Q 8.144. - Elementary Statistics

Question

Let $0<\alpha <1$. For a $t-$curve, determine

a. the $t-$value having area $\alpha$ to its right in terms of $t_{\alpha }$

b. the $t-$value having area $\alpha$ to its left in terms of $t_{\alpha }$

c. the two $t-$values that divide the area under the curve into a middle $1-\alpha$ area and two outside $\alpha /2$ areas.

d. Draw graphs to illustrate your results in parts (a)-(c).

Step-by-Step Solution

Answer

Part (a) The graph is 

Part (b) The graph is 

Part (c) The graph is 

Part (a) Step 1: Given information

$0<\alpha <1$

Part (a) Step 2: Explanation

For a $t=$curve, the $t-$value having area $\alpha$ to its right is $t_{\alpha }$

Part (b) Step 1: Explanation

Because the $t-$curve is symmetric about $0$, the $t-$value with area $\alpha$ to its left is equal to the negative of the $t-$value with area $\alpha$ to its right.

$\therefore$ The $t$-value having area $\alpha$ to its left

=-($t-$value having area $\alpha$to its right )

$=-t_{\alpha }$ 

Part (c) Step 1: Explanation

To obtain the two $t-$values that divide the curve's area into two halves $1-\alpha$ area and two outside $\frac{\alpha }{2}$ areas i.e., to obtain two $t-$values such that one of it, has area $\frac{\alpha }{2}$ to its right and the other has area $\frac{\alpha }{2}$ to its left.