Question

Suppose X1,...,X20 are a random sample of size 20 from N(100,100) population. What are the distributions of the following quantities?

(a) sample mean: X(bar) = (1/20) (X1 + . . . + X20);

(b) a scaled sample variance: (19/100)S^2, where S^2 = (1/19) *SIGMA from i=1 to 20* (Xi - X(bar))^2;

(c) standardized mean: (X(bar) - 100) / (10/sqrt(20));

(d) studentized mean: (X(bar) - 100) / (S/sqrt(20))

Answer 1

Result:

Suppose X1,...,X20 are a random sample of size 20 from N(100,100) population. What are the distributions of the following quantities?

(a) sample mean: X(bar) = (1/20) (X1 + . . . + X20);

Normal distribution

(b) a scaled sample variance: (19/100)S^2, where S^2 = (1/19) *SIGMA from i=1 to 20* (Xi - X(bar))^2;

Chi square distribution

(c) standardized mean: (X(bar) - 100) / (10/sqrt(20));

Standard normal distribution

(d) studentized mean: (X(bar) - 100) / (S/sqrt(20))

t distribution with degrees of freedom19.