Statistics and Probability

1. Testing
Hypothesis

2. According to a survey, the average
daily usage of social media
worldwide of global internet users
amounts to 142 minutes per day.
Sofia conducts her own survey
among her friends to find out if their
time spent on social media is
significantly higher than the global
survey.

3. Claim A: The average daily usage of
social media of her friends is the
same as the global average usage.
Claim B: The average daily usage
of social media of her friends is
higher than the global average
usage.

5. 1. What statistical data is/are
needed to prove which of Sofia’s
claims is accepted or rejected?
2. What is the average daily usage of social
media of her friends? Compare it with the
previous average usage.
3. Which of the two claims
could probably be true? Why?

6. 4. If Sofia computed the average daily internet usage of
her friends to be higher than the global survey, do you
think it would be significantly higher?
5. What is your idea of an average value being
significantly higher than the global average
value?
6. What do you think is the difference between
simple comparison of data and hypothesis
testing?

7. Hypothesis testing is a statistical
method applied in making decisions
using experimental data. Hypothesis
testing is basically testing an
assumption that we make about a
population.

8. A hypothesis is a proposed
explanation, assertion, or assumption
about a population parameter or about
the distribution of a random variable.

9. Statistical HYPOTHESIS
Statistical hypothesis is a guess
or prediction made by a
researcher regarding the possible
outcome of the study.

10. Types of Statistical Hypothesis
1.Null Hypothesis (Ho)
It serves as the working hypothesis.
It is that which one hopes to accept or
reject.
It must always express the idea of no
significant difference.

11. Types of Statistical
Hypothesis
2. Alternative Hypothesis (Ha)
It generally represents the
hypothetical statement that the
researcher wants to prove.

12. Here are the examples of questions you can
answer with a hypothesis test:
Does the mean height of Grade 12
students differ from 66 inches?
Do male and female Grade 7 and Grade
12 students differ in height on average?
Is the proportion of senior male students’
height significantly higher than that of
senior female students?

14. •Statistical hypothesis is a guess or
prediction made by a researcher
regarding the possible outcome of the
study.
•A hypothesis is a conjecture or
statement or assertion which aims to
explain certain phenomena in the real
world.

15. HYPOTHESIS TESTING
•Hypothesis testing is the process of making an
inference or generalization on a population
based on the results of the study on samples.
•A test of hypothesis is a method to determine
whether the statistical hypothesis is true or not.
•A hypothesis test is a statistical method that uses
sample data to evaluate a hypothesis about a
population.

16. •1. Null Hypothesis (Ho)
•It serves as the working hypothesis.
•It is that which one hopes to accept or reject.
•It must always express the idea of no significant
difference.
•The null hypothesis (denoted by Ho) is a statement
that the value of a population parameter (such as
proportion, mean, or standard deviation) is equal to
some claimed value.

17. •2. Alternative Hypothesis (Ha)
•It generally represents the hypothetical
statement that the researcher wants to
prove.
•The alternative hypothesis (denoted by H1 or
Ha) is the statement that the parameter has
a value that somehow differs from the null
hypothesis.

18. Example
You are the manager of a fast-food restaurant. You want
to determine whether the waiting time to place an
order has changed in the past month from its previous
population mean value of 4.5 minutes. State the null
and alternative hypotheses.
Ho: The population mean waiting time has not
changed from its previous value of 4.5 minutes.
Ha: The population mean waiting time has
changed from its previous value of 4.5
minutes.
Ho: 𝝁 = 𝟒. 𝟓
Ha: 𝝁 ≠ 𝟒. 𝟓

19. Example 2
The Philippine Statistics Authority (PSA) reported that in
Cagayan Valley region, the average household size was
4.3 persons in 2015. You conducted a similar study in
Cagayan Valley and believed that the regional average
household size is lower than the PSA report.
Ho: The average household size
in Cagayan Valley is 4.3 persons.
Ho: 𝝁 = 𝟒. 𝟑
Ha: The average household size in
Cagayan Valley is less than 4.3
persons.
Ha: 𝝁 < 𝟒. 𝟑

20. Example 3
The Social Weather Survey (SWS) conducted a poll on
Job Satisfaction and found out that more than 75% of
workers are satisfied with their job. You conducted a
similar study which include 580 employed adults, with
516 of them saying that they are satisfied with their job.
State the null and alternative hypotheses.
Ho: The proportion of workers who are satisfied
with their job is 75%.
Ho: p = 𝟕𝟓%
Ha: The proportion of workers who are satisfied
with their job is more than 75%.
Ha: p > 𝟕𝟓%

