Republic of the Philippines

Department of Education
Regional Office IX, Zamboanga Peninsula

11 Zest for Progress

Z Peal of artnership

4th QUARTER – Module 1:

HYPOTHESIS TESTING

Name of Learner: ___________________________

Grade & Section: ___________________________
Name of School: ___________________________
Mathematics – Grade 11
Alternative Delivery Mode
Quarter 4 - Module 1: Hypothesis Testing
First Edition, 2020
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Published by the Department of Education

Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writer: Dickenson L. Canizares
Editors: Mercedita B. Tarog
Illustrator: Edward S. Duarte
Layout Artist: Abdurauf J. Baldomero
Reviewers: EPS, Mathematics Vilma A. Brown, Ed. D.
SHS Master Teacher Enrico P. Minao
Management Team: SDS Roy C. Tuballa, EMD, JD, CESO VI
ASDS Jay S. Montealto, CESO VI
ASDS Norma T. Francisco, DM, CESE
EPS Mathematics Vilma A. Brown, Ed. D.
EPS LRMS Aida F. Coyme, Ed. D.

Printed in the Philippines

Department of Education – Region IX, Zamboanga Peninsula
Office Address: Tiguma, Airport Road, Pagadian City
Telefax: (062) – 215 – 3751; 991 – 5975
E-mail Address: region9@deped.gov.ph

1
Introductory Message
This Self – Learning Module (SLM) is prepared so that you, our dear learners, can continue
your studies and learn while at home. Activities, questions, directions, exercises, and
discussions are carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you step-by-step as you
discover and understand the lesson prepared for you.
Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will
tell you if you can proceed on completing this module or if you need to ask your facilitator or
your teacher’s assistance for better understanding of the lesson. At the end of each module,
you need to answer the post-test to self-check your learning. Answer keys are provided for
each activity and test. We trust that you will be honest in using these.
In addition to the material in the main text, notes to the Teacher are also provided to our
facilitators and parents for strategies and reminders on how they can best help you on your
home-based learning.
Please use this module with care. Do not put unnecessary marks on any part of this SLM.
Use a separate sheet of paper in answering the exercises and tests. Read the instructions
carefully before performing each task.
If you have any questions in using this SLM or any difficulty in answering the tasks in this
module, do not hesitate to consult your teacher or facilitator.
Thank you.

What I Need to Know

This module was written as an aid in understanding Hypothesis testing knowing that
Hypothesis testing is a widely used statistical technique. It forces you to think ahead about
what you might find. By forcing you to think ahead, it often helps with decision-making by
forcing you to think about what goes into your decision. All of statistics requires clear
thinking, and clear thinking generally makes better decisions. Hypothesis testing requires
very clear thinking and often leads to better decision-making.
The module follows a step – by – step approach to understand the concept of
Hypothesis testing and it is supported by examples and exercises. It covers the key
concepts in testing hypothesis and probability of committing a type I or Type II error.
This module was designed to cater the academic needs of diverse learners in
achieving and improving the twin goals of mathematics in basic education levels which are
critical thinking and problem solving. The language used recognizes the vocabulary level of
senior high school students. The lessons followed a developmentally sequenced teaching
and learning processes to meet the curriculum requirement.
After going through the module, you are expected to:
• Illustrate (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);

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 • Calculates the probabilities of committing a Type I and Type II error. (M11/12SP-
IVa-2); and
• Identify the parameter to be tested given a real-life problem.
(M11/12SP-IVa-3)
Believe that learning can continue amid the health crisis. Good luck, stay safe, and
God bless.

What I Know
Directions: Answer the following questions by choosing the letter of the correct answer.
Write your answer on a separate sheet.

