MODULE 7: INFERENTIAL STATISTICS

INTRODUCTION
This module will discuss inferential statistics, the topics under this module will go deeper
than plain descriptive statistics. You will learn the reasons why two or more groups are different,
why one group behaves differently, whether a group performs better than the other and or why the
sample is different from the population.
This module is designed for you to: define the basic concepts related to hypothesis testing,
steps in hypothesis testing, tests concerning means
You are expected to study this lesson and accomplish your tasks within two weeks, upon receiving
this module (5th and 6th week of this cluster).

LEARNING OUTCOMES
By the end of this module, you should have been able to:
a. define basic concepts in hypothesis testing;
b. formulate a good hypothesis;
c. acquire techniques to test hypothesis with the use of a software to analyze data.

MOTIVATION
Test yourself
1. The hypothesis which is hoped to be rejected.
a. null hypothesis c. both a and b
b. alternative hypothesis d. neither a nor b
2. if 𝐻𝑜 is rejected, it means
a. it is wrong c. not enough information was gathered
b. there is insufficient evidence to accept it d. both b and c are correct
3. if 𝐻𝑜 is accepted, it means
a. it is correct c. more than enough information was gathered
b. there is insufficient evidence to reject it d. both a and c are correct

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 4. if 𝐻𝑜 is rejected when it is true, then it is a
a. Type I error c. type III error
b. Type II error d. no error
5. What determines if a test is a One-tailed or two-tailed test?
a. 𝐻𝑜 b. 𝐻𝑎 c. 𝛼 d. 𝛽

CONTENT
HYPOTHESIS TESTING and the STATISTICAL HYPOTHESES
Hypothesis testing is a method for testing a claim or hypothesis about a parameter in a
population, using data measured in a sample. In this method, we test some hypothesis by determining
the likelihood that a sample statistic could have been selected, if the hypothesis regarding the
population parameter were true.
In testing the hypothesis there are 4 steps involved:
1. State the null 𝐻𝑜 and alternative 𝐻𝑎 hypothesis
2. Identify the:
a. level of significance (𝛼 ),
b. the type of test (one-tailed left directional, one-tailed right directional or two-
tailed non directional)
c. identify the critical value or the tabular value
3. Decision rule, data analysis and the Decision (either to reject or accept the null
hypothesis)
4. Conclusion

Remember these steps in testing the hypothesis because we will be discussing them in this module
one-by-one.

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 Let us start with the 1st step:

Statistical hypothesis – is a guess or a prediction made by a researcher

regarding the possible outcome of the study.

The actual test begins by considering two hypotheses. They are called the
null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints.
Stating the null hypothesis is not much of a problem, since it only uses the “ = ” or it tells
us that there is no significant difference between quantities. The only difficulty will be on the
alternative hypothesis since you are given three choices; the “≠ " meaning that there is no
difference between the groups being compared, the " < " meaning that the first is inferior than the
other, and the " > " meaning that the first is greater or superior than the second.

null hypothesis (𝑯𝟎 ) – it is a kind of statistical hypothesis in which it is hoped to be

rejected. It always shows equality between two or more subjects, it considers that there
is no significant difference and or relationship between and among variables.

alternative hypothesis (𝑯𝒂 ) - it is a kind of statistical hypothesis in which it

contradicts and challenges the null hypothesis. It uses the symbols “≠, <, > ".

Example: the average monthly salary of Teachers in the public and the private
schools.
𝐇𝟎 : 𝝁𝟏 = 𝝁𝟐 ; there is no significant difference between the average
monthly salary of Teachers in the public and the private
schools.

𝐇𝐚 : 𝝁𝟏 ≠ 𝝁𝟐 ; there is a significant difference between the average

monthly salary of Teachers in the public and the private
schools.
𝐇𝐚 : 𝝁𝟏 > 𝝁𝟐 ; the average monthly salary of the Teachers in the public
school is greater than those in the private school.
𝐇𝐚 : 𝝁𝟏 < 𝝁𝟐 ; the average monthly salary of the Teachers in the public
school is less than those in the private school.

