Here are the steps to solve this problem: 1) State the null and alternative hypotheses: H0: μ1 = μ2 H1: μ1 ≠ μ2 2) Calculate the test statistic: t = (x̅1 - x̅2)/√(s12/n1 + s22/n2) = (62 - 60)/√(4.52/20 + 3.52/30) = 2/√(0.225 + 0.12) = 2/0.345 = 5.8 3) Degrees of freedom = n1 + n2 - 2 = 20 + 30 - 2 = 48 4) Find

1. Stat-1163: Statistics in Environmental Science
Section B,
Chapter: Test of Significance
Md. Menhazul Abedin
Lecturer
Statistics Discipline
Khulna University, Khulna-9208
Email: menhaz70@gmail.com

2. Acknowledgment
• Dr. K C Bhuyan
• M. Nurul Islam
• Rabindra Nath Shil & Subash Chandra Debnath
• Internet

3. Tools To Study The Environment
• The nature of
environmental science
• The scientific method
and the scientific
process
• Natural resources and
their importance
• Culture and worldviews
• Environmental ethics
• Sustainability

4. Environmental science
• How does the natural world work?
• How does our environment affect us?
• How do we affect our environment?
• Goal: Developing solutions to environmental
problems.

5. Environmental Science: Multidisciplinary Science
Environmental
Science
Ethics
Histrory
Geology
Oceanogr
aphy
Atomospheric
science
Biology
Ecology
Statistics
Political
Science
Philosophy

6. • We are constantly being bombarded with
statistics and statistical information. For
example:
– Environmental data, Market data, Medical News,
Demographics, Economic Predictions, Sports Statistics etc.
• How can we make sense out of all this data?
• How do we differentiate valid from flawed
claims?
Initial talk

7. Statistics? (Rough concept)
“Statistics is a way to get information from data.”
Statistics is a tool for creating an understanding
from a set of numbers or text or…
Statistics
Data
Facts, especially
numerical facts, collected
together for reference or
information.
Information
Knowledge
communicated concerning
some particular fact.

8. Statistics? (Rough concept)
Statistics
Data
List of last term’s marks.
95, 89, 70, 65, 78, 57…
Information
Class average, Proportion
of class receiving A’s Most
frequent mark, Marks
distribution, etc.

9. Statistics? (Precise)
• Statistics is concerned with scientific methods
for collecting, summarizing, presenting and
analyzing sample data as well as drawing valid
conclusions about population characteristics
and making reasonable decisions on the basis
of such analysis.
• According to Fisher, the science statistics is
essentially a branch of applied mathematics
and may be rgarded as mathematics, applied
to observational data.

10. Population and Sample
• Population: A population is the collection or
agregate of all elements or items of interest in
a particular study about which we wish to
make an inference.
• Sample: A sample is a collection of sampling
units hopefully representative of the total
population or universe that one desire to
study.

11. Population and Sample

12. Parameter and Statistic
• Parameter: The unknown population
characteristics is called parameter.
• Statistic: Function of sample observations is
called statistic.

13. Descriptive statistics
• Descriptive statistics are statistics that
quantitatively describe or summarize the data.
The aim of descriptive statistics is to summarize
a sample, rather than use the data to learn
about the population.
• Graphs, Measure of Central tendency, Measure
of dispersion, frequency distribution…

14. Inferential Statistics

15. Inferential Statistics
• Is hypothesis true?? Million dollar question.
Mean income
20000 Taka
hypothetically
Mean
income
19500 taka
Estimated

16. Inferential Statistics
• Inferential Statistics: Mathematical methods
that employ probability theory for deducing
(inferring) the properties of a population from
the analysis of the properties of a data sample
drawn from it.
– Estimation (point estimation & interval estimation)
– Hypothesis testing
How??????
Test of hypothesis

17. Why inferencial statistics?
• Large populations make investigating each
member impractical and expensive.
• Easier and cheaper to take a sample and make
estimates about the population from the
sample.

18. Process…
• Researchers use guesses called hypotheses to draw
predictions
that are then tested experimentally.
• Results may reject or fail to reject the hypothesis.
• Results never confirm a hypothesis, but only lend
support to it by failing to reject it. This means we
never prove anything with this method.

