STATISTICS AND PROBABILITY
Quarter 4- Week 1:
Testing Hypothesis (Introduction)
Content Standards:
The learner demonstrates understanding of key concepts of tests of hypotheses on the
population mean and population proportion.
Performance Standards:
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.
Most Essential Learning Competency:
1. 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
2. Identify the parameter to be tested given a real-life problem.M11/12SP-IVa-3
3. Formulate the appropriate null and alternative hypotheses on a population mean.
M11/12SP-IVb-1
4. Identify the appropriate form of the test-statistic when: (a) the population variance
is assumed to be known; (b) the population variance is assumed be unknown; and
(c) the Central Limit Theorem is to be used. M11/12SP-IVb-2
Lesson
1
Testing Hypothesis
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.
Hypothesis is a proposed explanation, assertion, or assumption about a population
parameter or about the distribution of a random variable.
The Null and Alternative Hypothesis
The null hypothesis denoted by 𝐻0 states that there is no difference between a
parameter and specific value, or that there is no difference between two parameters.
It can be written as follows:
𝐇𝟎 : 𝜇 = 𝜇𝟎
𝐇𝟎 : 𝜇 ≤ 𝜇𝟎
𝐇𝟎 : 𝜇 ≥ 𝜇𝟎
The alternative hypothesis denoted by 𝐻1 or 𝐻𝑎 states that there is a difference between
a parameter and specific value, or that there is a difference between two parameters.
It can be written as follows:
𝐇𝟏 : 𝜇 ≠ 𝜇𝟎
𝐇𝟏 : 𝜇 > 𝜇𝟎
𝐇𝟏 : 𝜇 < 𝜇𝟎
=
>
≥
Hypothesis-Testing Common Phrases
is equal to
is not equal to
≠
is the same as
is not the same
is exactly the same as
is different from
has not changed from
has changed from
is decreased
is increased
is less than
is greater than
is lower than
is higher than
<
is below
is above
is smaller than
is bigger than
is decreased or reduced
is longer than
from
is not more than
is at least
is at most
≤
is not less than
is not more than
is greater than or equal to
is less than or equal to
Example 1.
The owner of factory sells a particular bottled fruit juice claims that the average
capacity of their product is 250 ml. Is the claim true?
Solutions:
The parameter of interest is the mean μ = 250.
𝑯𝟎 : The bottled drinks contain 250 ml per bottle. (This is the claim)
In symbols: 𝑯𝟎 : μ = 250
𝑯𝟏 : The bottled drinks do not contain 250 ml per bottle. (This is the opposite of the
claim)
In symbols: 𝑯𝟏 : μ ≠ 250
Example 2.
A farmer believes that using organic fertilizers on his plants will yield greater income.
His average income from the past was P 200, 000.00 per year. State the hypotheses in
symbols.
Solutions:
𝑯𝟎 : μ = 200, 000.00
The phrase ‘greater income’ is associated with the greater than direction. So,
𝑯𝟏 : μ >200, 000.00
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.
 Most of the significant levels are 0.10, 0.05 and 0.01 level.
Two-Tailed Test vs One-Tailed Test
 Two-Tailed Test
-it is non-directional test with the region lying on both tails of the normal curve. It is
used when the alternative hypothesis uses words such as not equal to, significantly
different, etc.
𝑯𝟎 : μ = μ𝟎
𝑯𝟏 : μ ≠ μ𝟎
 One-Tailed Test
-it is a directional test with the rejection, lying on either left or right tail of the normal
curve.
a. Right directional test. The region of rejection is on the right tail. It is used when
the alternative hypothesis uses comparatives such as greater than, higher than,
better than, superior to, exceeds, etc.
𝑯𝟎 : μ = μ𝟎
𝑯𝟏 : μ > μ𝟎
b. Left directional test. The region of rejection is on the left tail. It is used when
the alternative hypothesis uses comparatives such as less than, smaller than,
inferior to, lower than, below, etc.
