Statistical Inference

1. Dr. Abdul Aziz, Ph.D. in Business
Administration from University of Sindh.
abdul.aziz@fuuast.edu.pk

2. Dr. Abdul Aziz, Ph.D. in Business
Administration from University of Sindh.
abdul.aziz@fuuast.edu.pk

3. The Normal Distribution
Handout Two-Chapter-6
(Practice Problems)
Dr. Abdul Aziz, Ph.D. in Business
Administration from University of Sindh.
abdul.aziz@fuuast.edu.pk

4. 3 1 2 0 4 6 1 0 1 5 2 3 3 0 7 2 2 0 3 1 2 2 4 2 4 0 4 3 1 4
Q-4 From Handout One
50%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 6 7
25%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 6 7
75%
0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 6 7

5. Q-1: Explain why being able to obtain areas under the
standard normal curve is important.
Answer1: Refer Book and internet
Q-2: With which normal distribution is the standard
normal curve associated?
Answer2: Refer Book and internet
Q-3: Without consulting
z-table, explain why the
area under the standard
normal curve that lies to
the right of 0 is 0.5.
Answer3: Because the area under
the standard normal curve is equal
to 1 or 100% and 0 falls at the center
of standard normal curve and divides
into two parts.

6. Q-4: According to z-
Table, the area under the
standard normal curve
that lies to the left of
−2.08 is 0.0188.Without
further consulting z-table;
determine the area under
the standard normal curve
that lies to the right of
2.08. Explain your
reasoning.
Answer4: Above graphs can clearly be interpreted that 0 is in the middle of the
curve which divides the curve in two parts and each part represents 0.5 or 50% of
the area of the curve. Similarly we can see that area lies to the left of -3 is 0.0013
and area lies to right of 3 is also 0.0013. This pattern can be checked for -2 and 2, -1
and 1. It verifies the that area lies to the left of any minus number and same area lies
to the right of any positive number. We can say area lies to the right of 2.08 is

7. Q-5: According to z-Table, the area under the standa
curve that lies to the left of 0.43 is 0.6664. Withou
consulting z-table, determine the area under the
normal curve that lies to the right of 0.43. Exp
reasoning.
`
Answer5:
As discussed in the previous questions, the
area under the standard normal curve is
equal to 1 or 100%. If the area lies to the left
of 0.43 is 0.6664 (see graph yellow shaded
area) it is obvious that area lies to the right of
0.43 would 1-0.6664 = 0.3336 (see graph
green shaded area).
0.43

8. Dr. Abdul Aziz, Ph.D. in Business
Administration from University of Sindh.
abdul.aziz@fuuast.edu.pk
Q-6: According to z-Table, the area under the standard normal curve
that lies to the left of 1.96 is 0.975.Without further consulting z-table;
determine the area under the standard normal curve that lies to the
left of −1.96. Explain your reasoning.
Answer6:
Z-table summarizes the area of the
standard normal curve from left to
right. The area under the standard
normal curve that lies to the left of -3
is 0.0013, similarly area lies to the left
of -2 is 0.0228, -1 is 0.1587, 0 is 0.50,
1 is 0.8413, 2 is 0.9772, and 3 is
0.9987 respectively (See Graph).
We can see area lies to the left of 3 is
0.9987 and area lies to left of -3 is
0.0013 which is equal to 1-
0.9987=0.0013. The same you find
with 2 and -2 and 1 and -1. So area
lies to left of 1.96 is 0.975 then area
lies to the left of -1.96 would 1-0.975
= 0.025.
0.0013
0.0228
0.1587
0.50
0.8413
0.9772
0.9987

9. Q-7: Most of the area under the standard normal curve
lies between −3 and 3. Use Table II to determine
precisely the percentage of the area under the standard
normal curve that lies between −3 and 3.
Answer7:
As discussed in the previous questions,
the area under the standard normal
curve is equal to 1 or 100%. If we
subtract the area that lies to the left of -
3 and the area that lies to the right of +3
from 1 or 100% then we find area
between -3 and 3. According to graph
the area that lies to the left of -3 is
0.0013 and the area that lies to right of 3
is also 0.0013 now we subtract them
from 1 so 1 – (0.0013 + 0.0013) = 0.9974.
The area lies between -3 and 3 is (Green
shaded area) 0.9974 or 99.74%.
0.9974