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 TYPES OF HYPOTHESIS TESTING

2nd step in testing the hypothesis; (a) types of hypothesis testing, (b) level of significance, (c)
critical value.
Let us start with the types of hypothesis test.

In testing the hypothesis, you may either use a ONE-TAILED TEST or a

TWO-TAILED TEST.
the type of
hypothesis test
to be used I. ONE-TAILED TEST: this is a directional test with the region of rejection lies
depends on the either on the left or right tail of the normal curve.
alternative
hypothesis (𝐇𝐚 ) A. Right Directional Test – the region of rejection is on the right tail of the normal
curve. To decide whether a one-tailed right directional test will be used, refer to the
Alternative hypothesis. If the 𝐇𝐚 uses " > " or comparatives like greater than, higher than, superior
to, etc.

𝜷  Acceptance Rejection (𝜷)

∝  Region of Rejection (∝)

𝒄𝒗
 Critical Value

B. Left Directional Test - the region of rejection is on the left tail of the normal curve. To decide whether
a one-tailed left directional test will be used, refer to the Alternative hypothesis. If the 𝐇𝐚 uses " < " or
comparatives like less than, lower than, inferior to, etc.

𝜷  Acceptance Rejection (𝜷)

∝  Region of Rejection (∝)

𝒄𝒗
 Critical Value

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 C. Two-tailed Non Directional Test - the region of rejection lies on both tail
Critical Value – of the normal curve. To decide whether a Two-tailed Non Directional Test will be
the point that
separates the
used, refer to the Alternative hypothesis. If the 𝐇𝐚 uses " ≠ " or comparatives like not
region of rejection equal to, there is a significant difference, etc.
and the acceptance
region.
It is essential to
know and identify 𝜷  Acceptance Rejection (𝜷)
the critical value
because it plays an
important role in
deciding whether 𝜶 𝜶
 Region of Rejection (∝)
the 𝐇𝐨 will be 𝟐 𝟐

accepted or 𝒄𝒗 𝒄𝒗
rejected.  Critical Value

Level of significance – the area of the rejection region (∝) and the acceptance region (𝛽)

 if ∝ = 𝟎. 𝟎𝟏,
then 𝜷 = 𝟎. 𝟗𝟗
 if ∝ = 𝟎. 𝟎𝟓,
then 𝜷 = 𝟎. 𝟗𝟓
𝛽 = 0.99 𝛼 = 0.01
 if ∝ = 𝟎. 𝟏𝟎,
then 𝜷 = 𝟎. 𝟗𝟎

𝛽 = 0.95 𝛼 = 0.05 𝛽 = 0.90 𝛼 = 0.10

For this illustration I only used a one-tailed right directional test, the same rule is applied to a one-tailed
left directional test except for a two-tailed non directional in which the 𝛼 will be divided by two

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 Task 1. Formulate the Null (𝐻𝑜 ) and alternative (𝐻𝑎 ) hypotheses of the following research
problems. Identify if the hypothesis test is one-tailed left directional test, one-tailed right
directional test or a two-tailed non directional test.:
.
1. A teacher wants to know if there is a significant difference in the academic performance
between FM and HM students.
𝐻𝑜 : 𝜇𝐹𝑀 _______𝜇𝐻𝑀 ; _________________________________________________
__________________________________________________
𝐻𝑎 : 𝜇𝐹𝑀 _______𝜇𝐻𝑀 ;_________________________________________________

_________________________________________________

THE DECISION RULE AND THE DECISION

The 3rd step in testing the hypothesis is to decide whether to reject or to accept the null hypothesis.
To aide us in deciding and in analyzing the data we need to use the appropriate statistical tools.
In this lesson you will be presented two of the most commonly used statistical tools in testing the
significance of difference between means. Take note that there are a lot of statistical tools but as a researcher
you need to decide which of these statistical tools are appropriate to use for a particular situation.
The Z-test is used when “n is large” or when 𝑛 ≥ 30.
Following are the three formulas. There are clues so that you will not be confused as to what
formula to use in the problem. (Blay, 2007)