𝑯𝟎 : μ = μ𝟎
𝑯𝟏 : μ < μ𝟎
Example 3. Determine whether the test is one-tailed or two-tailed.
a. Given: The owner of factory sells a particular bottled fruit juice claims that the
average capacity of their product is 250 ml.
Answer: two-tailed test
a. Given: A farmer believes that using organic fertilizers on his plants will yield
greater income. His average income from the past was P 200, 000.00 per year.
Answer: one-tailed test (right)
Illustration of the Rejection Region
 The rejection region (or critical region) is the set of all values of the test statistic
that causes us to reject the null hypothesis.
 The non-rejection region (or acceptance region) is the set of all values of the test
statistic that causes us to fail to reject the null hypothesis.
 The critical value is a point (boundary) on the test distribution that is compared to
the test statistic to determine if the null hypothesis would be rejected.
Commonly used Level of Significance and
its Corresponding Critical Values of z-distribution
Test Types
Level of Significance 𝜶
One-Tailed
Two-Tailed
0.010
+2.33 or -2.33
± 2.575
0.025
+1.96 or -1.96
± 2.24
0.05
+1.645 or -1.645
±1.96
0.100
+1.28 or -1.28
± 1.645
Example 4
Write the critical value of the following:
a. two- tailed test 𝛼 = 0.01 n= 67
Solutions:
two tailed
n ≥30, (use the z-distribution)
z = ± 2.575
b. right-tailed test 𝛼 = 0.05 n= 25
Solutions:
one tailed (positive)
n < 30 (use the t-distribution)
Identify the level of significance
𝛼 = 0.05
Identify the degree of freedom
df = n-1
= 25 – 1 = 24
Find the critical value using t-distribution in the row with n-1 df.
t = 1.711
Example 5
Illustrate the rejection region given the critical value and identify if the t-values lie
in the non-rejection region or rejection region.
a. critical t-value of -2.33
computed t-value of -1.38
The computed t-value is at the
non-rejection region.
Type I and Type II Errors
 If the null hypothesis is true and rejected, then it is a Type I error. The probability
of committing a Type I error is denoted by α (alpha).
 If null hypothesis is false and accepted, then it is a Type II error. The probability of
committing a Type II error is denoted by β (beta).
Example 6.
a. Maria’s Age
Maria insists that she is 30 years old when, in fact, she is 32 years old. What error is
Mary committing?
Solutions:
Mary is rejecting the truth. She is committing a Type I error.
b. 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?
Solutions:
Hunting Philippine eagle is prohibited by law. Thus, it is not a good sport. It is a Type
II error.
To summarize the difference between the Type I and Type II errors, take a look at the
table below:
Null
Hypothesis,
𝑯𝟎
True
False
Lesson
2
Failed to Reject 𝑯𝟎
or Accept 𝑯𝟎
Reject 𝑯𝟎
Correct decision
Failed to reject 𝐻0 when it is true
Type II Error
Failed to reject 𝐻0 when it is false
Type I Error
Rejected 𝐻0 when it is true
Correct decision
Rejected 𝐻0 when it is false
Identifying Appropriate
Test Statistics Involving
Population Mean
A test statistic is a value used to determine the probability needed in decision making.
It is a random variable that is calculated from sample data and used in a hypothesis.
z- test.
In a z-test, the sample is assumed to be normally distributed. A z-score is calculated
with population parameters such as population mean and population standard
deviation. The normal or sample size is large.
t- test.
The sample is also assumed to be normally distributed. A t-test is used when the
population variance or standard deviation are not known. The sample size is less than
30.
Central Limit Theorem
If the population is normally distributed or the sample size is large and the true
population mean μ = μ𝟎 , then z has a standard normal distribution.
When population standard deviation 𝜎 is not known, we may still use z-score by
replacing the population standard deviation 𝜎 by its estimate, sample standard
deviation s.
When the value of sample size (n)…
n ≥ 30
𝜎 is known
z-test
n < 30
𝜎 is unknown
z-test
𝜎 is known
𝜎 is unknown
z-test
t-test
Example7
Identify the appropriate test statistic to be used in the given problem.
a. The average test score for an entire school is 75 with a standard deviation of 10. What
is the probability that a random sample of 5 students scored above 80?