10. Z-Table-Standard Normal Distribution

11. Q-8: Why is the standard normal curve sometimes referred to as the z-curve?
Answer8:
Z-curve follows the standard normal distribution, therefore standard normal curve
is also called z-curve.
Q-9: Explain how z-table is used to determine the area
under the standard normal curve that lies
A. to the left of a specified z-score.
B. to the right of a specified z-score.
C. Between two specified z-scores.
1
0.8413 or
84.13%
Answer9:
Follow the table in previous slide:
A. Z-table summarizes the area of the standard
normal curve from left to right. If someone finds
the area of any z-score that area would be
considered for left side area. For example area
of z-score 1 is 0.8413 given in the table will be
shown left side in graph(Orgage shaded area).

12. Q-9: Explain how z-table is used to determine the area
under the standard normal curve that lies
A. to the left of a specified z-score.
B. to the right of a specified z-score.
C. Between two specified z-scores.
1
Answer9:
Follow the table in previous slide:
B. Z-table summarizes the area of the standard
normal curve from left to right. If someone finds
the area of any z-score that area would be
considered for left side area. If you find the right
side area, first find the area from z-table then
subtract it from 1 that would be considered right
side area. For example right side area of z-score
1 is 1 - 0.8413= 0.1587 (Orange shaded area).
0.1587 or
15.87%
1 – 0.8413=0.1587

13. Q-9: Continued: C. Between two specified z-scores.
0.8413 or 84.13%
0.1587
0.8413 or 84.13%
(0.1587) or (15.87%)
0.6826 or 68.26%
Answer9:
Follow the table in previous slide:
C. As we know table summarizes left
side area. The area of -1 is 0.1587 (See
Graph) and the area of 1 is 0.8413
(See Graph). To find the area between
1 and -1, we would subtract -1 area
(0.1587) from +1 area (0.8413) we find
0.6826 i.e. the area between 1 to -1.

14. Use z-table to obtain the areas under the standard normal curve
required in Q-10 to Q-17. Sketch a standard normal curve and shade
the area of interest in each problem.
Q-10: Determine the area under the standard normal
curve that lies to the left of (Solve P-11)
A. 2.24. B. −1.56. C. 0. D. −4.
Answer4: Area of 2.24 is 0.9875, area of -1.56 is 0.0594, areas of 0 is 0.5 and -4 is 0.

15. Q-12: Find the area under the standard normal curve
that lies to the right of (Solve Q-13)
A. −1.07. B. 0.6. C. 0. D. 4.2.
Answer12: Z-table summarizes left side area, we know total area is equal to 1 or
100%. To find right side area we subtract left side area from 1. The right side area of -
1.07 is 1 – 0.1423 = 0.8577, area of 0.6 is 1 – 0.7257 = 0.2743, areas of 0 is 0.5 and 4.2
is 0.

16. Q-14: Determine the area under standard normal curve that lies between
A. −2.18 and 1.44. B. −2 and −1.5.C. 0.59 and 1.51. D. 1.1 and 4.2.
Answer14: Z-table summarizes left side area, to find middle area first we find the
area of largest number i.e. 1.44. The area of 1.44 is 0.9251, then find area of smallest
number -2.18 which is 0.0146. Now subtract 0.0146 (Smallest) from 0.9251(largest)
we find 0.9251 – 0.0146 = 0.9105. The area lie between -2.18 and 1.44 is 0.9105.
(Solve Parts B, C, and D. Refer Part A). Solve Q-15 to 21, refer Q-1 to Q-14.

17. Answer22: Z-table summarizes left side area, we have to
find z-score of 0.025 (left area). With the help of table we
found -1.96. This value is also shown graphically. (Solve
Q-23 to Q-25)
In Q-22 to Q-35, use z-table to obtain the required z-scores. Illustrate
your work with graphs.
Q-22: Obtain the z-score for which the area under the standard normal curve to its left is 0.025.