1. Hypothesis tests are designed so that the _____ hypothesis will be rejected.
a. null b. alternative c. incorrect d. alpha
2. Which of the following is a statement about a population developed for the purpose of
testing?
a. Hypothesis b. Hypothesis testing
c. Level of significance d. Test-statistic
3. Which of the following is a statement whose validity is tested based on a sample?
a. Null hypothesis b. Alternative hypothesis
c. Statistical hypothesis d. Simple hypothesis
4. Which of the following is the dividing point between the region where the null hypothesis is
rejected and the region where it is not rejected?
a. Critical region b. Critical value
c. Acceptance region d. Significant region
5. The choice of one-tailed test and two-tailed test depends upon
a. Null hypothesis b. Alternative hypothesis
c. None of these d. Composite hypotheses
6. What type of test is described by the hypothesis below?
Ho: µ = 50 and H1: µ > 50
a. Left-tailed test b. Right-tailed test
c. Two-tailed test d. No-tailed test
7. Which type of statistical error occurred when Ho is true, and it is rejected?
a. Type-I error b. Type-II error
c. Standard error d. Sampling error
8. What type of statistical error occurred when a failing student is passed by an examiner?
a. Type-I error b. Type-II error c. Unbiased decision d. Fair decision
9. 1 – α is also called ___________ .
a. Confidence coefficient b. Power of the test
c. Size of the test d. Level of significance
10. Which region of the distribution should the test statistic lie to reject the null hypothesis?
a. Rejection region b. Acceptance region
c. Both (a) and (b) d. Neither (a) nor (b)

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 LESSON
HYPOTHESIS TESTING
1
What’s In
ACTIVITY DID I MAKE THE RIGHT DECISION?
1. What would you wish to have with you if you were stranded in an island? Now consider
the what ifs?

Everybody can benefit from having good decision-making skills as we all encounter
situations where decision-making is at most vital daily. Some problems or situations are
obviously more severe or complex than others.
It would be wonderful to have the ability to solve all problems/situations efficiently
and in a timely fashion without difficulty, unfortunately though there is no one way in which
all problems can be solved.
However, well prepared we are for a situation, there is always an element of the
unknown. Although planning and structuring will help make the decision-making process
more likely to be successful, good judgement and an element of good luck ultimately
determine whether the decision making was a success.

What’s New

Sometimes we hear claims on social media that we find unbelievable. Such as:
• a whitening product advertisement stating that if you use their whitening product,
then your skin is like snow white.
• The weatherman stating that there is a 90% chance of rain tomorrow.

We might feel compelled to challenge such claims. To challenge claims, we must

run a research study upon a sample (since the surveying the entire population would be
impossible). To test a claim, you must write two hypotheses. The Null Hypothesis (no
change) and the Alternative Hypothesis (change occurs). Then you will assume the Null
Hypothesis is true and find evidence against the Alternative Hypothesis. You will either find
evidence to reject the Null Hypothesis (it is not true) or fail to reject the Null Hypothesis
(there is not enough evidence to disprove the Null Hypothesis).

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 What is It

Hypothesis testing is a decision-making process for evaluating claims about a

population. In this process, the researchers must define the population under study, state the
particular hypotheses to be investigated, give the significance level, select a sample from the
population, collect the data, perform the required test, and reach a conclusion. There are two
specific statistical tests for hypothesis testing on means: the z-test and the t-test. On the
other hand, the chi-square test is used for testing the standard deviation.

As you will learn in this module, you need to choose between two statements about
the population. These two statements are the hypotheses.

• Null hypothesis (Ho), is basically, “The population is like this.” It states, in formal
terms, that the population is no different than usual.
• Alternative hypothesis (Ha), is, “The population is like something else.” It states
that the population is different than the usual, that something has happened to this
population, and as a result it has a different mean, or different shape than the usual
case.

Between the two hypotheses, all possibilities must be covered. Remember that you
are making an inference about a population from a sample. Keeping this inference in mind,
you can informally translate the two hypotheses into “I am almost positive that the sample
came from a population like this” and “I really doubt that the sample came from a population
like this, so it probably came from a population that is like something else”. Notice that you
are never entirely sure, even after you have chosen the hypothesis, which is best. Though
the formal hypotheses are written as though you will choose with certainty between the one
that is true and the one that is false, the informal translation of the hypotheses, with “almost
positive” or “probably came”, is a better reflection of what you find.