19. Experiments
• Manipulative experiments
are strongest.
Figure 1.10
Natural or correlational ones are often
necessary.

20. • Test Hypothesis or Statistical hypothesis: The
process of making inference about parameter
using sample statistic of estimator is known as
test of hypothesis or statistical hypothesis.
• Basically two types test
– Parametric test
– Non-parametric test
Test of hypothesis

21. Test of hypothesis
• Parametric test: Any specification about one
or more parameters.
• 𝑛(𝜇, 𝜎2
) any sprcification about 𝜇, or 𝜎2
or
both.
• Non-parametric test: Any specification of
population or sample of the population
except the parameter.
• Randomness, distribuition etc

22. Test of hypothesis
• Hypothesis: Any specification regarding a
parameter or a population which is under
investigation isknown as hypothesis. The
hypothesis is of two types
– Null hypothesis
– Alternative hypothesis

23. Examples
• A physician may hypothesize that the
recommended drug is effective in 90% cases
• The court assumes that the indicted person is
innocent
• Green house gas is responsible for global
warming
• 𝐶𝑂2 responsible for high birth rate
Test this statement whether true or false

24. Test of hypothesis
• Null hypothesis: The hypothesis which is used
for possible rejection under verification on the
basis of sample statistics is known as null
hypothesis. It is denoted by 𝐻0.
• 𝑋~𝑁(𝜇, 𝜎2
) Thus null hypothesis can be
defined as 𝐻0: 𝜇 = 𝜇0 or about variance.

25. Test of hypothesis
• Alternative hypothesis: Any statement against
the null hypohesis. Denoted by 𝐻1 or 𝐻𝐴
• 𝐻1: 𝜇 > 𝜇0 or 𝐻1: 𝜇 < 𝜇0 or 𝐻1: 𝜇 ≠ 𝜇0

26. Test of hypothesis
• Simple hypothesis: If in investigting the
significance of a hypothesis all parameters of
the distribution or the distribution itself are
specified, then the null hypothesisis known as
simple hypothesis. Let 𝑋~𝑁(𝜇, 𝜎2
). The
object is to test the significance of 𝐻0: 𝜇 = 𝜇0
against 𝐻1: 𝜇 ≠ 𝜇0 and 𝜎2
is known. If 𝜎2
is
unknown, thus both parameter are
unspecified and hypothesis is composite

27. Test of hypothesis
• Composite hypothesis: if by a hypothesis all
the parameter of a distribution or distribution
itself are not specified, the hypothesis is called
composite hypothesis.

28. Test of hypothesis
• Test statistic: The function of sample
observation which is used to verify the null
hypothesis is known as test statistic.
– Thus, if the phenomenon under investigation is to
test the null hypothesis 𝐻0: 𝜇 = 𝜇0( a known
value), where 𝜇 is the mean of amount of 𝑂2.
Then the test statistic is 𝑧 =
𝑥−𝜇0
𝑠2
𝑛
~𝑁(0,1)

29. Test of hypothesis
• Critical region: The area constituted with the
values of sample statistic due to which the null
hypothesis is rejected even when it is true is
called critical region. It is denoted by W.

30. Test of hypothesis
• Acceptance region: The area constituted with
the values of sample statistic due to which the
null hypothesis is accepted when it is true is
called critical region. It is denoted by W.

31. Test of hypothesis
• Type-I error & Type-II error:
True
Statement
Fin
din
gs

32. Test of hypothesis
• Type-I error: The error that creeps in taking
decision to reject the null hypothesis when
the null hypothesis is true is called type-I
error. The probability of this error is denoted
by 𝛼.
• Type-II error: The errror that creeps in taking
decision to accept the null hypothesis even
when it is not true is known as type-II error.
The probability of his error is denoted by 𝛽

33. Test of hypothesis
• Level of significance: The error that creeps in
taking decision to reject the null hypothesis
when the null hypothesis is true is called type-
I error. The probability of this error is denoted
by 𝛼 is known as level of significance.
Mathematically, 𝑃 𝑋 ∈ 𝑤|𝐻0 = 𝛼 . Here X denote the
test statistics and W is the critical region.

