This presentation identifies regions under the normal curve corresponding to different standard normal values. Week 4 M11/12SP-IIIc-3 and converts a normal random variable to a standard normal variable and vice versa.

1. 3rd Quarter
Statistics and
Probability

2. Module 4 :
Standard Normal Variable

3. OBJECTIVES:
After going through this module, you are
expected to:
1. convert a normal random variable to a
standard normal variable and vice versa
(M11/12 SP-IIIc-4)
2. compute probabilities and percentiles using
the standard normal table (M11/12 SP-IIIc-d-1)

4. Multiple Choice. Choose the letter
of the best answer.

5. 1. It is a distribution with a mean of 0 and a standard
deviation of 1.
a. Discrete Distribution
b. Probability Distribution
c. Normal Random Distribution
d. Standard Normal Distribution

6. 2. Which of the following formulas will you use to
convert a random variable into a standard normally
distributed variable?
a. =
𝑧 +
𝑥 𝜇 c. =
𝑧 𝜇−𝑥
𝜎 𝜎
b. =
𝑧 𝑥−𝜇 d. =
𝑧 +
𝜇 𝑥
𝜎 𝜎

7. 2. Which of the following formulas will you use to
convert a random variable into a standard normally
distributed variable?
a. =
𝑧 +
𝑥 𝜇 c. =
𝑧 𝜇−𝑥
𝜎 𝜎
b. =
𝑧 𝑥−𝜇 d. =
𝑧 +
𝜇 𝑥
𝜎 𝜎

8. 3. If a random variable x is greater than the mean,
then which of the following is TRUE about the z-score?
a. The z-score is at the left side of the mean.
b. The z-score is at the right side of the mean.
c. The z-score is the same position as the mean.
d. The z-score is either on the left side or on the right
side of the mean.

9. 4. Find the equivalent z-score of a random variable x =
85 when =75 =15.
𝜇 𝑎𝑛𝑑 𝜎
a. 0.6667
b. 0.667
c. 0.67
d. 0.7

10. 5. You have observed that it takes an average of 15 minutes with
a standard deviation of 1.5 minutes when you commute from
your house to your first period class in the morning. If the
computed z-score is -1.5, then what does it mean?
a. It means that you are late for the first-period class.
b. It means that you come early to the first-period class.
c. It means that you are just on time for the first-period class.
d. It cannot be determined since there is no time given.

11. Standard
Normal
Variable
01.

12. Standard Normal Variable
• The normal distribution focuses on its mean
and standard deviation to determine the
area under the normal curve.
• The curve of the distribution may vary to
its standard deviation.

13. Standard Normal Variable
• The larger the standard deviation, the
wider or disperse its distribution.
• The standard normal distribution is a
distribution with a mean of 0 and a
standard deviation of 1.

14. Standard Normal Variable
• The standard normal distribution is a
distribution with a mean of 0 and a
standard deviation of 1.

15. Z-Score or Z-Value
• All random variables or raw scores (x) can be
standardized using the formula for the standard
score or z-score:

16. Z-Score or Z-Value
In a Math Test, given mean is 77, and
standard deviation is 14. Find the z-score
value that corresponds to each of the
following scores ( in two decimal places).
1. X=63 2. X= 50 3. X= 112 4. X=84

18. Region Under Standard Normal Curve (CASE 1)
The region under the standard normal curve can be
determined using the z-table. This table shows the
four-decimal-place that describes the area from 0 to
a specified value of z.

19. Region Under Standard Normal Curve ( Case 1)
Example . Find the area under the standard normal
curve between 0 to 0.75.

20. Example . Find the area under the standard normal
curve between 0 to 0.75.
SOLUTION: Find the intersection of the two values to
determine the area under the normal curve.

21. Region Under Standard Normal
Curve
Example . Find the area under the standard
normal curve between -1.17 to 0.

22. Region Under Standard Normal
Curve
Example . Find the area under the standard
normal curve between -1.17 to 0.

23. Region Under Standard Normal Curve (CASE 2)
Example. Find the area under the standard normal
curve between -0.87 to 1.63.

24. Region Under Standard Normal Curve (CASE 2)

25. Region Under Standard Normal Curve (CASE 2)

26. Region Under Standard Normal Curve (CASE 2)

27. Region Under Standard Normal Curve (CASE 3)
Example. Find the area under the standard normal
curve between 0.42 to 1.28.

28. Region Under Standard Normal Curve (CASE 3)
Example. Find the area under the standard normal
curve between 0.42 to 1.28.

29. Region Under Standard Normal Curve (CASE 3)
Example. Find the area under the standard normal
curve between -2.16 to -0.33.

30. Region Under Standard Normal Curve (CASE 3)
Example. Find the area under the standard normal
curve between -2.16 to -0.33.

31. Region Under Standard Normal Curve (CASE 4)
Example . Find the area under the standard normal
curve to the right of z = 1.31 (z > 1.31)

32. Region Under Standard Normal Curve (CASE 4)
Example . Find the area under the standard normal
curve to the right of z = 1.31 (z > 1.31)

33. Region Under Standard Normal Curve (CASE 4)
Example. Find the area under the standard normal
curve to the left of z = -0.89 (z < -0.89)

34. Region Under Standard Normal Curve (CASE 4)
Example. Find the area under the standard normal
curve to the left of z = -0.89 (z < -0.89)

35. Region Under Standard Normal Curve (CASE 4)
Example. Find the area under the standard normal
curve to the left of z = 0.89 (z < 0.89)

36. Finding the Z-score,
Raw score, Mean ,and
SD

37. Problem: In a math exam, the mean score is
75 and the standard deviation is 10. If a
student scored 85 on the exam, what is their
Z-score?
Finding the Z-score

38. A student scored 92 in a chemistry test
where the mean score is 80 and the
standard deviation is 8. What is their Z-
score?
Finding the Z-score

39. In a statistics exam, the mean score is 60
with a standard deviation of 7. If a
student's Z-score is -1.2, what is their raw
score?
Finding the Raw Score (X)

40. In a science test, the mean score is 70
with a standard deviation of 5. If a
student's Z-score is 1.5, what is their raw
score?
Finding the Raw Score (X)

41. In an English test, a student scored 90, and the
standard deviation of the scores is 8. If the
student's Z-score is 2.5, what is the mean score
of the test?
Finding the Mean (μ)

42. A student received a score of 85 in a literature exam.
The standard deviation of the scores is 6. If the
student's Z-score is 0.5, what is the mean score of the
test?
Finding the Mean (μ)

43. In a history exam, the mean score is 65, and a student
has a Z-score of -1.2 with a raw score of 50. What is the
standard deviation of the scores?
Finding the Standard Deviation (σ)

44. In a physics test, the mean score is 78, and a student
has a Z-score of -0.75 with a raw score of 70. What is the
standard deviation of the scores?
Finding the Standard Deviation (σ)

45. 1. In an economics exam, the mean score is 68 and the standard
deviation is 6. If a student scored 80, what is their Z-score?
2. In a history exam, the mean score is 55 with a standard deviation
of 9. If a student's Z-score is -0.5, what is their raw score?
3. A student received a score of 95 in a geography test. The standard
deviation of the scores is 7. If the student's Z-score is 3, what is
the mean score of the test?
4. In a computer science exam, the mean score is 82, and a student
has a Z-score of 1.2 with a raw score of 88. What is the standard
deviation of the scores?
Assessment

46. Assessment

47. Assessment

48. Assessment

49. Assessment