SEPT.

21-25, 2020 | Page 1 of 8

MANDAUE CHRISTIAN SCHOOL, INC.

P. Burgos St. Alang-Alang Mandaue City 6014
School ID 404573

Second Semester
All Strands

STATISTICS &
PROBABILITY
GRADE 11

Prepared by:
Mme. Maristela Ruelo
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TOPIC: CONSTRUCT A NORMAL CURVE DATE GIVEN: January 16, 2021

MIDTERM | MODULE 3 FOR WEEK 3 DUE DATE: January 21, 2021

MOST ESSENTIAL LEARNING COMPETENCIES:

✓ illustrate a normal random variable and its characteristics
✓ construct a normal curve
✓ identify regions under the normal curve corresponding to different standard normal
values
✓ converts a normal random variable to a standard normal variable and vice versa.
BIBLICAL INTEGRATION:
In making decisions, it is always right to call on to God for it is God who will guide us
to do great things that are good for people and would glorify God Jeremiah 33:3
James 3:13 The wisdom from God is what we truly need.
Proverbs 15:22, Proverbs 16:9, Proverbs 16:33

TEACHER’S INTRUCTIONS:
1. Read and understand the next lessons.
2. Follow the instructions given in each activity. Answer all questions indicating the
corresponding number and/or activity title. Write all your answers in your own
answer sheet.
3. Do all activities below, take a photo of your answers and send it via Gakkou.
4. Follow the deadline given in each activity and must be submitted on or before 8 PM
Keep learning for God’s glory! God bless and keep safe always 😊

LEARNING ACTIVITIES:

TASK 1:

How can you accurately solve real life problems involving normal distribution?

Introduction: Follow the instructions carefully and give what are asked. Write all your answers in a
file entitled “GR 11 ANSWER SHEET STAT & PROB MODULE 3” (Sample: “RUELO GR 11 ANSWER SHEET
STAT & PROB MODULE 3” in an organized manner.

A) Read about normal distribution and its properties/characteristics and give what are asked
below. Watch the video for more learning and better comprehension.
https://www.youtube.com/watch?v=cBz70MUnAYw

1) In your own words, define and describe a normal distribution.

2) Illustrate the following characteristics of normal distribution by drawing it. One drawing for each
letter.
a) The curve is symmetrical
b) The curve is asymptotic to the baseline
c) Approximately 68.3% of all values of a X lie within 1 standard deviation from the mean.
d) approximately 95.4% of all values of X lie within 2 standard deviations from the mean
e) About 99.7% of all values of a X lie within 3 standard deviations from the mean.

Normal Distribution
• also known as Gaussian Distribution in honor of Carl Friedrich Gauss because he was the
first to develop normal distribution.
• is a probability function that describes how the values of a variable are distributed.
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• It is a symmetric distribution where most of the observations cluster around the central peak
and the probabilities for values further away from the mean taper off equally in both
directions. Extreme values in both tails of the distribution are similarly unlikely.
• it is simply called Normal Curve or the Bell Curve
• it has a very important role in inferential statistics. It provides a graphical representation of
statistical values that are needed in describing the characteristics of a population as well as
in making decisions.
• it is defined by an equation that uses the population mean and the standard deviation.
• is the most important probability distribution in statistics because it fits many natural
phenomena. For example, heights, blood pressure, measurement error, and IQ scores follow
the normal distribution.
• In addition, there are several other reasons why the normal distribution is crucial in
statistics.
• Some statistical hypothesis tests assume that the data follow a normal distribution.
• Linear and nonlinear regression both assume that the residuals follow a normal distribution.
• The central limit theorem states that as the sample size increases, the sampling distribution
of the mean follows a normal distribution even when the underlying distribution of the
original variable is non-normal.

CHARACTERISTICS OF NORMAL CURVE

1) The distribution curve is bell-shaped

-when the data is presented through a bar graph or histogram, the picture will look like a bell.

2) The curve is symmetrical

-the left side of the bell curve is identical to the right side.

3) The mean, median and the mode coincide at the center.

- the mean, median and mode are located at the center of the curve and are equal.

4) The width of the curve is determined by the standard deviation of the distribution.

5) The curve is asymptotic to the baseline

- mean the tails of the curve flatten out indefinitely along the horizontal axis always approaching
the axis but will never touch it.

