X_i,Y_i)\}_{i=1}^n \] is a sequence of i.i.d. Gaussian random vectors.

Construct a test for the hypothesis \[ H_0:\quad a_1=a_2 \] and prove its consistency.

Problem 5

Let \(X_1,\ldots,X_n\) be a sample from \(\mathcal N(a_1,\sigma^2)\), \(Y_1,\ldots,Y_n\) a sample from \(\mathcal N(a_2,\sigma^2)\), and \(Z_1,\ldots,Z_n\) a sample from \(\mathcal N(a_3,\sigma^2)\).

Construct an \(F\)-test of size \(\alpha\) for testing the hypothesis \[ H_0:\quad a_1=a_2,\qquad a_1+a_2=a_3. \]

Problem 6

Let \(X_1,\ldots,X_n\) and \(Y_1,\ldots,Y_m\) be two independent samples.

Using the Wald test, construct a test of significance level \(\alpha\) for \[ H_0:\quad \mathbb E X_1=\mathbb E Y_1 \] against the alternative \[ H_1:\quad \mathbb E X_1\neq\mathbb E Y_1. \]

You may assume that \[ \frac{n}{m}=k, \] where \(k\) is a constant as \(n,m\to\infty\).

Problem 7

Construct an \(F\)-test of significance level \(\alpha\) for testing the hypothesis \[ H_0:\quad \beta_2=\beta_1 \] in Problem 4 from Week 8.

Problem 8

Seeds of a certain plant species were randomly assigned either to enriched soil (treatment group) or to standard soil (control group). After a fixed period of time, all plants were harvested, dried, and weighed. The resulting weights, in grams, are shown below.

The control observations \(X_1,\ldots,X_n\) are i.i.d. random variables with distribution \[ \mathcal N(\mu_X,\sigma^2), \] while the treatment observations \(Y_1,\ldots,Y_n\) are i.i.d. random variables with distribution \[ \mathcal N(\mu_Y,\sigma^2). \]

Test the hypothesis \[ H_0:\quad \mu_X=\mu_Y \] against the alternative \[ H_1:\quad \mu_X\neq\mu_Y. \]

Group	Observations
Control group	\(4.17,\ 5.58,\ 5.18,\ 6.11,\ 4.50,\ 4.61,\ 5.17,\ 4.53,\ 5.33,\ 5.14\)
Treatment group	\(4.81,\ 4.17,\ 4.41,\ 3.59,\ 5.87,\ 3.83,\ 6.03,\ 4.89,\ 4.32,\ 4.69\)