34. Test of hypothesis
• p-value: The minimum value of 𝛼 for which
the null hypothesis is rejected is known as p-
value. If the p-value is less than or equal to 𝛼,
the null hypothesis is rejected, otherwise it is
accepted.

35. Test of hypothesis
• Degrees of freedom: The number of degrees
of freedom generally refers to the number of
independent observations in a sample minus
the number of population parameters that
must be estimated from sample data.
• Sample size= 𝑛
• Number of population parameter = 𝑘
• Thus the df = 𝑛 − 𝑘

36. Test of hypothesis
• One sided test: If the critical region is
considered at one end of the probability
distribution of the test statistic, the test is
called one sided test.
• 𝐻0: 𝜇 = 𝜇0 against 𝐻1: 𝜇 > 𝜇0 or 𝐻1: 𝜇 < 𝜇0
is one sided test

37. Test of hypothesis
• Two sided test: If in statistical test , the critical
region is considered in left as well as right side
of the distribution of the test statistic, then
the test is called two sided test.
• 𝐻0: 𝜇 = 𝜇0 against 𝐻1: 𝜇 ≠ 𝜇0 is two sided
test
• In this case level of significance is divided both
side as
𝛼
2

38. Test of hypothesis
• One sided test & Two sided test:
Two tailed test

39. Test of hypothesis
• Power of test: The probability of rejecting null
hypothesis when alternative hyothesis is realy
true. It is enoted by 1 − 𝛽. This is the
probability of right decision regarding the null
hypothesis.
• Mathematically,
Power = 1 − 𝛽 = 𝑃(𝑋 ∈ 𝑊/𝐻1)

40. Test of hypothesis
• Some methods of test:
– Most powerful (MP) test
– Uniformly most powerful (UMP) test
– Similar region test
– Likelihood ratio (LR) test
– Sequential probability ratio test (SPRT) etc

41. Steps of Test of Hypothesis
• Explicit knowledge of the nature of the population
distribution and the parameter of interest.
• Set null hypothesis (𝐻0) and alternative hypothesis
𝐻1 .
• Choice a suitable test statistics 𝑡 = 𝑡(𝑥1, 𝑥2, … , 𝑥 𝑛)
• Partitioning the set of possible values of the test
statistic 𝑡 into two disjoint sets 𝑊 (critical region) and
𝑊 (acceptance region)
– Reject 𝐻0 if the value of 𝑡 falls in 𝑊
– Accept 𝐻0 if thevalue of 𝑡 falls in 𝑊

42. Test of hypothesis
• We discuss here about
– Mean test
• Single or paired mean test → t-test or z- test
• Several mean test → F-test
– Variance test
• Single variance test → Chi-square test
• Paired variance test → F-test
• Several variance test → Chi-square test
– Proportion test
– Correlation test

43. Test of hypothesis
Single/ Paired mean test
Large sample
Sample size >
30
Variance known or
unknown
Use normal test
(z-test)
Small sample
Sample size ≤ 30
Variance unknown
Use t-test
Vaiance known
Use normal test
(z-test)

44. Example-1
• A bulb manufacture company claims that the average longevity of
their bulb is 4 years with a standard deviation of 0.16 years. A random
sample of 40 bulbs gave a mean longevity of 3.45 years. Does the
sample mean justify the claim of the manufacturer? Use a 5 percent
level of significance.
• Sol: We test the following hypothesis with 𝛼 = 0.05
1. 𝐻0: 𝜇 = 4 𝑉𝑠 𝐻1: 𝜇 ≠ 4
2. 𝑥 = 3.45, 𝑛 = 40, 𝜇0 = 4 𝑎𝑛𝑑 𝜎 = 0.16
3. 𝑧 =
𝑥−𝜇0
𝜎
𝑛
=
3.45−4
0.16/ 40
= −21.7
4. Critical region −1.96 < 𝑧 < 1.96 or|𝑧 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙| = 1.96
5. Decision: Null hypothesis rejected that is claim is flawed.
Sample size large
Variance known
If sample size 25 ????

