Q. 23 - Elementary Statistics

Question

For the standard normal curve, find the z- score(s)

a. that has area 0.30 to its left. 

b. that has area 0.10 to its right.

c. $z_{0.025} , z_{0.05} , z_{0.01} and z_{0.005}.$

d. that divide the area under the curve into a middle 0.99 area and two outside 0.005 areas

Step-by-Step Solution

Answer

a): the z-score having area 0. 30 to its left under the standard normal curve is roughly -0.52 

b) :  the z-score having area 0 .10 to its right under the standard normal curve is roughly 1.28, 

c) : the z-score having area 0.025 to its right under the standard normal curve is roughly 1.96

      the z-score having area 0.05 to its right under the standard normal curve is roughly 1.645.

      the z-score having area 0.01 to its right under the standard normal curve is roughly 2.33.

      the z-score having area 0.005 to its right under the standard normal curve is roughly 2.575 .

d) : The required two z-scores are -2 575 and 2 575. 

Step 1. Part ( a )

Search the body of the table for the area 0.30. There is no such area, so use the area closest to 0.30, which is 0.3015. The z-score corresponding to that area is -0.52.

Thus the z-score having area 0. 30 to its left under the standard normal curve is roughly -0.52, as shown in figure.

Step 2. Part ( b )

The z-score that has an area of 0.10 to its right is equal to the z-score that has an area of 1

0.10 0.90 to its left Search the body of the table for the area 0.90. There is no such area, so use area closest to 0.8997, which is 1.28. Thus the z-score having area 0 .10 to its right under the standard normal curve is roughly 1.28, as shown in figure.

Step 3. part (c )

For $z_{0.025}$

Search the body of the table for the area 0.025. The z-score corresponding to that area is 196 Thus the z-score having area 0.025 to its left under the standard normal curve is-1.96 By applying the symmetry property the z-score having area 0.025 to its right under the standard normal curve is roughly 1.96

For $z_{0.005}$

Search the body of the table for the area 0.05. There is no such area, so use area closest to 0.05, which are 0.0495 and 0.0505. Here, we use the average of the two z-scores-1.64 and 1.65, which is-1.645,

Thus the z-score having area 0.05 to its left under the standard normal curve is roughly -1.645 By applying the symmetry property the z-score having area 0.05 to its right under the standard normal curve is roughly 1.645.

For $z_{0.01}$

Search the body of the table for the area 0.01 There is no such area, so use the area closest to 0.01, which is 0.0099. The z-score corresponding to that area is-2.33 Thus the 2-score having area 0.01 to its left under the standard normal curve is roughly-2.33. By applying the symmetry property the z-score having area 0.01 to its right under the standard normal curve is roughly 2.33

For $z_{0.005}$

Search the body of the table for the area 0.005. There is no such area, so use area closest to 0.005, which are 0 0049 and 0.0051 Here, we use the average of the two z-scores-2.57 and - 258, which is 2.575

Thus the z-score having area 0.005 to its left under the standard normal curve is roughly -2.575.

By applying the symmetry property the z-score having area 0.005 to its right under the standard normal curve is roughly 2.575

Step 4. Part ( d )

Search the body of the table for the area 0.005 There is no such area, so use area closest to 0.005, which are 0.0049 and 0.0051 Here, we use the average of the two z-scores-2:57 and 2.58, which is-2.575

thus the 2-score having area 0.005 to its left under the standard normal curve is roughly -2.575, as shown in figure. By applying the symmetry property the z-score having area 0.005 to its right under the standard normal curve is roughly 2.575, as shown in figure. The required two z-scores are -2 575 and 2 575.