Chapter 12, Problem 26PT4

To determine

To identify: the correct option among options (a) to (d) about statements I, II, III given below.

For a t distribution, with df degrees of freedom,

1. It is symmetric.
2. It has more variability than a t distribution with df+1 degrees of freedom
3. As df increases, the t distribution approaches the standard Normal distribution.

The options are given by

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

Answer to Problem 26PT4

Correct option is (e) I, II and III. All statements given above are correct.

Explanation of Solution

1. The statement is correct becuase
t distribution is symmetric about zero as it’s probability distribution function is symmetric about zero.

The probability function of t distribution is a function of the form below which is symmetric about zero

  

2. the statement is correct because
The variance of t distribution with df degrees of freedom is . Hence variance of t distribution with df+1 degree of freedom is . Let df be equal to x. This means that is to be proved consider,

  

Hence above statement is true that variance of t distribution with df degrees of freedom is more than that with df+1 degree of freedom.

3. This statement is also correct as the curve becomes mesokurtic as degrees of freedom increase and approaches normal distribution.

Conclusion: All three statements are correct and correct option is (e)