DAILY LESSON LOG OF M11/12SP-IVa-1 (Week One- Day One)
School
Teacher
Teaching Date
and Time
I. OBJECTIVES
A. Content Standards
B. Performance Standards
C.
Learning Competencies
/ Objectives
II. CONTENT
Grade Level
Learning
Area
Quarter
11/12
Statistics and
Probability
Fourth
Objectives must be met over the week and connected to the curriculum standards. To meet the
objectives, necessary procedures must be followed and if needed, additional lessons, exercises
and remedial activities may be done for developing content knowledge and competencies. These
are assessed using Formative Assessment Strategies. Valuing objectives support the learning of
content and competencies and enable children to find significance and joy in learning the lessons.
Weekly objectives shall be derived from the curriculum guides.
The learner demonstrates understanding of key concepts of tests of
hypotheses on the population mean and population proportion.
The learner is able to perform appropriate tests of hypotheses involving the
population mean and population proportion to make inferences in real-life
problems in different disciplines.
Learning Competency:
Illustrates: (a) null hypothesis (b) alternative hypothesis (c) level of
significance (d) rejection region; and (e) types of errors in hypothesis
testing. M11/12SP-IVa-1
Learning Objectives:
1. Illustrates the null hypothesis and alternative hypothesis in words
and symbols.
2. Identify the level of significance, rejection region and types of errors
in hypothesis testing.
3. Show appreciation to the concept of hypothesis testing
Null and Alternative Hypothesis, Level of Significance, Rejection Region
and Types of Errors
III.LEARNING RESOURCES
A. References
1. Teacher’s Guide pages
Statistics and Probability (Belecina, Baccay, Mateo) by Rex Bookstore pp. 6770
2.Learner’s Materials pages
3.Textbook pages
Statistics and Probability (Belecina, Baccay, Mateo) by Rex Bookstore pp.
216-232
4.Additional Materials from
Learning Resource (LR) portal
B. Other Learning
Resources
IV.PROCEDURES
These steps should be done across the week. Spread out the activities appropriately so that
pupils/students will learn well. Always be guided by demonstration of learning by the pupils/
students which you can infer from formative assessment activities. Sustain learning systematically
by providing pupils/students with multiple ways to learn new things, practice the learning,
question their learning processes, and draw conclusions about what they learned in relation to
their life experiences and previous knowledge. Indicate the time allotment for each step.
The teacher states the following to the class:
We make decisions every day. Some of these are important while others are not. In decisionmaking, we usually follow certain processes: weigh alternatives, collect evidence, and make a
decision. After a decision is made, an appropriate interpretation is made (or an action is
undertaken). We follow these basic processes in testing hypothesis in Statistics.
The teacher uses a PowerPoint Presentation in presenting the following
data/facts.
A. Review previous lesson
or presenting the new
lesson (8 minutes)
You hear claims involving statistics all the time; the media has no shortage of them:

Twenty-five percent of all women in our community have varicose veins.

A 6-month-old baby sleeps an average of 14 to 15 hours in a 24-hour period.

A name-brand ready-mix pie takes only 5 minutes to make.
In today’s age information (and big money), a great deal rides on being able to back up your
claims. Companies that say their products are better than the leading brand had better be able
to prove it, or they could face lawsuits. Drugs that are approved by the BFAD have to show strong
evidence that their products actually work without producing life-threatening side effects.
Manufacturers have to make sure their products are being produced according to specifications
to avoid recalls, customer complaints, and loss of business.
When we make decisions in real life, we usually associate them with probability values. For
example, when we buy a lottery ticket, we hope for a 100% win. Of course, we do not look forward
to a loss. In statistical inference, we also make statements of probability when we decide about
the null hypothesis.
ACTIVITY 1 What mistakes do people make?
Read the following statements and identify the errors. Discuss the situations
with your classmates.
1. Bryan thinks that he is a six-footer. His actual height is 156 cm.
2. On a moonlit night, a young man declares that there are two moons.
3. Mark says “I am virtuous!” In the next moment, he finds himself in jail.
4. Thousands of years ago, Ptolemy declared that the earth is flat.
5. On a beachfront, a signage reads, “No littering of plastic wrappers, empty
bottles, and cans.” A few yards away, environmentalists are picking up
the rubbish left behind by picnic lovers.
B. Establishing a purpose for the
lesson (2 minutes)
B. Presenting examples/
instances of the new
lesson
(15minutes)
The teacher presents the objectives of the lesson to the class and let the
students illustrate: (a) null hypothesis (b) alternative hypothesis (c) level of
significance (d) rejection region; and (e) types of errors in hypothesis
The teachers discuss the concepts null hypothesis, alternative hypothesis,
level of significance, rejection region and types of errors in hypothesis. (See
appendix A.)
The teacher may opt to discuss one concept then proceed to the item
activity related to it or to discuss all then have the students answer all the
items activity below.