Hypothesis testing has many applications in business, though few managers are
aware that is what they are doing. As you will see, hypothesis testing, though disguised, is
used in quality control, marketing, and other business applications. Many decisions are
made by thinking as though a hypothesis is being tested, even though the manager is not
aware of it. Learning the formal details of hypothesis testing will help you make better
decisions and better understand the decisions made by others.

Every hypothesis testing begins with the statement of a hypothesis. A statistical

hypothesis is an inference about a population parameter. This inference may or may not be
true. Anyone who has watched commercial TV cannot fail to be aware of the constant
barrage of claims. Brand X detergent will wash white clothes sparkling white. With a certain
gasoline, your car will get more kilometers to the liter than before. And so on and so on.

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 The only sure way of finding the truth or falsify a hypothesis is by examining the
entire population. Because this is not always feasible, a sample is instead examined for the
purpose of drawing conclusion.

In order to state the hypothesis correctly, the researcher must translate the claim into
mathematical symbols. There are three possible sets of statistical hypotheses.

1. Ho : parameter = specific value

TWO-TAILED TEST
Ha : parameter ≠ specific value
2. Ho : parameter = specific value
LEFT-TAILED TEST
Ha : parameter < specific value
3. Ho : parameter = specific value
RIGHT-TAILED TEST
Ha : parameter > specific value

In the hypothesis testing, there are four possible outcomes.

Reject Ho Do not Reject Ho

Ho is True Type I Error Correct Decision

Ho is False Correct Decision Type II Error

• A type I error occurs if one rejects the null hypothesis when it is true.
• A type II error occurs if one does not reject the null hypothesis when it is false.

The decision is made based on probabilities, that is, if there is a large difference
between the value of the parameter obtained from the sample and the hypothesized
parameter, the null hypothesis is probably not true. The next question the researcher would
ask is “How large a difference is necessary to reject the null hypothesis?” here is where the
level of significance is used.

The level of significance is the maximum probability of committing a type I error.

This probability is symbolized by α (Greek letter alpha). That is, P(type I error) = α. the
probability of type II error is symbolized by β (Greek letter beta), that is, P(type II error) = β,
although in most hypothesis testing situations, b cannot be computed.

Generally, statisticians agree on using three arbitrary significance levels: the 0.10,
0.05, and 0.01 level. That is, if the null hypothesis is rejected, the probability of type I error
will be 10%, 5% and 1%, and the probability of a correct decision will be 90%, 95% and
99%, depending on which level of significance is used. In other words, when α = 0.05, there
is a 5% chance of rejecting a true null hypothesis.

In a hypothesis-testing situation, the researcher decides what level of significance to

use. It does not have to be the levels mentioned above. It can be any level, depending on
the seriousness of the type I error.

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 After a significance level is chosen, a critical value is selected from a table for the
appropriate test.

• The critical value determines the critical and non-critical regions.

• The critical region or the rejection region is the range of values of the test value
that indicates that there is a significant difference and that the null hypothesis should
be rejected.
• The non-critical or non-rejection region is the range of values of the test value
that indicates that the difference was probably due to chance and that the null
hypothesis should be rejected.

The rejection region can be located on both sides with the non-rejection region in the
middle, or it can be on the left side or right side of the non—rejection region. A test with two
rejection regions is called a two-tailed test. In this test, the null hypothesis should be rejected
when the test value is in either of the two critical regions. A one-tailed test indicates that the
null hypothesis should be rejected when the test values are in the critical region on one side
of the parameter. A one-tailed test is either right-tailed when inequality in the alternative
hypothesis is greater than or left-tailed when the inequality is less than.

