Q 8.117. - Elementary Statistics

Question

For a $t-$curve with $\mathrm{df}=21$, find each $f-$value, and illustrate your results graphically.

a. The $t-$value having an area $0.10$ to its right

b. $t_0$ a $1$

c. The $t-$value has an area $0.025$ to its left (Hint: A $t-$curve is symmetric at about 0.)

d. The two $t-$values that divide the area under the curve into a middle $0.90$ area and two outside areas of $0.05$

Step-by-Step Solution

Answer

Part (a) $df=21, t_{0.10}=1.323$

Part (b) $df=21, t_{0.01}=2.518$

Part (c) $=-t_{0.025}$ $=-2.080$

Part (d) two $t-$values are $-1.721$ and $1.721$

Part (a) Step 1: Given information

For a $t-$curve with $\mathrm{df}=21$

Part (a) Step 2: Calculation

Degrees of freedom of the $t-$curve is $21$

We need to acquire the $t-$ value with the area $0.10$ to its right, i.e. $t_{0.10}$, from table-IV of APPENDIX-A.

For $df=21, t_{0.10}=1.323$

Part (a) Step 3: Explanation

Create the Graph plot 

Part (b) Step 1: Calculation

From table-IV of APPENDIX - A, we get For $df=21, t_{0.01}=2.518$

Create the Graph plot 

Part (c) Step 1: Calculation

We need to get $t-$the value with the area $0.025$ to its left.

Curve is symmetric around $t$ As a result, the negative of the $t-$ value with area $0.025$ to its left is identical to the $t-$ value with area $0.025$ to its right.

i.e. With $t-$ value having area $0.025$ to its left $=-t_{0.025}$

$=-2.080$

Part (c) Step 2: Explanation

Create the Graph plot

Part (d) Step 1: Calculation

The two $t-$ values that split the area under the curve into a middle $0.90$ area and outer $0.05$ areas must be obtained. i.e. We have to obtain $-t_{0.05}$ and $t_{0.05}$ for $d f=21$

Because $t-$curve is symmetric about $0$

$\therefore$ From table-IV,

For $df=21,t_{0.05}=1.721$

$\therefore$ The required two $t-$values are $-1.721$ and $1.721$

Part (d) Step 2: Explanation

Create the Graph plot