Testing the significance of difference between means

“n is large or when 𝒏 ≥ 𝟑𝟎 and 𝝈 is known”
Testing the significance of difference between means
1. Hypothesized/population mean VS Sample mean and population standard deviation
“n is large or when 𝒏 ≥ 𝟑𝟎 and 𝝈 is known”
is known.
2. Sample mean 1 VS Sample mean 2 and 2 sample standard deviations are known.
(𝒙
̅ − 𝝁)√𝒏
𝒛= 𝒙 ̅𝟏 − 𝒙̅𝟐
𝒛= 𝝈
(𝒔 ) 𝟐 (𝒔 )𝟐
̅ is the sample mean
Where: 𝒙 √ 𝟏 + 𝟐
𝒏𝟏 𝒏𝟐
𝝁 is the population mean
̅𝟏 is the mean of sample 1
Where: 𝒙
𝒏 is the sample size
̅𝟐 is the mean of sample 2
𝒙
𝝈 is the population standard deviation.
𝒏𝟏 and 𝒏𝟐 are the sample sizes
(Blay, 2007)
𝒔 and 𝒔𝟐 are the sample standard deviations
(Blay, 2007)

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 Testing the significance of difference between means
“n is large or when 𝒏 ≥ 𝟑𝟎 and 𝝈 is known”
3. Sample mean 1 VS Sample mean 2 and population standard deviations is known.
̅𝟏 − 𝒙
𝒙 ̅𝟐
𝒛=
𝟏 𝟏
𝝈√ +
𝒏𝟏 𝒏 𝟐
̅𝟏 is the mean of sample 1
Where: 𝒙
̅𝟐 is the mean of sample 2
𝒙
𝒏𝟏 and 𝒏𝟐 are the sample sizes
𝝈 is the population standard deviation
(Blay, 2007)

T-TEST
The t-test is used if n is small, it will be 𝑛 < 30 and 𝜎 is unknown. If the sample size is small, 𝑛 <
30, the values of the mean and standard deviation fluctuate from sample to sample. The sampling
distribution of the sample mean and the standard deviation is no longer a standard normal
distribution, thus you call it a t-distribution.
Degree of If the mean and standard deviation are computed from samples of size n, the values
freedom (df) – it
is the number of of t are said to belong to a t-distribution with degree of freedom (df) equals 𝑛 − 1.
variables which
are free to vary. (Blay, 2007)
𝒅𝒇 = 𝒏 − 𝟏 Testing the significance of difference between means
(Blay, 2007) “n is small or when 𝒏 < 𝟑𝟎 and 𝝈 is unknown”
̅𝟏 − 𝒙
𝒙 ̅𝟐
𝒕=
(𝒏 − 𝟏)(𝒔𝟏 )𝟐 + (𝒏𝟐 − 𝟏)(𝒔𝟐 )𝟐 𝟏 𝟏
√ 𝟏 √ +
𝒏𝟏 + 𝒏𝟐 − 𝟐 𝒏𝟏 𝒏𝟐

𝒅𝒇 = 𝒏𝟏 + 𝒏𝟐 − 𝟐
̅𝟏 is the mean of sample 1
Where: 𝒙
̅𝟐 is the mean of sample 2
𝒙
𝒏𝟏 and 𝒏𝟐 are the sample sizes
𝒔𝟏 and 𝒔𝟐 are the sample standard deviation
(Blay, 2007)

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 Dependent samples are drawn from the same population or the same set of samples
subjected to different experimental conditions, such as weight before and after
aerobics, pretest and posttest, etc.
Correlated samples are two different sets of samples which are so related, like
mother and daughter, brother and sister, old and new machine, etc.
(Blay, 2007)

How do you determine which t-test to use?