21. Exercises
Examine the given statement, then express the null
hypothesis Ho and alternative hypothesis Ha in symbolic
form. Be sure to use the correct symbol (𝜇, p, 𝜎) for the
indicated parameter.
1. Ten percent of Filipinos 6 to 24 years old were out-of-
school children and youth (OSCY). (PSA Report, 2015)
Ho: p = 𝟏𝟎 % Ha: p ≠ 𝟏𝟎%

22. Exercises
2. You claim that the Philippine national average IQ is
higher than 86, which (in the Binet Scale) qualifies as
dull. (2002’s IQ and the Wealth of Nations by University of Ulster’s Dr. Richard Lynn
and University of Tampere’s Dr. Tatu Vanhanen.)
3. IQ scores of college professors have a standard deviation
less than 15, which is the standard deviation for the general
population.
Ho: 𝝁 = 𝟖𝟔
Ho: 𝝈 = 𝟏𝟓
Ha: 𝝁 > 𝟖𝟔
Ha: 𝝈 < 𝟏𝟓

23. Exercises
4. According to a study that the Association of Southeast Asian Nations
DNA (ASEAN DNA) conducted, Filipino females are the second shortest race
in the region with an average height of 150 cm. (http://globalnation.inquirer.net, April
25, 2014)
5. The prevalence of undernutrition among children, 10-19
years old in the province of Isabela is more than 23%. (FNRI Report,
2015)
Ho: 𝝁 = 𝟏𝟓𝟎 Ha: 𝝁 ≠ 𝟏𝟓𝟎
Ho: p = 𝟐𝟑% Ha: p > 𝟐𝟑%

24. Take a look at this example.
The school record claims that the
mean score in Math of the incoming
Grade 11 students is 81. The teacher
wishes to find out if the claim is
true. She tests if there is a
significant difference between the
batch mean score and the mean
score of students in her class.

25. Option 1:
𝐻𝑜: The mean score of the
incoming Grade 11 students is 81
or 𝜇 = 81.
𝐻𝑎: The mean score of the
incoming Grade 11 students is
not 81 or 𝜇 ≠ 81.

26. Option 2:
𝐻𝑜: The mean score of the incoming
Grade 11 students has no significant
difference with the mean score of her
students or 𝜇 = 𝑥.
𝐻𝑎: The mean score of the incoming
Grade 11 students has a significant
difference with the mean score of her
students or 𝜇 ≠ 𝑥 .

27. Based on the first claim of Sofia in Activity 2 that “the
average daily usage of social media of her friends is
the same as the global average usage”, formulate two
hypotheses about the global average usage (𝜇) and the
average usage of her friends (𝑥) on the blanks
provided below.
𝐻𝑜:
___________________________________________
𝐻𝑎:
___________________________________________

31. Directions: State the null and the alternative
hypotheses of the following statements.
1. A medical trial is conducted to
test whether or not a new
medicine reduces uric acid.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________

32. Directions: State the null and the alternative
hypotheses of the following statements.
2. We want to test whether
the general average of
students in Math is different
from 80%.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________

33. Directions: State the null and the alternative
hypotheses of the following statements.
3. We want to test whether
the mean height of Grade 8
students is 58 inches.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________

34. Directions: State the null and the alternative
hypotheses of the following statements.
4. We want to test if LPIHS
students take more than four
years to graduate from high
school, on the average.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________

35. Directions: State the null and the alternative
hypotheses of the following statements.
5. We want to test if it takes
less than 60 minutes to
answer the quarterly test in
Calculus.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________

36. Directions: State the null and the alternative
hypotheses of the following statements.
6. A medical test is conducted to
determine whether or not a new
vaccine reduces the
complications of dengue fever.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________

37. Directions: State the null and the alternative
hypotheses of the following statements.
7. The enrolment in high
school this school year
increases by 10%.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________

38. Directions: State the null and the alternative
hypotheses of the following statements.
8. The intelligence quotient of
male grade 11 students is the
same as the female students.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________

39. Directions: State the null and the alternative
hypotheses of the following statements.
9. The school want to test if
the students in Grade 7 prefer
online distance learning as
the method of instruction.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________

40. Directions: State the null and the alternative
hypotheses of the following statements.
10. The school librarian wants to
find out if there was an increase
in the number of students
accessing the school library.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________

42. Directions: State the null and the alternative
hypotheses of the following statements.
1. A medical trial is conducted to
test whether or not a new
medicine reduces uric acid by
50%.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________
1.𝐻𝑜: The new medicine cannot reduce
uric acid.
𝐻𝑎: The new medicine can reduce uric
acid.

