Q 8.118. - Elementary Statistics

Question

For a $t-$curve with $\mathrm{df}=8$, find each $t-$value, and illustrate your results graphically.

a. The $t-$value having area $0.05$ to its right

b. $t_{0.10}$

c. The $t-$value having area $0.01$ to its left (Hint: A $t-$curve is symmetric about $0$

d. The two $t-$values that divide the area under the curve into a middle $0.95$ area and two outside $0.025$ areas

Step-by-Step Solution

Answer

Part (a) $df=8, t_{0.05}=1.860$

Part (b) $df=8, t_{0.10}=1.397$ 

Part (c) $=-t_{0.01}$ $=-2.896$

Part (d) $df=8, t_{0.025}=2.306$

Part (a) Step 1: Given information

For a $t-$curve with $\mathrm{df}=8$

Part (a) Step 2: Calculation

The degrees of freedom of the $t-$ curve is 8

We have to obtain the $t-$value having area $0.05$ to its right i.e. to obtain $t_{0.05}$ for $d f=8$

$\therefore$ For $df=8, t_{0.05}=1.860$ [From table IV of APPENDIX $-\mathrm{A}$ ]

Part (a) Step 2: Calculation

Create the Graph plot  

Part (b) Step 1: Calculation

We have to obtain $t_{0.10}$ for $d f=8$ For $df=8, t_{0.10}=1.397$

Part (c) Step 1: Calculation

We have to obtain $t-$ value having area $0.01$ to its left.

$t-$Curve is symmetric about 0 . So, $t-$value having area $0.01$ to its left is equal to the negative of the $t-$value having area $0.01$ to its right.

$\therefore$ With $d f=8, t$ - value having area $0.01$ to its left $=-t_{0.01}$

$=-2.896$

Part (c) Step 2: Explanation

Create the Graph plot  

Part (d) Step 1: Calculation

We must acquire the two $t-$values that divide the area under the curve into a $0.95$ area in the middle and two $0.025$ areas on either side.

i.e. We have to obtain $-t_{0.025}$ and $t_{0.025}$ for $d f=8$

$[\because t$-curve is symmetric about $0$]

From table IV,

For $df=8,t_{0.025}=2.306$

$\therefore$ The required two $t-$ values are $-2.306$ and $2.306$

Part (d) Step 2: Explanation

Create the Graph plot