Exam MAS-I Practice Question 167

Study guide: Exam MAS-I

$X_1, \dots, X_n$ is a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$ , with sample mean $\bar X$ and sample variance $S^2 = \sum (X_i - \bar X)^2/(n - 1)$ . Match each statistic to its exact distribution. An option may be used more than once or not at all.

Part a (1 point)

$\frac{1}{\sigma^2}\sum_{i=1}^{n} (X_i - \mu)^2$

A) $\chi^2(1)$
B) $\chi^2(n - 1)$
C) $\chi^2(n)$
D) $t(n - 1)$
E) $F(1,\ n - 1)$

Part b (1 point)

$\frac{(n - 1)S^2}{\sigma^2}$

A) $\chi^2(1)$
B) $\chi^2(n - 1)$
C) $\chi^2(n)$
D) $t(n - 1)$
E) $F(1,\ n - 1)$

Part c (1 point)

$\frac{n(\bar X - \mu)^2}{S^2}$

A) $\chi^2(1)$
B) $\chi^2(n - 1)$
C) $\chi^2(n)$
D) $t(n - 1)$
E) $F(1,\ n - 1)$

Part d (1 point)

$\frac{\bar X - \mu}{S/\sqrt{n}}$

A) $\chi^2(1)$
B) $\chi^2(n - 1)$
C) $\chi^2(n)$
D) $t(n - 1)$
E) $F(1,\ n - 1)$

Part e (1 point)

$\frac{n(\bar X - \mu)^2}{\sigma^2}$

A) $\chi^2(1)$
B) $\chi^2(n - 1)$
C) $\chi^2(n)$
D) $t(n - 1)$
E) $F(1,\ n - 1)$

Answer this question and see the worked solution