43. Directions: State the null and the alternative
hypotheses of the following statements.
2. We want to test whether
the general average of
students in Math is different
from 80%.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________
2.𝐻𝑜: The general average of students in
Math is equal to 80%.
𝐻𝑎: The general average of students in
Math is different from 80%.

44. Directions: State the null and the alternative
hypotheses of the following statements.
3. We want to test whether
the mean height of Grade 8
students is 58 inches.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________
3.𝐻𝑜: The mean height of students is 58
inches.
𝐻𝑎: The mean height of students is not (or
less than or higher than) 58 inches

45. Directions: State the null and the alternative
hypotheses of the following statements.
4. We want to test if LPIHS
students take more than four
years to graduate from high
school, on the average.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________
4.𝐻𝑜: Students takes an average of four years to
graduate from high school.
𝐻𝑎: Students takes an average of more than four
years to graduate from high school (less than 4 years
if not possible).

46. Directions: State the null and the alternative
hypotheses of the following statements.
5. We want to test if it takes
less than 60 minutes to
answer the quarterly test in
Calculus.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________
5.𝐻𝑜: It takes 60 minutes to answer the
quarterly test in calculus.
𝐻𝑎: It takes less than (or more than) 60 minutes
to answer the quarterly test in calculus.

47. Directions: State the null and the alternative
hypotheses of the following statements.
6. A medical test is conducted to
determine whether or not a new
vaccine reduces the
complications of dengue fever.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________
6.𝐻𝑜: The new vaccine cannot reduces the
complications of dengue fever.
𝐻𝑎: The new vaccine reduces the complications
of dengue fever.

48. Directions: State the null and the alternative
hypotheses of the following statements.
7. The enrolment in high
school this school year
increases by 10%.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________
7.𝐻𝑜: The enrolment in high school this school
year did not increase by 10%.
𝐻𝑎: The enrolment in high school this school
year increases by 10%.

49. Directions: State the null and the alternative
hypotheses of the following statements.
8. The intelligence quotient of
male grade 11 students is the
same as the female students.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________
8.𝐻𝑜: The intelligence quotient of male grade 11
students is the same as the female students.
𝐻𝑎: The intelligence quotient of male grade 11
students is higher (or lower) than the female
students.

50. Directions: State the null and the alternative
hypotheses of the following statements.
9. The school want to test if
the students in Grade 7 prefer
online distance learning as
the method of instruction.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________
9.𝐻𝑜: Grade 7 students prefer online distance
learning as the method of instructions.
𝐻𝑎: Grade 7 students did not prefer online
distance learning as the method of instructions.

51. Directions: State the null and the alternative
hypotheses of the following statements.
10. The school librarian wants to
find out if there was an increase
in the number of students
accessing the school library.
𝐻𝑜: ____________________________________________________
𝐻𝑎: ____________________________________________________
10.𝐻𝑜: There was an no increase in the number
of students accessing the school library.
𝐻𝑎: There was an increase in the number of
students accessing the school library.

53. Here is another key term you should know!
Level of Significance
 The level of significance denoted by
alpha or 𝛂 refers to the degree of
significance in which we accept or reject the
null hypothesis.
 100% accuracy is not possible in
accepting or rejecting a hypothesis.
 The significance level α is also the
probability of making the wrong decision
when the null hypothesis is true.

55. Here is another key term you should know!
Two-Tailed Test vs One-Tailed Test
• When the alternative hypothesis is two-
sided like 𝐻𝑎: 𝜇 ≠ 𝜇0, it is called two-
tailed test.
• When the given statistics hypothesis
assumes a less than or greater than
value, it is called one-tailed test.

62. According to a survey, the average
daily usage of social media
worldwide of global internet users
amounts to 142 minutes per day.
Sofia conducts her own survey
among her friends to find out if their
time spent on social media is
significantly higher than the global
survey.

63. Claim A: The average daily usage of
social media of her friends is the
same as the global average usage.
Claim B: The average daily usage
of social media of her friends is
higher than the global average
usage.

67. As you can see from your
previous illustration, the
computed t-value of 1.593 is at
the left of the critical value
1.833. So, in which region do
you think the computed value
falls?

69. Illustrative Example 2:
A medical trial is conducted to
test whether or not a certain
drug reduces cholesterol level.
Upon trial, the computed z-
value of 2.715 lies in the
rejection area.

71. Illustrative Example 3:
Sketch the rejection region
of the test hypothesis with
critical values of ±1.753 and
determine if the computed
t-value of –1.52 lies in that
region.

95. 5. In terms of safety,
which error has more
dangerous
consequences in this
setting?