REJECTION REGIONS If the test is two-tailed, the critical

value will be either positive or negative. If the
test is left-tailed, the critical value will be
negative. If the test is right-tailed, the critical
value will be positive.

CRITICAL VALUES

The z-test is a statistical test for the mean of a population. It can be used when the
sample size is greater than 30, or when the population is normally distributed and population
deviation is known.

̅− µ
𝒙 where, 𝑥̅ –sample mean 𝞂–population deviation
𝒛=
𝝈/√𝒏 µ–hypothesized mean n–sample size

The t-test is a statistical test for the mean of a population and is used when the
population is normally or approximately normally distributed, population is less than 30 and
population deviation is unknown.

̅− µ
𝒙 where, 𝑥̅ –sample mean s–sample deviation
𝒕=
𝒔/√𝒏 µ–hypothesized mean n–sample size

The degrees of freedom are df = n-1.

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 What’s More
ACTIVITY 1
PROBLEM 1. A manufacturer claims that the average lifetime of his lightbulbs is three 3
years or 36 months. The standard deviation is 8 months. Fifty bulbs are selected, and
the average lifetime is found to be 32 months. Should the manufacturer’s statement be
rejected at α = 0.01?
SOLUTION:
STEP 1: State the hypotheses
Ho: µ = 36 months Ha: µ ≠ 36 months
STEP 2: Level of significance 𝛼 = 0.01
STEP 3: Determine the critical values and rejection region.

The significance level is 0.01. the ≠ sign in

the alternative hypothesis indicates that the test is
two-tailed with two rejection regions, one in each tail
of the normal distribution curve of µ. Because the
total area of both rejections is 0.01 (Significance
level), the area of rejection in each tail is 0.005, since
the sample population is more than 30, z-test is to
be utilized. The area to the right of µ is 0.5, the area
between 0 and the critical value 𝑧 is 0.495. looking at
the z-table the critical value is 𝑧 = ±2.575.

STEP 4: State the decision rule.

Reject the null hypothesis if 𝑧𝑐 > 2.575 or 𝑧𝑐 < −2.575.
STEP 5: Compute the test statistic.
̅
𝒙− µ 𝟑𝟐− 𝟑𝟔
𝒛= = = −3.54
𝝈/√𝒏 𝟖/√𝟓𝟎
STEP 6: Make a decision.
The test statistic 𝑧𝑐 = −3.54 is less than the critical value 𝑧 = −2.575 and it falls in
the rejection region in the left tail. Therefore, reject Ho and conclude that the
average lifetime of the lightbulb is not equal to 36 months.

PROBLEM 2. In order to increase customer service, a muffler repair shop claims its
mechanics can replace a muffler in 12 minutes. A time management specialist selected
six repair jobs and found his mean time to be 11.6 minutes. The standard deviation of
the sample was 2.1 minutes. At α = 0.025, is there enough evidence to conclude that the
mean time in changing a muffler is less than 12 minutes?

Since the sample population is less than 30, t-test is to be utilized

SOLUTION:
STEP 1: Ho : µ = 12 Ha : µ < 12
STEP 2: α = 0.025

8
STEP 3. Since α = 0.025 and df = 6 – 1 = 5, then tα = -2.571.
STEP 4: Reject Ho if tc < -2.571.
STEP 5: Compute for the test statistic.
̅
𝒙− µ 𝟏𝟏.𝟔 − 𝟏𝟐
𝒕= = = −0.47
𝒔/√𝒏 𝟐.𝟏/√𝟔
STEP 6: Since the critical value falls within the noncritical region, do not reject Ho.