Are the two means from the same subject

or related subjects?

yes no

t-test: paired two sample means Are there the same number of sample
(dependent samples) in the two groups?

yes no

t-test: two-sample assuming equal Are the variances of the two groups
variances (independent samples) different?

no yes

t-test: two-sample assuming equal t-test: two-sample assuming unequal

variances (independent samples) variances

(Blay, 2007)

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 Looking at the different formula of the z-test and the t-test, using them when analyzing the data will be a
tedious work. Microsoft excel can process the data and generate the results without computing manually.
There are other programs or software that you can use to analyze the data some of which is SPSS,

To start working with Microsoft excel, follow these steps:

1. Open your Microsoft excel

2. Click DATA
3. Find DATA ANALYSIS

If there is no data analysis, then follow these steps:

1. Go to FILE
2. Click OPTIONS
3. Click ADD-INS
4. lick ANALYSIS TOOL-PAK
5. Click GO
6. Click ANALYSIS TOOL-PAK
7. Click OK

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 Example: using the Microsoft excel, find the mean, median mode, standard deviation, range, sum,
sample size and use the z-test: two sample for means of the following sets of data

Set A

79 80 83 90 70 65 60 71 85 89 80 87 85 85 75 73 74 71 70 68
88 78 81 85 87 91 93 90 83 84 81 80 74 73 71 66 65 60 78 78

Set B
88 80 81 66 75 75 78 61 85 90 89 88 91 81 85 88 84 86 83 84
85 88 81 80 83 81 95 90 99 98 65 60 61 90 91 93 83 80 82 85
91 90 88 81 80 80 87 87 86 83

Follow these steps

1. Open Microsoft Excel
2. Enter the data into separate columns. Set A for column 1 and Set B for column 2
3. Go to tools, then DATA ANALYSIS,

4. Select DESCRIPTIVE STATISTICS

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 5. In the dialogue box, the cursor
must be in the INPUT RANGE
then highlight all the data under
Set A,
Do not forget to click either
columns or rows(depending on
the data) and the labels in first
row
Click output range
Click anywhere in the active
sheet for the output
Click Summary Statistics
You may change the confidence
level
Then click Ok
(Note that the same process will
be used for SET B)

result of the descriptive statistics

6. Go to tools, then data

analysis. Select Z-TEST:
TWO SAMPLES FOR
MEANS

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 7. Variable 1:
highlight Set A
Variable 2: highlight
Set B
Hypothesized Mean:
input 0
Variable 1: Variance (known):
input the computed variance of
set A
Variable 1: Variance (known):
input the computed variance of
set B
Input the 𝛼
Click labels
Click on output range, then select
any cell
Click ok

8. Z-computed
p-value for one-
tailed test
Z-tabular for one-
tailed test (critical
value)
p-value for two-
tailed test
Z-tabular for two-
tailed test (critical
value)

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 For the T-TEST: follow these steps
1. Open Microsoft Excel
2. Enter the data into separate columns
3. Go to tools, then DATA ANALYSIS, select T-TEST: TWO
SAMPLE ASSUMING UNEQUAL VARIANCES
4. variable 1 Range: highlight set A
variable 2 Range: highlight set B
Hypothesized Mean: input 0
Click labels
Input 𝛼
Output range: click any cell

5. result of the descriptive statistics

degree of freedom
t-computed (computed value)
p-value for one-tailed test
t-value for one-tailed test
p-value for two-tailed test
t-value for two-tailed test

in deciding whether to accept or reject the null hypothesis, be guided by the following rules:
when using the z-test:

if |𝑍𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 | ≥ |𝑍𝑡𝑎𝑏𝑢𝑙𝑎𝑟 |, then reject the null hypothesis.

if |𝑍𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 | < |𝑍𝑡𝑎𝑏𝑢𝑙𝑎𝑟 |, then accept the null hypothesis.