18. Answer26: We know z-table summarizes left side area, to
find 0.95 or 95% right area first we find 0.05 or 5% left area
z-score which is -1.645 (we took average of 0.05 and 0.04
and added with -1.6 because table does not contain exact
0.05 area rather it contains 0.0495 and 0.0505 areas). The
left area of -1.645 is 0.05, it is understood right side of -1.645
would be 0.95 or 95% because total area of curve is 1 or
100% This value is also shown graphically. (Solve Q-27 to Q-30)
Q-26: Obtain the z-score that has an area of 0.95 to its right.

19. Q-31: Determine the two z-scores that divide the area under the standard
normal curve into a middle 0.90 area and two outside 0.05 areas.
Answer31: We know z-table summarizes left side area, we
need to find z-score of 0.05 left area which is -1.645. But for
0.05 right area we need to find z-score of 0.95 left area
which is 1.645 (we took average of 0.05 and 0.04 and added
with ±1.6 because table does not contain exact 0.05 or 0.95
areas rather it contains 0.0495&0.0505 and 0.9495&0.9505
areas). The area between -1.645 to 1.645 is 0.90 or 90%. It is
also shown graphically. (Solve Q-32 to Q-34)

20. Q-35: A variable is normally distributed with mean 6 and standard
deviation 2. Find the percentage of all possible values of the variable that
A. lie between 1 and 7. B. exceed 5. C. are less than 4.
Answer35: First find z-scores by using the formula,
A. and , area of -2.5 is 0.0062 and the area of 0.5 is
0.6915. The area between -2.5 and 0.5 or 1 and 7 is 0.6915 – 0.0062 = 0.6853.
B. , z-score of 5 is -0.5. The area exceed 5 means right side area,
first find area of -0.5 which 0.3085(left area). Now find right area 1- 0.3085 = 0.6915.
C. , Less than 4 means left area. Area of -1 z-score is 0.1587 (Solve Q-26 to Q-38)

21. Q-39: A variable is normally distributed
with mean 6 and standard deviation 2.
A. Determine and interpret the quartiles
of the variable.
B Obtain and interpret the 85th
percentile.
C. Find the value that 65% of all possible
values of the variable exceed.
D. Find the two values that divide the
area under the corresponding normal
curve into a middle area of 0.95 and two
outside areas of 0.025. Interpret your
answer.
Answer39:
A. Quartile is the value at 25th percent
position when data is arranged in order
(smallest to largest). First we find z-score
of 0.25 which is -0.675. Now put it in z-
score formula:
Quartile of the given data is 4.65.

22. Q-39: A variable is normally distributed
with mean 6 and standard deviation 2.
B Obtain and interpret the 85th
percentile.
C. Find the value that 65% of all possible
values of the variable exceed.
D. Find the two values that divide the
area under the corresponding normal
curve into a middle area of 0.95 and two
outside areas of 0.025. Interpret your
answer.
Answer39:
B. First we find z-score of 0.85 which is 1.035. Now put it in z-score formula:
85th percentile of the given data is 4.65.
C. First we find z-score of 0.35 which is -0.47. Now put it in z-score formula:
65% of all possible values of variable exceed from 5.53.
D. Solve Part D. Refer previous problems.
(Solve Q-40 to Q-52, refer Q-1 to Q-39)

23. Answer53:
The symbol zα is used to denote the z-
score that has an area of α (alpha) to its
right under the standard normal curve.
68.26% area of the curve lie between ±1,
we can check area between -1 and 0 is
0.3413 or 34.13% and similarly area
between 0 and +1 is also 0.3413 or
34.13%. If we add both of them the area
will be 0.3413+0.3413 = 0.6826 or 68.26%,
see graph.
95.44% area of the curve lie between ±2,
we can check area between -2 and 0 is
0.4772 (between -1 and 0 is 0.3413 and
between -2 and -1 is 0.1359) and same for
0 and +2. If we add both of them the area
will be 0.4772 + 0.4772 = 0.9544 or
95.44%.
99.74% area of the curve lie between ±3.
For verification check graphs.
Q-53: What does the symbol zα signify? State the 68.26, 95.44, 99.74
rule. (Solve Q-54 to Q-60, refer Q-1 to Q-52)

24. Lynn Connaway
connawal@oclc.org
Thank YOU
Questions &
Suggestions
abdul.aziz@fuuast.edu.pk
abdulaziz2004@gmail.com
24