6) The total area under the curve is equal to 1. It is equivalent to 100%.

- the total area of the region that lies under the curve and above the horizontal axis is equal to 1.
- it represents the probability or proportion of the percentage associated with specific sets of
measurement values.

7) Empirical Rule of normally distributed random variable:

a) approximately 68.3% of all values of a X lie within 1 standard deviation from the mean.
- approximately this percent of all values lie in the interval (mean-standard deviation, mean
plus standard deviation) with about 34.1% on each side of the mean.

b) approximately 95.4% of all values of X lie within 2 standard deviations from the mean
- it is within the interval (mean-2 standard deviation, mean plus 2 standard deviation)

c) About 99.7% of all values of a X lie within 3 standard deviations from the mean.
- it is within the interval (mean-3 standard deviation, mean plus 3 standard deviation)

In the illustration below, the mean is equal to 0, the horizontal axis -4 to +4 refers to the
standard deviations. Those regions with the same color refer to the specific region that
contains the X within the specific interval and specific standard deviations of 1, 2, or 3. The
standard deviations of 4 contains very small number of x which is 0.1% to the left and to the
right. It refers to scores that are too big or too small. The region of the normal curve contains
100% in all.
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B) Read the information below and watch the video about z-scores, standardization, normal
distribution
https://www.youtube.com/watch?v=2tuBREK_mgE and answer the following questions.

3) What is standard normal curve? How is it different from normal distribution?

4) What are standard scores?
5) What is the importance of standard scores?
6) How can you standardize an observation x?

STANDARD NORMAL CURVE/ STANDARD NORMAL DISTRIBUTION

• is a special case of normal distribution where the mean is equal to 0 and standard deviation
also equal to 1.
• this distribution is also known as the Z-distribution
• a value on the standard normal distribution is known as a standard score or a Z-score.

Standard scores

• represent the number of standard deviations above or below the mean that a specific
observation falls. For example, a standard score of 1.5 indicates that the observation is 1.5
standard deviations above the mean. On the other hand, a negative score represents a value
below the average. The mean has a Z-score of 0.

• are a great way to understand where a specific observation falls relative to the entire
distribution. They also allow you to take observations drawn from normally distributed
populations that have different means and standard deviations and place them on a standard
scale. This standard scale enables you to compare observations that would otherwise be
difficult.

• came from a process called standardization, and it allows you to compare observations and
calculate probabilities across different populations. In other words, it permits you to compare
apples to oranges.
• To standardize your data, you need to convert the raw measurements into Z-scores.
• To calculate the standard score for an observation, take the raw measurement, subtract the
mean, and divide by the standard deviation. Mathematically, the formula for that process is the
following:
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X in the formula represents the raw value of the measurement of interest. Mu and sigma
represent the parameters for the population from which the observation was drawn.
After you standardize your data, you can place them within the standard normal distribution. In
this manner, standardization allows you to compare different types of observations based on
where each observation falls within its own distribution.

Performance Task:
Mini Task: Each student will analyze his/her performance in one subject during the midterm and final
exams of the first semester. He/she will compare which exam did he/she performed better. He/she will
provide reasons why his/her performance is like that and tell what must he/she do to have better
performance for the second semester. Be guided by the rubrics. This will be passed on February 18, 2021.

DRILL # 3: Read and answer the questions via Gakkou carefully. If you encounter problems while
answering, please notify me right away. Please write all your answers in the answer sheet entitled “GR
11 STAT & PROB DRILL 3” (Sample: “RUELO GR 11 STAT & PROB DRILL 3” in an organized manner and
send it to me via Gakkou. God bless you!

TOPIC: SOLVE REAL LIFE PROBLEMS ABOUT Z-SCORES DATE GIVEN: January 16, 2021
MIDTERM | MODULE 4 FOR WEEK 4 DUE DATE: January 28, 2021

MOST ESSENTIAL LEARNING COMPETENCIES:

✓ compute probabilities and percentiles using the standard normal table.
✓ formulate and solve real-life problems in different disciplines involving normal
distribution

C. Watch the video and explain how you can calculate the z-score and sketch the distribution.
https://www.youtube.com/watch?v=Wp2nVIzBsE8&t=268s find the z scores and construct the
normal curve

Once we have the general idea of the Normal Distribution, the next step is to learn how to
find areas under the curve. We'll learn two different ways - using a table and using
technology.
Since every normally distributed random variable has a slightly different distribution shape,
the only way to find areas using a table is to standardize the variable - transform our variable
so it has a mean of 0 and a standard deviation of 1. How do we do that? Use the z-score!