Example are given in order for the students to identify and rewrite the null
and alternative hypotheses into symbols.
Example # 1 : Bottled Fruit Juice Content
The owner of a factory that sells a particular bottled fruit juice claims that the
average capacity of a bottle of their product is 250mL.
In the example, the owner’s statement (called claim) is a general statement.
The claim is that the capacity of all their bottled products is 250mL per bottle.
A consumer group may generalize that the bottled product is short of the
claim. If this can be proven, then the factory owner is lying. The evidence has
to be established. So the consumer group gets interested to know if, in reality,
each bottle contains 250 mL.
Thus, the two hypotheses would be:
H0: The bottled drinks contain 250 mL per bottle. (This is the claim.)
H1: The bottled drinks do not contain 250 mL per bottle. (This is the opposite
of the claim.)
C.
Discussing new
concepts and practicing
new skills #1
(10 minutes)
But these statements should be written in symbols. For now, let us drop the
unit measure and simply write:
𝐻0: 𝜇1 = 250 and H1: 𝜇 ≠ 250
Example # 2 : Working Students
A university claims that working students earn an average of Php 20 per hour.
H0: The working students earn an average of Php 20 per hour. (This is the
claim.)
H1: The working students do not earn an average of Php 20 per hour. (This is
the opposite of the claim.)
Ask the students to write the null and alternative hypotheses in symbols.
Answer: H0: 𝜇 = 250 and H1: 𝜇 ≠ 250
Example # 3 : Songs on an MP3 player
Suppose that 𝜇 is the average number of songs on an MP3 player owned by a
student. Write down the description of the null hypothesis H0: 𝜇 = 228.
Answer:
H0: The average number of songs on an MP3 player is 228.
Example # 4 : Songs on an MP3 player
In example number 3, write down the description of the alternative
hypothesis H1: 𝜇 ≠ 228.
Answer:
H0: The average number of songs on an MP3 player is not 228.
Understanding Errors
Task: Study the following examples carefully and the notes that follow.
Discuss for better understanding of hypothesis testing.
Example 1: Maria’s Age
Maria insists that she is 3o years old when, in fact, she is 32 years old. What
error is Mary committing?
Solution: Mary is rejecting the truth. She is committing a Type I error.
Example 2: Stephen’s Hairline
Stephen says that he is not bald. His hairline is just receding. Is he
committing an error? If so, what type of error?
Solution: Yes. A receding hairline indicates balding. This is a Type I error.
Stephen’s action may be to find remedial measures to stop falling hair.
Example 3: Monkey-Eating Eagle Hunt
A man plans to go hunting the Philippine monkey-eating eagle believing that
it is a proof of his mettle. What type of error is this?
Solution: Hunting the Philippine eagle is prohibited by law. Thus, it is not a
good sport. It is a Type II error. Since hunting the Philippine monkey-eating
eagle is against the law, the man may find himself in jail if he goes out of his
way hunting endangered species.
D. Discussing new
concepts and practicing
new skills #2
(10 minutes)
In decision that we make, we form conclusions and these conclusions are the
bases of our actions. But this is not always the case in Statistics because we
make decisions based on sample information. The best that we can do is to
control the probability with which an error occurs.
The probability of committing a Type I error is denoted by the Greek letter 𝛼
(alpha) while the probability of committing a Type II error is denoted by 𝛽
(beta).
The following table shows the probability with which decisions occur.
Table 5.2. Types of Errors
The teacher shows the decision errors under the normal curve. Afterwhich, she
explains the following:
Under the normal curve, the rejection region refers to the region where the
value of the test statistic lies for which we will reject the null hypothesis. This
region is also called critical region.
So, if your computed statistic is found in the rejection region, then you reject
H0. If it is found outside the rejection region, you accept H0.
The teacher gives an input on what level of significance is. He/she gives
examples to deepen the concept.
ACTIVITY 3
1. A study claims that adults watch television an average of 1.8 hours
per day. Identify which of the following statements is the null and
alternative hypotheses.
a.
b.
The average hours per day of adults watching television is 1.8.
The average hours per day of adults watching television is not 1.8.
After identifying, write the null and alternative hypotheses in symbols.
2.
E.
Developing mastery
(leads to formative
assessment 3)
(5 minutes)
Individually, give an erroneous statement and identify the type of
error being emphasized.
3. Complete the following summary table of critical values.
Confidence Level
2-tailed
1-tailed
−𝑧𝛼⁄2 =
95%
-z=
+z=
+𝑧𝛼⁄ =
2
99%
−𝑧𝛼⁄2 =
+𝑧𝛼⁄2 =
Answer Key
1. A. Ho, Ho=1.8 B. Ha,Ha≠1.8
2. Answers may vary
3.