What I Have Learned

A hypothesis is essentially an idea about the population that you think might be true,
but which you cannot prove to be true. While you usually have good reasons to think it is
true, and you often hope that it is true, you need to show that the sample data support your
idea. Hypothesis testing allows you to find out, in a formal manner, if the sample supports
your idea about the population. Because the samples drawn from any population vary, you
can never be positive of your finding, but by following generally accepted hypothesis testing
procedures, you can limit the uncertainty of your results.
In hypothesis testing the following steps should be considered
1. state the null and alternative hypotheses.
2. Select the level of significance.
3. determine the critical value and the rejection region/s.
4. state the decision rule.
5. compute the test statistic.
6. make a decision, whether to reject or not to reject the null hypothesis.

What I Can Do
ACTIVITY LET’S CHECK
Directions: Answer the problem applying the steps in testing hypothesis.

A diet clinic states that there is an average loss of 24 pounds for those who stay
on the program for 20 weeks. The standard deviation is 5 pounds. The clinic tries a new
diet, reducing salt intake to see whether that strategy will produce a = greater weight
loss. A group of 40 volunteers loses an average of 16.3 pounds each over 20 weeks.
Should the clinic change the new diet? Use α= 0.05.

STEP 1: Ho : ____________
Ha : ____________
STEP 2: α= _____________
STEP 3: __ = _____________
STEP 4: Decision Rule:____________________________________________
STEP 5: Compute the test Statistic:

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STEP 6: Conclusion:______________________________________________
_______________________________________________

Assessment
A. Directions: Choose the letter that corresponds to your answer. Write your answer on a
separate sheet.

1. Which of the following is TRUE?

(a) the alternative hypothesis represents the conclusion for which evidence is sought
(b) the statement of the null hypothesis never contains an equality
(c) an increase in the risk of type I error also increases the risk of a type II error
(d) the computed test statistic is also known as the critical value
2. What is the dividing point between the region where the null hypothesis is rejected and the
region where it is not rejected?
(a) Critical region (b) Critical value
(c) Acceptance region (d) Significant region
3. Which of the following is a statement about a population developed for the purpose of
testing?
(a) Hypothesis (b) Hypothesis testing
(c) Level of significance (d) Test-statistic
4. What do you call any hypothesis which is tested for the purpose of rejection under the
assumption that it is true?
(a) Null hypothesis (b) Alternative hypothesis
(c) Statistical hypothesis (d) Composite hypothesis
5. Which of the following is a statement about the value of a population parameter?
(a) Null hypothesis (b) Alternative hypothesis
(c) Simple hypothesis (d) Composite hypothesis
6. Which of the following is a statement whose validity is tested based on a sample?
(a) Null hypothesis (b) Alternative hypothesis
(c) Statistical hypothesis (d) Simple hypothesis
7. Which of the following is the basis in choosing between one-tailed test and two-tailed test?
(a) Null hypothesis (b) Alternative hypothesis
(c) Composite hypothesis (d) None of these
8. What type of test is described by the hypothesis below?
Ho: µ = 50 and H1: µ > 50
(a) Left-tailed test (b) Right-tailed test (c) Two-tailed test (d) No-tailed test
10. Which of the following is a rule or formula that provides a basis for testing a null
hypothesis?
(a) Test-statistic (b) Population statistic
(c) Both of these (d) None of the above

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 11. Which statistical error occurred if Ho is true and we reject it?
(a) Type-I error (b) Type-II error
(c) Standard error (d) Sampling error
12. What probability is associated with committing type-I error?
(a) β (b) α (c) 1 – β (d) 1 – α
13. Which statistical error occurred if a failing student is passed by an examiner?
(a) Type-I error (b) Type-II error
(c) Unbiased decision (d) Fair decision

B. Directions: Perform hypothesis testing. (5 Points Each)

1. A biologist knows that the average length of a leaf of a certain plant is 4 inches. The
standard deviation of the population is 0.6 inch. A sample of 20 leaves of that type of plant
given a new type of plant food had an average length of 4.2 inches. At α = 0.01, is there
reason to believe that the new food is responsible for a change in the growth of leaves?