Answer: Here, the sample size (n) is 5 which is less than 30 and population
standard deviation (10) is known, then the appropriate test statistical to be used
is z-test.
b. From a random sample of 100 students who have passed a statistic course, the
average score was 71.8. Assuming that the population standard deviation is 8.9, with a
significance level of 0.05, does it seem to signify that the average score is more than 70?
Answer: Here, the sample size (n) is 100 which is greater than 30 and population
standard deviation (8.9) is known, then the appropriate test statistical to be used
is z-test.
c. An English teacher wanted to test whether the mean reading speed of students
is 550 words per minute. A sample of 12 students revealed a sample mean of 540
words per minute with a standard deviation of 5 words per minute. At 0.05
significance level, is the reading speed different from 550 words per minute?
Answer: The sample size (n) is 12 which is less than 30 and sample
standard deviation (5 words per minute) was given. Therefore, the
appropriate test is t-test.
ASSESSMENT: (30 points)
I. State the null and the alternative hypotheses of the following statements. (4 points each)
1. A car dealership announces that the mean time for an oil change is less than 15
minutes.
2. A company advertises that the mean life of its furnaces is more than 18 years.
3. A consumer analyst reports that the mean life of a certain type of automobile battery
is not 74 months.
4. A transportation network company claims that the mean travel time between two
destinations is about 16 minutes.
II. Determine if one-tailed test or two-tailed test fits the given alternative hypothesis.
(2 points each)
1. The average age of doctors in Las Piñas is 35 years.
2. The proportion of senior male students’ height is significantly higher than that of
senior female students.
III. Illustrate the rejection region given the critical value and identify if the t-values lie
in the non-rejection region or rejection region. (2points each)
1.
critical t-value of -2.086
computed t-value of -2.096
2.
critical t-value of ±1.071
computed t-value of 1.01
IV. Identify the appropriate test statistic to be used in each problem. (2 points each)
1. Based on the report of the school nurse, the average height of Grade 11 students has
increased. Five years ago, the average height of Grade 11 students was 170cm with
standard deviation of 38cm. She took a random sample of 150 students and derived
the average height of 165cm.
2. A manufacturer of tires claim that their tire has a mean life of at least 50,000kms. A
random sample of 28 of these tires is tested and the sample mean is 33,000kms.
Assume that the population standard deviation is 3,000kms and the lives of the tires
are approximately normally distributed.
3. In the population, the average IQ is 100. A team of scientists wants to test a new
medication to see if it has either a positive or a negative effect on intelligence, or no
effect at all. A sample of 30 participants who have taken the medication has a mean
of 140 with a standard deviation of 20. Did the medication affect intelligence?
Alpha=0.05.
REFERENCES:
Textbooks:
Belecina, R. R., Baccay, E. S., & Mateo, E. B, (2016).
Statistics and Probability. Rex Book Store.
Mangaran, A. J., Santos E. M. (2005)
Probability and Statistics: A Comprehensive Approach
Online Resources:
Wow Math. (2021, April 23). Null and alternative hypotheses||hypothesis
testing||statistics and probability q4 [Video]. Youtube.
https://www.youtube.com/watch?v=8IxJaU06qJA&t=3s
BYJU’S (2021). Retrieved from:
https://byjus.com/maths/t-test-table/
Alcantara A. [Teacher Ayhi]. (2021, April 13). Identifying Appropriate Test
Statistics involving Population Mean [Video]. Youtube.
https://www.youtube.com/watch?v=Vk5mJeROzME
Rai University (2015). Unit 4 Tests of Significance. Slideshare.
https://www.slideshare.net/raiuniversity/unit-4-45983025
socratic.org/statistics (n.d). Retrieved from:
https://socratic.org/questions/how-do-you-find-the-area-under-thenormal-distribution-curve-to-the-right-of-z-3
/rrsa