Confidence Level
2-tailed
−𝑧𝛼⁄2 =-1.96
95%
+𝑧𝛼⁄2 = +1.96
−𝑧𝛼⁄2 =-2.58
99%
+𝑧𝛼⁄2 = +2.58
G.Finding practical applications
of concepts and skills in daily
living
H.Making generalizations and
abstractions about the lesson (2
minutes)
-z=
+z=
1-tailed
-z=-1.65
+z=+1.65
-z=-2.33
+z=+2.33
The teacher emphasizes on the mathematical concepts of (a) null hypothesis
(b) alternative hypothesis (c) level of significance (d) rejection region; and
(e) types of errors in hypothesis
A. Identify if the given hypothesis about the statement is the null or the
alternative hypothesis and write it in symbols.
1. An airline company claims that its flight arrive early 92% of the time.
H? = All of the flights arrive early 92% of the time.
In symbols, _________________________________________
2.
An investment company claims that its clients make an average of 8%
return on investments every year.
H? = Clients do not make an average of 8% return on investments every
year.
In symbols, _________________________________________
I.
Evaluating
Learning
(8 minutes)
3.
The General Manager of Fresh’n’Easy air conditioners tells an
investigative reporter that at least 85% of its customers are “completely
satisfied” with their overall purchase performance.
H? = Customers who are completely satisfied with their overall purchase
performance is lesser than 85%.
In symbols, _________________________________________
4.
A student counselor claims that Grade 7 Math students spend an
average 3 hours per week doing exercises in each subject.
H? = The average number of hours per week spent by students in doing
exercises in each subject is 3 hours.
In symbols, ___________________________________________
B.
1.
Answer the following questions.
Which of the following describes a Type I error?
a. Accepting the null hypothesis when it is true
b. Failing to accept the alternative hypothesis when it is true
2.
C.
c. Rejecting the null hypothesis when it is true
d. Failing to reject the alternative hypothesis when it is false
e. None of the above
Which of the following is a description of the power of the test?
a. The probability of accepting the alternative hypothesis when it
is true
b. The probability of failing to accept the alternative hypothesis
when it is true
c. The probability of rejecting the null hypothesis when it is false
d. None of the above
For each of the given, do the following:
First, draw the normal curve. Second, locate the z-value. Third,
indicate if the z-value is in the rejection region or in the acceptance
region.
1. z=2, 95% confidence, two-tailed
2. z=-2.68, 95% confidence, two-tailed
3. z=1, 95% confidence, one-tailed
4. z=1.33, 99% confidence, one-tailed
5. z=-4, 99% confidence, two-tailed
The students exchange their papers afterwards. Clarifications are
done while checking.
Answer Key
A.
1. Ho=92%
2. Ha≠8%
3. Ha<85%
4. Ho=3
B.
1. c. Rejecting the null hypothesis when it is true
2. c. The probability of rejecting the null hypothesis when it is false
C.
1.
reject Ho
2.
reject Ho
3.
or
accept Ho
4.
or
accept Ho
5.
reject Ho
J.Additional activities or
remediation
V.REMARKS
VI.REFLECTION
Reflect on your teaching and assess yourself as a teacher. Think about your
students’ progress. What works? What else needs to be done to help the
pupils/students learn? Identify what help your instructional supervisors can
provide for you so when you meet them, you can ask them relevant questions.
A.No. of learners who earned
80% of the evaluation
B.No. of learners who require
additional
activities
for
remediation who scored below
80%
C.Did the remedial lesson work?
No. of learners who have caught
up with the lesson.
D.No. of learners who continue
to require remediation
E.Which
of
my
teaching
strategies worked well? Why did
these work?
F.What
difficulties
did
I
encounter which my principal or
supervisor can help me solve?
G.What innovation or localized
materials did I use/ discover
which I wish to share with other
teachers
APPENDIX A
Discussion through PowerPoint Presentation:
Typically in a hypothesis test, the claim being made is about a population parameter (one number that characterizes the entire population).
Because parameters tend to be unknown quantities, everyone wants to make claims about what their values may be. For example, the claim
that 25% (or 0.25) of all women in our community have varicose veins is a claim about the proportion (that’s the parameter) of all the
women in the community (that’s the population) who have varicose veins (that’s the variable – having or not having varicose veins).
Researchers often challenge claims about population parameters. You may hypothesize, for example, that the actual proportion of women
who have varicose veins is lower than 0.25, based on your observations. Or you may hypothesize that due to the popularity of high heeled
shoes, the proportion may be higher than 0.25. or if you’re simply questioning whether the actual proportion is 0.25, your alternative hypothesis
is: “No, it isn’t 0.25.”
There are two types of hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis is what we want to test. It states
an exact value about the parameter. When the null hypothesis is accepted, the buck stops there! But when the null hypothesis is rejected, this
leads to another option, which is the alternative hypothesis that allows for the possibility of many values.
