Question

Which of the following statements about the t distribution with degrees of freedom df is (are) true?

I. It is symmetric.

II. It has more variability than the t distribution with $df+1$degrees of freedom.

III. As df increases, the t distribution approaches the standard Normal distribution.

(a) I only (c) III only (e) I, II, and III

(b) II only (d) I and II

Short answer

(e) I, II and III are true.

Step-by-step solution

Given information

we have been given that

$t$distribution with degree of freedom $df$

Explanation

$I:$True, because all $t-$distributions are symmetric.

$II:$True, because as the degree of freedom decreases, the $t-$distribution becomes wider and thus the $t-$distribution has more variability (when it has a lower degrees of freedom).

$III:$True, because the larger the degree of freedom, the more closely the $t-$distribution resembles the standard Normal distribution.

Since, $I,IIandIII$are all correct, the correct answer option is$\left(e\right)I,IIandIII$