2. During the pandemic people have become more health conscious, especially in eating red
meat. In March, the average consumption of red meat per person was 51 kilos. A sample of
100 persons showed that they consumed, on average, 48 kilos of red meat in April with a
standard deviation of 12 kilos. Using α = 0.05, is there enough evidence to conclude that the
mean consumption of red meat decreased in April.

Z – DISTRIBUTION TABLE
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549
0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015
1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177
1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441
1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857
2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890

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2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916
2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936
2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964
2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990
T-Distribution Table
Confidence
0.500 0.800 0.900 0.950 0.980 0.990
interval
d.f. One-tail t.250 t.100 t.050 t.025 t.010 t.005
d.f. Two-tails t.500 t.200 t.100 t.050 t.020 t.010
1 1.000 3.078 6.314 12.706 31.821 63.657
2 0.816 1.886 2.920 4.303 6.965 9.925
3 0.765 1.638 2.353 3.182 4.541 5.841
4 0.741 1.533 2.132 2.776 3.747 4.604
5 0.727 1.476 2.015 2.571 3.365 4.032
6 0.718 1.440 1.943 2.447 3.143 3.707
7 0.711 1.415 1.895 2.365 2.998 3.499
8 0.706 1.397 1.860 2.306 2.896 3.355
9 0.703 1.383 1.833 2.262 2.821 3.250
10 0.700 1.372 1.812 2.228 2.764 3.169
11 0.697 1.363 1.796 2.201 2.718 3.106
12 0.695 1.356 1.782 2.179 2.681 3.055
13 0.694 1.350 1.771 2.160 2.650 3.012
14 0.692 1.345 1.761 2.145 2.624 2.977
15 0.691 1.341 1.753 2.131 2.602 2.947
16 0.690 1.337 1.746 2.120 2.583 2.921
17 0.689 1.333 1.740 2.110 2.567 2.898
18 0.688 1.330 1.734 2.101 2.552 2.878
19 0.688 1.328 1.729 2.093 2.539 2.861
20 0.687 1.325 1.725 2.086 2.528 2.845
21 0.686 1.323 1.721 2.080 2.518 2.831
22 0.686 1.321 1.717 2.074 2.508 2.819
23 0.685 1.319 1.714 2.069 2.500 2.807
24 0.685 1.318 1.711 2.064 2.492 2.797
25 0.684 1.316 1.708 2.060 2.485 2.787
26 0.684 1.315 1.706 2.056 2.479 2.779
27 0.684 1.314 1.703 2.052 2.473 2.771
28 0.683 1.313 1.701 2.048 2.467 2.763
29 0.683 1.311 1.699 2.045 2.462 2.756
Inf. 0.674 1.282 1.645 1.960 2.326 2.576

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Assessment:
1. C. 2. B. 3. B. 4. A 5. B
6. B 7. A 8. A 9. B 10. B
What I Can Do:
Step 1: Ha: µ = 24
Ho: µ > 24
Step 2: α = 0.05
Step 3: z-test
Step 4: Reject Ho if z-computed value is greater than 1.64
Step 5: Compute test-statistic
z-computed is -9.74
Step 6: Since z-computed = -9.74 < the critical value 1.64, we fail to reject the Ho and
conclude that at α 0.05 level of significance there is insufficient evidence to support a
claim of a true mean greater than 24.
What I Know:
1. A 2. B 3. C 4. B 5.B
6. B 7. A 8. B 9. D 10.A
Answer Key
References:
Bluman, A. G., (2012). Elementary Statistics: A Step-by-Step Approach, Eight Edition. New
York: McGraw – Hill Companies, Inc., 400 – 411.

