statistics and probability lesson 4

1. Lesson 4
Solving Problems
Involving the Normal
Random Variable

2. Objectives
At the end of this lesson, the learner should be able to
 correctly recall how to locate the given scores on a
normal curve;
 correctly recall how to find the area of a region
under the normal curve; and
 solve problems involving the normal random
variable.

3. Essential Questions
 How can we easily solve problems involving the normal
random variable?
 How can the empirical rule help you in solving problems
involving the normal random variable?

4. Warm Up!
Before we learn about Solving Problems Involving the Normal
Random Variable, le us construct a normal curve by using an
interactive normal distribution.
Show the following areas:
1. 0.5
2. less than one standard deviation below the mean
3. more than two standard deviations above the mean

5. Warm Up!
(Click on the link to access the
website.)
Bognar, Matt. “Normal
Distribution.” Homepage Stat.
Retrieved 1 July 2019 from
http://bit.ly/2XTQZdF

6. Guide Questions
 How are you able to assign a mean and standard deviation
to show the given area in the normal curve?
 What rules do you follow in showing the given areas in the
normal curve?

7. Learn about It!
Normal Curve (or Bell Curve)
a graph that represents the probability density function of a normal probability
distribution. It is also called a Gaussian curve named after the Mathematician Karl
Friedrich Gauss.
1
Example:
The graph below represents the normal curve where 𝜇 is the
mean and 𝜎 is the standard deviation.

8. Learn about It!
Normal Curve Areas under the Empirical Rule
To solve problems involving the normal random variable, we
can use the following areas in the normal curve based from
the empirical rule.

9. Learn about It!
Normal Curve Areas under the Empirical Rule
A. Areas between the mean and one standard deviation
above or below the mean

10. Learn about It!
Normal Curve Areas Based on the Empirical Rule
B. Area between the mean and two standard deviations
above or below the mean

11. Learn about It!
Normal Curve Areas Based on the Empirical Rule
C. Areas between the mean and three standard
deviations above or below the mean

12. Try It!
Example 1: A teacher recorded the weight of his students
and got an average of 132.5 kg with a standard deviation of
13.4. Assuming that the weights are normally distributed,
what percent of the students are between 119.1 kg and 159.3
kg?

13. Try It!
Solution:
1. Construct the normal curve of the distribution and locate
the given weights.
Example 1: A teacher recorded the weight of his students
and got an average of 132.5 kg with a standard deviation of
13.4. Assuming that the weights are normally distributed,
what percent of the students are between 119.1 kg and 159.3
kg?

14. Try It!
2. Shade the area of the normal curve based on the problem.
The problem asks for the percent of the students between
119.1 kg and 159.3 kg. Thus, shade the area between these
two weights in the normal curve.

15. Try It!
3. Find the area of the region.
To find the area of the region, add the given areas below.
0.3413 + 0.4772 = 0.8185

16. Try It!
4. Convert the decimal to percent.
0.8185 100 = 81.85%
Thus, there are 𝟖𝟏. 𝟖𝟓% of the students who are between
119.1 kg and 159.3 kg.

17. Try It!
Example 2: The average price of carrots was ₱60.5 per
kilogram with a standard deviation of 2.4 for the last three
months. If the price is normally distributed for 90 days, what
percent of 90 days when the price was above ₱62.9?

18. Try It!
Solution:
1. Construct the normal curve of the distribution and locate
the given price.
Example 2: The average price of carrots was ₱60.5 per
kilogram with a standard deviation of 2.4 for the last three
months. If the price is normally distributed for 90 days, what
percent of 90 days when the price was above ₱62.9?

19. Try It!
2. Shade the area of the normal curve based on the problem.
The problem asks for the percent of 90 days that the price
was above ₱62.9. Thus, shade the area from 62.9 and above.

20. Try It!
3. Find the area of the region.
To find the area of the region, subtract the given area below
from 0.5.
0.5 − 0.3413 = 0.1587

21. Try It!
4. Convert the decimal to percent.
0.1587 100 = 15.87%
Thus, there are 𝟏𝟓. 𝟖𝟕% of 90 days that the price was above
₱62.9.

22. Let’s Practice!
Individual Practice:
1. The average weight of students in a school is 134.6 kg with a
standard deviation of 3.8. Mrs. Antazo wants to identify how
many students have weight between 123.2 and 138.4 kg.
Assuming that the weights are normally distributed, what
percent of the students can Mrs. Antazo identify?

23. Let’s Practice!
Individual Practice:
2. A mechanic can finish repairing a car in an average of 2.3
hours with a standard deviation of 0.12. Given that the
number of hours to repair a car is normally distributed, what
is the percentage of the cars repaired in less than 2.54 hours?

24. Let’s Practice!
Group Practice: To be done in groups of five.
Carlo saves money every day for 60 days. His savings are
normally distributed with a mean of ₱34.8 and a standard
deviation of 8.2. How many days was he able to save money
less than ₱43?

25. Key Points
Normal Curve (or Bell Curve)
a graph that represents the probability density function of a normal probability
distribution. It is also called a Gaussian curve named after the Mathematician Karl
Friedrich Gauss.
1

26. Key Points
Normal Curve Areas Based on the Empirical Rule
A. Area between the mean and one standard deviation
above or below the mean

27. Key Points
Normal Curve Areas Based on the Empirical Rule
B. Area between the mean and two standard deviations
above or below the mean

28. Key Points
Normal Curve Areas Based on the Empirical Rule
C. Area between the mean and three standard deviations
above or below the mean

29. Synthesis
 What strategies did you use in solving each problem
involving the normal random variable?
 Does planning play a vital role in solving problems? Why or
why not?
 How can we convert a random variable to a standard
normal variable and vice versa?