Question

Which of the following has the greatest probability?

a.$P(t>2)$if t has 5 degrees of freedom.

b. $P(t>2)$ if t has 2 degrees of freedom.

c. $P(z>2)$ if z is a standard Normal random variable.

d.$P(t<2)$if t has 5 degrees of freedom.

e.$P(z<2)$ if z is a standard Normal random variable.

Short answer

Option e)$\mathrm{P}(\mathrm{Z}<2)\text{if}\mathrm{z}\text{is a standard normal random variable}$

Step-by-step solution

Step 1:Given information

Normal random variable

Step 2:Explaination

Find the corresponding probability $P(z>2)andP(z<2)$ using the normal probability table

$P(z>2)=1-P(z<2)$

$=1-0.9772$

$=0.0228$

Although the total probability requires to be $P(t<2)$ is $0.95$when $P(t<2)$is $0.05$and $P(t<2)$ is

$0.90\text{when}P(t>2)\text{is}0.10$

$0.90<P(t<2)<0.95$

The probability$P(t>2)$ at $2$ degrees of freedom is the number (or interval) in the column title of the Student's T table in the appendix that contains the $t$-value in the row $df=2$

$\text{It is observed that the highest probability is}P(z<2)=0.9772\text{.}$

$\text{Hence, the correct option is (e)}$