The null hypothesis, is denoted by H0, is a statement that there is no difference between a parameter and a specific value, or that there is no
difference between two parameters.
The alternative hypothesis, denoted by H1, is a statement that there is a difference between a parameter and a specific value, or that there is
a difference between two parameters.
In short, the process of hypothesis testing involves making a decision between two opposing hypotheses. These two hypotheses are formulated
in such a way that one is a negation of the other. If one is true, the other must be false. That is why one hypothesis is tested to show that it
cannot happen. If the improbability of occurrence can be established, then the other hypothesis is likely to occur. Between the null hypothesis
and the alternative hypothesis, it is usually the null hypothesis that is subjected to the rigor of a statistical test.
Suppose the two parameters of interest are denoted by 𝜇1 and 𝜇2 . If there is no difference between the two values, the relationship is written
in symbols as:
𝜇1 − 𝜇2 = 0
So, the null hypothesis would be written in symbols as:
𝐻0: 𝜇1 = 𝜇2
The null hypothesis is the starting point of investigation. Thus, it is the first statements to be made. You might ask: why start with the null
hypothesis? The sequence of the arguments is like the situation of a case brought to court where the accused is presumed ‘not guilty’ at the
start. Then, evidences are collected and evaluated following a standard procedure. At the end of the process, a decision is made as to whether
‘not guilty’ should be rejected or not rejected.
When the alternative hypothesis utilizes the ≠ symbol, the test is said to be non-directional. A non-directional test is also called a two-tailed
test.
When the alternative hypothesis utilizes the > 𝑜𝑟 𝑡ℎ𝑒 < symbol, the test is said to be directional. A directional test may be left-tailed or righttailed.
Toward the end of a hypothesis testing exercise, based on the evaluation of the data at hand, a decision is made about the null hypothesis:
Should the null hypothesis, H0 be rejected or not rejected (i.e., accepted)? It is logical to state that if there is evidence to warrant the rejection
of the null hypothesis, then there is a stand by hypothesis to be accepted. This is the role of the alternative hypothesis. Should you decide to
accept the null hypothesis after considering the evidences, then you stop there as there is no need for an alternative hypothesis. Remember:
No two things can be and cannot be at the same time. This is a mathematical principle.
In hypothesis testing, we make decisions about the null hypothesis. Of course, there are risks when we make decisions. When we conduct a
hypothesis test, there are four possible outcomes. The following decision grid shows these four outcomes (Table 5.1). Again, note that the
decision is focused on the null hypotheses.
Table 5.1 Four Possible Outcomes in Decision-Making
Reality
H0 is true.
H0 is false.
Decisions about the H0
Do not Reject
Reject
H0
(or Accept H0)
Correct
Type I error
decision
Correct decision
Type II error
If the null hypothesis is true and accepted, or if it is false and rejected, the decision is correct. If the null hypothesis is true and rejected, the
decision is incorrect and this is a Type I error. If the null hypothesis is false and accepted, the decision is incorrect and this is a Type II error.
In an ideal situation, there is no error when we accept the truth and reject what is false.
We can control the errors by assigning small probability values to each of them. The most frequently used probability values for 𝛼 𝑎𝑛𝑑 𝛽 are
0.05 and 0,01 respectively. The probability assigned to each depends on its seriousness. The more serious the errors, we are to have it occur.
So, a smaller probability will be assigned. The symbols 𝛼 𝑎𝑛𝑑 𝛽 are each probabilities of error, each under separate conditions, and they
cannot be combined. Therefore, there is no single probability for making an incorrect decision. In like manner, the two correct decisions are
distinct and each has its own probability. As can be seen from Table 5.2, 1 − 𝛼 is the probability of correct decision when the null hypothesis is
true, and 1 − 𝛽 is the probability of a correct decision when the null hypothesis is false. 1 − 𝛽 is called the power of the statistical test since it
is the measure of the ability of a hypothesis test to reject a false null hypothesis which is considered very important. (McClave & Sinich, 2003)
Graphically we can show the decision errors under the normal curve.
Note that the rejection region for directional test is in one tail (first figure) but distributed to the two tails in a non-directional test (second
figure).
Under the normal curve, the rejection region refers to the region where the value if the test statistic lies for which we will reject the null
hypothesis. This region is also called the critical region.
So if your computed statistic is found in the rejection region, then you must reject Ho. If it is found outside the rejection region, you accept Ho.
Note also that the line that separates the rejection region from the non-rejection region (1 − 𝛼) This line passes through the confidence
coefficients, which are also called critical values. The critical values can be obtained from the critical values table of the test statistic. For
example, if the test statistic is a z, the critical values can be obtained from the z-table. The critical values for a non-directional test are -1.96
and +1.96. When the confidence level is 99%, for a non-directional test, the critical values are -2.58 and +2.58.
(Attach Answer Key)