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 When using the t-test

if |𝑡𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 | ≥ |𝑡𝑡𝑎𝑏𝑢𝑙𝑎𝑟 |, then reject the null hypothesis.

if |𝑡𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 | < |𝑡𝑡𝑎𝑏𝑢𝑙𝑎𝑟 |, then accept the null hypothesis.

When using the p-value

p-value If p-value ≤ 𝛼, then reject the null hypothesis.
It is a tool in hypothesis
If p-value > 𝛼, then accept the null hypothesis.
testing, it is utilized as an
alternative and equivalent
way of conducting tests of
significance. Here, you
compare the p-value with
the level of significance
(𝜶).
p-value refers to the
probability or the
expected value that the
phenomenon is likely to
occur
(Blay, 2007)

ERRORS IN HYPOTHESIS TESTING

Since we know that testing the hypothesis is making a decision, a decision whether to accept or
reject the null hypothesis. In making a decision there are moments that you commit an error, in testing
the hypothesis a researcher can commit an error too, these errors are what we call TYPE I and TYPE
II ERRORS.
A type I error is committed when a researcher rejected a TRUE null hypothesis. It is often
times referred to as the alpha error (𝛼 error).
A type II error is committed when a researcher accepted a FALSE null hypothesis. It is often
times referred to as the beta error (𝛽 error).

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 Decision made regarding 𝑯𝒐

If you reject 𝑯𝒐 , it means it is wrong.

If you accept 𝑯𝒐 , it does not mean it is correct, you simply do not have
enough evidence to reject it.
(Blay, 2007)

Decision
facts Accept 𝐇𝐨 Reject 𝐇𝐨
𝐇𝐨 is true Correct Decision Type I error (𝛼 )
𝐇𝐨 is false Type II Error(𝛽) Correct Decision

Example 1: A Teacher wants to know if students without calculator got significantly lower scores in the
midterm exam than those with calculator. Verify his claim using the steps in hypothesis testing.
Use Microsoft excel. (Blay, 2007)
Midterm exam scores of 40 students without calculator
79 80 83 90 70 65 60 71 85 89 80 87 85 85 75 73 74 71 70 68
88 78 81 85 87 91 93 90 83 84 81 80 74 73 71 66 65 60 78 78

Midterm exam scores of 40 students without calculator

88 80 81 66 75 75 78 61 85 90 89 88 91 81 85 88 84 86 83 84
85 88 81 80 83 81 95 90 99 98 65 60 61 90 91 93 83 80 82 85
91 90 88 81 80 80 87 87 86 83

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 Solution:
1. state the hypotheses
𝐻0 : 𝜇1 = 𝜇2 ; There is no significant difference between the performance of students with and
without calculator.
𝐻𝑎 : 𝜇1 < 𝜇2 ; The performance of students without calculator is lower than those with calculator.
2. 𝛼 = 0.05, one-tailed left directional test, 𝑍𝑡𝑎𝑏𝑢𝑙𝑎𝑟 = 1.64
3. Microsoft Excel output
z-Test: Two Sample for Means

𝑍𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 = −2.75 without with

when using the z-test:
Mean 78.15 83.22
𝑍𝑡𝑎𝑏𝑢𝑙𝑎𝑟 = 1.64
if |𝑍𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 | ≥ |𝑍𝑡𝑎𝑏𝑢𝑙𝑎𝑟 |, Known Variance 76.23 74.87
then reject the null Observations 40 50
hypothesis. Hypothesized Mean Difference 0
Since |𝑍𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 | ≥ |𝑍𝑡𝑎𝑏𝑢𝑙𝑎𝑟 | or z -2.75
if |𝒁𝒄𝒐𝒎𝒑𝒖𝒕𝒆𝒅 | < |𝒁𝒕𝒂𝒃𝒖𝒍𝒂𝒓 |, P(Z<=z) one-tail 0.00
then accept the null 2.75 ≥ 1.64 z Critical one-tail 1.64
hypothesis. P(Z<=z) two-tail 0.01
Therefore, REJECT THE 𝐻𝑂 z Critical two-tail 1.96

4. Conclusion: Based from the analyzed data, the performance of students without calculator is lower
than those with calculator.