As we noted in this section, if the random variable X has a mean μ and standard deviation
σ, then transforming X using the z-score creates a random variable with mean 0 and standard
deviation 1! With that in mind, we just need to learn how to find areas under the standard
normal curve, which can then be applied to any normally distributed random variable.
Z-scores are related to the Empirical Rule from the standpoint of being a method of
evaluating how extreme a particular value is in a given set. You can think of a z-score as
the number of standard deviations there are between a given value and the mean of
the set. While the Empirical Rule allows you to associate the first three standard deviations
with the percentage of data that each SD includes, the z-score allows you to state (as
accurately as you like), just how many SDs a given value is above or below the mean.
Conceptually, the z-score calculation is just what you might expect, given that you are
calculating the number of SDs between a value and the mean. You calculate the z-score by
first calculating the difference between your value and the mean, and then dividing that
amount by the standard deviation of the set. The formula looks like this:

or
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Examples # 1.

SOLUTION:

NORMAL DISTRIBUTION STANDARD NORMAL DISTRIBUTION

Since the mean is 48, it should be written in the center of the normal distribution. Since the standard
deviation is 5, subtract 5 from 48 to get 43 then estimate 45 somewhere in between 48 and 43.
Then add 5 to 48 to get 53 which is the symmetrical to the right of the mean or the center. This is for
Normal distribution curve.

For standard distribution curve, the mean which is 48 corresponds to the mean that is 0 and the
standard deviation 5 corresponds to 1 standard deviation that is why 0-1 = -1 corresponds to 43 and
0+1=1 corresponds to 53. Then when you solve for the z-score using the formula, the answer -0.60
corresponds to the score 45.

Examples # 2.
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FINAL ANSWER:

Example # 3.

Example # 4.

Why Standardize ... ?

It can help us make decisions about our data.

1) Example: Professor Willoughby is marking a test.

Here are the student's results (out of 60 points):

20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17
Most students didn't even get 30 out of 60, and most will fail.
The test must have been really hard, so the Prof decides to Standardize all
the scores and only fail people 1 standard deviation below the mean.
The Mean is 23, and the Standard Deviation is 6.6, and these are the
Standard Scores:
-0.45, -1.21, 0.45, 1.36, -0.76, 0.76, 1.82, -1.36, 0.45, -0.15, -0.91
Now only 2 students will fail (the ones lower than −1 standard deviation)
Much fairer!

Example # 4.
Practical Applications of the Standard Normal Model

The standard normal distribution could help you figure out which subject you are getting good
grades in and which subjects you have to exert more effort into due to low scoring percentages.
Once you get a score in one subject that is higher than your score in another subject, you might
think that you are better in the subject where you got the higher score. This is not always true.
You can only say that you are better in a particular subject if you get a score with a certain
number of standard deviations above the mean. The standard deviation tells you how tightly
your data is clustered around the mean; It allows you to compare different distributions that
have different types of data — including different means.
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Read the following carefully and give what are asked. Write all your answers in a file entitled
“GR 11 ANSWER SHEET STAT & PROB MODULE 4” (Sample: “RUELO GR 11 ANSWER SHEET
STAT & PROB MODULE 4” in an organized manner. Show your solution.

1)

2)

3) The marks of a student across two subjects were recorded. General Math = 62, Calculus =
60. The mean and standard deviation for General Math were 70 and 15. The mean and
standard deviation for Calculus were 61 and 8. In which subject did the student perform
better in? Why?

4)

Construct the normal distribution curve and the standard distribution curve of this.

DRILL # 4: Read and answer the questions via Gakkou carefully. If you encounter problems
while answering, please notify me right away. Please write all your answers in the answer sheet
entitled “GR 11 STAT & PROB DRILL 4” (Sample: “RUELO GR 11 STAT & PROB DRILL 4” in an
organized manner and send it to me via Gakkou. God bless you!