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 I AM A FILIPINO
by Carlos P. Romulo
I am a Filipino – inheritor of a glorious past, hostage to the It is the mark of my manhood, the symbol of my dignity as
uncertain future. As such, I must prove equal to a two-fold a human being. Like the seeds that were once buried in the
task – the task of meeting my responsibility to the past, and tomb of Tutankhamen many thousands of years ago, it
the task of performing my obligation to the future. shall grow and flower and bear fruit again. It is the insigne
I am sprung from a hardy race – child many generations of my race, and my generation is but a stage in the
removed of ancient Malayan pioneers. Across the centuries, unending search of my people for freedom and happiness.
the memory comes rushing back to me: of brown-skinned I am a Filipino, child of the marriage of the East and the
men putting out to sea in ships that were as frail as their West. The East, with its languor and mysticism, its
hearts were stout. Over the sea I see them come, borne upon passivity and endurance, was my mother, and my sire was
the billowing wave and the whistling wind, carried upon the the West that came thundering across the seas with the
mighty swell of hope – hope in the free abundance of the Cross and Sword and the Machine. I am of the East, an
new land that was to be their home and their children’s eager participant in its struggles for liberation from the
forever. imperialist yoke. But I know also that the East must awake
This is the land they sought and found. Every inch of shore from its centuried sleep, shake off the lethargy that has
that their eyes first set upon, every hill and mountain that bound its limbs, and start moving where destiny awaits.
beckoned to them with a green and purple invitation, every For I, too, am of the West, and the vigorous peoples of the
mile of rolling plain that their view encompassed, every West have destroyed forever the peace and quiet that once
river and lake that promised a plentiful living and the were ours. I can no longer live, a being apart from those
fruitfulness of commerce, is a hollowed spot to me. whose world now trembles to the roar of bomb and cannon
By the strength of their hearts and hands, by every right of shot. For no man and no nation is an island, but a part of
law, human and divine, this land and all the appurtenances the main, and there is no longer any East and West – only
thereof – the black and fertile soil, the seas and lakes and individuals and nations making those momentous choices
rivers teeming with fish, the forests with their inexhaustible that are the hinges upon which history revolves. At the
wealth in wild and timber, the mountains with their bowels vanguard of progress in this part of the world I stand – a
swollen with minerals – the whole of this rich and happy forlorn figure in the eyes of some, but not one defeated
land has been for centuries without number, the land of my and lost. For through the thick, interlacing branches of
fathers. This land I received in trust from them, and in trust habit and custom above me I have seen the light of the
will pass it to my children, and so on until the world is no sun, and I know that it is good. I have seen the light of
more. justice and equality and freedom, my heart has been lifted
I am a Filipino. In my blood runs the immortal seed of by the vision of democracy, and I shall not rest until my
heroes – seed that flowered down the centuries in deeds of land and my people shall have been blessed by these,
courage and defiance. In my veins yet pulses the same hot beyond the power of any man or nation to subvert or
blood that sent Lapulapu to battle against the alien foe, that destroy.
drove Diego Silang and Dagohoy into rebellion against the I am a Filipino, and this is my inheritance. What pledge
foreign oppressor. shall I give that I may prove worthy of my inheritance? I
That seed is immortal. It is the self-same seed that flowered shall give the pledge that has come ringing down the
in the heart of Jose Rizal that morning in Bagumbayan corridors of the centuries, and it shall be compounded of
when a volley of shots put an end to all that was mortal of the joyous cries of my Malayan forebears when first they
him and made his spirit deathless forever; the same that saw the contours of this land loom before their eyes, of the
flowered in the hearts of Bonifacio in Balintawak, of battle cries that have resounded in every field of combat
Gregorio del Pilar at Tirad Pass, of Antonio Luna at from Mactan to Tirad Pass, of the voices of my people
Calumpit, that bloomed in flowers of frustration in the sad when they sing:
heart of Emilio Aguinaldo at Palanan, and yet burst forth “I am a Filipino born to freedom, and I shall not rest until
royally again in the proud heart of Manuel L. Quezon when freedom shall have been added unto my inheritance—for
he stood at last on the threshold of ancient Malacanang myself and my children and my children’s children—
Palace, in the symbolic act of possession and racial forever.”
vindication. The seed I bear within me is an immortal seed.

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