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 Example 2: Below are the test results of the Team-Based Instruction Method (TBI) and the Individually-
Guided Instruction Method (IGI).
Student 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
TBI 30 28 29 20 18 19 16 27 22 24 26 28 30 29 18
IGI 25 27 20 30 16 21 15 25 28 21 19 17 18 13
Based on the result of the test, can you say that the TBI is more effective than IGI? Use 𝛼 = 0.05.
Solution:
1. 𝐻0 : 𝜇 𝑇𝐵𝐼 = 𝜇𝐼𝐺𝐼 ; There is no significant difference between the TBI method of teaching and the
IGI method of teaching.
𝐻𝑎 : 𝜇 𝑇𝐵𝐼 > 𝜇𝐼𝐺𝐼 ; The TBI method of teaching is more effective than the IGI method of teaching
2. 𝛼 = 0.05, One-tailed right directional test, 𝑑𝑓 = 27, 𝑡𝑡𝑎𝑏 = 1.7
3. Use t-test: two-sample assuming unequal variances

t-Test: Two-Sample Assuming Unequal Variances

when using the t-test Since |𝒕𝒄𝒐𝒎𝒑𝒖𝒕𝒆𝒅 | < |𝒕𝒕𝒂𝒃𝒖𝒍𝒂𝒓 | or
TBI IGI
if |𝒕𝒄𝒐𝒎𝒑𝒖𝒕𝒆𝒅 | ≥ |𝒕𝒕𝒂𝒃𝒖𝒍𝒂𝒓|, 1.69 < 1.70 Mean 24.27 21.07
then reject the null Variance 24.78 27.15
hypothesis. therefore, Observations 15.00 14.00
if |𝒕𝒄𝒐𝒎𝒑𝒖𝒕𝒆𝒅 | < |𝒕𝒕𝒂𝒃𝒖𝒍𝒂𝒓|, Hypothesized Mean Difference 0.00
ACCEPT THE 𝐻𝑂
then accept the null df 27.00
hypothesis t Stat 1.69
P(T<=t) one-tail 0.05
t Critical one-tail 1.70
P(T<=t) two-tail 0.10
t Critical two-tail 2.05

Conclusion: Based from the analyzed data, there is no significant difference between the
TBI method of teaching and the IGI method of teaching.

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 REFERENCES
Bluman, Allan G (2012). Elementary Statistics: a step by step approach. (8 th Ed) New York: McGraw-
Hill,
Blay, Basilia e. (2007). Elementary Statistics. Pasig City: Anvil Publishing, Inc.,
Calmorin, Laurentina P.,Pledad Ma. Lauremelch (2008). Nursing Biostatistics with Computer. Manila:
Rex Bookstore,
Baltazar, E.C, Ragasa, C, Evangelista, J.(2018). Mathematics in the Modern World. C & E.
Publishing:Quezon City Philippines.
Concepcio, Benjamin P. et.al. Business Statistics with Computer Applications. Sta. Monica Printing
Corp.: Manila, Philippines.
Calano, Roel B., et.al. (2009). Biostatistics. (1 st ed) Educational Publishing House: Ermita, Manila,
Philippines

THIS MODULE IS FOR THE EXCLUSIVE USE OF THE UNIVERSITY OF LA SALETTE, INC. ANY FORM OF REPRODUCTION, DISTRIBUTION, UPLOADING, 1
OR POSTING ONLINE IN ANY FORM OR BY ANY MEANS WITHOUT THE WRITTEN PERMISSION OF THE UNIVERSITY IS STRICTLY PROHIBITED.
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