45. Example-1
• 95% Cofidence interval :
• |
𝑥−𝜇0
𝜎
𝑛
| ≤ 𝑧 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
⇒ 𝑥 − 𝑧 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
𝜎
𝑛
≤ 𝜇0 ≤ 𝑥 + 𝑧 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
𝜎
𝑛
⇒ Pr 𝑥 − 𝑧 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
𝜎
𝑛
≤ 𝜇0 ≤ 𝑥 + 𝑧 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
𝜎
𝑛
= 0.95
• Putting the values from previous slide lower limit 3.95 and uper
limit 4.05
• But our mean 𝑥 = 3.45 is outside of the interval, hence hypothesi
is rejected at 5% level of significance or with 95% confidence.
If 𝛼 = 0.01
Critical region -
2.58<z<2.58
99% Confidence
interval ?????

46. Example-2
• Suppose that a steel manufacturing company wishes to know whether the
tensile strength of the steel wire has an overall average of 120 pounds. A
sample of 25 units of steel wire produced by the company yields a mean
strength of 110 pounds and variance 144. Should the company conclude
that the strength is not 120 pounds with 𝛼 = 0.05 ? Obtain 95%
confidence interval for the true mean.
• Sol:
– We would like to test 𝐻0: 𝜇 = 120 𝑉𝑠 𝐻1: 𝜇 ≠ 120
– 𝑥 = 110, 𝑛 = 25, 𝜇0 = 120, 𝑠2
= 144 𝑡ℎ𝑢𝑠 𝑠 = 12
– Under null hypothesis test statistics t =
𝑥−𝜇0
𝑠
𝑛
~𝑡( 0.05,24)
⇒ t = −4.17
– Critical value of t at 5% level of significane is 2.064 thus null hypothesis is rejectedt hat
is the strength is not 120.

47. Example-2
• 95% Confidence interval
𝑥 − 𝑡 𝛼/2(𝑛−1)
𝑠
𝑛
< 𝜇 < 𝑥 + 𝑡 𝛼/2(𝑛−1)
𝑠
𝑛
110 − 2.064
12
25
< 𝜇 < 110 + 2.064
12
25
105.05 < 𝜇 < 114.95
• We see that 120 is outside of the interval thus null hypothwes is wrong at
5% level of significance or with 95% confidence .
Do this math at 1% level of
significance and find 99% level
of significance

48. Example-3
• In Bangladesh a delivery record as following
Test the hypothesis that there is no difference between the mean heights of
the groups of women. Use 5%level of significance.
• Sol:
– 𝐻0: 𝜇1 = 𝜇2 Vs 𝐻1: 𝜇1 ≠ 𝜇2
– 𝑥1 = 156, 𝑛1 = 60, 𝑠1 = 3.1, 𝑥2 = 154, 𝑛2 = 52, 𝑠1 = 2.8
– 𝑧 =
𝑥1− 𝑥2
𝑠1
2
𝑛1
+
𝑠2
2
𝑛2
= 3.6
Delivery type No of women Mean height (inch) SD
Normal 60 156 3.1
Ceasearean 52 154 2.8

49. Example-3
• Critical value =1.96
• Decision: Since the critical value is less than
calculated value thus null hypothesis is
rejected and thus differencre is significant.

50. Double(piared) mean test
Let 𝑥1 and 𝑥2 mean of two sample 𝑠1
2 and 𝑠2
2 are variances .
To test there equality the test statistics is
𝑡 =
𝑥1 − 𝑥2
𝑠1
2
𝑛1
+
𝑠2
2
𝑛2
=
𝑥1 − 𝑥2
𝑠 𝑝
2
𝑛1
+
𝑠 𝑝
2
𝑛2
=
𝑥1 − 𝑥2
𝑠 𝑝
1
𝑛1
+
1
𝑛2
• Where 𝑠 𝑝 is pooled variance
𝑠 𝑝 =
𝑛1 − 1 𝑠1
2
+ (𝑛2 − 1)𝑠2
2
𝑛1 + 𝑛1 − 2

51. Example-4
• A professional Bangladesh batsman averaged 70.2 runs with a standard
deviation 8.4 runs in 16 randomly selected one-day match in Dhaka, and
62.7 runs with a standard deviation of 7.7 in 14 randomly selected such
matches in Chittagong. Can we conclude at 5 percent level of significance
that the observed difference 7.5 in bating average could be attributed to
regional difference or it is a real difference?
• Solution:
– 𝐻0: 𝜇1 = 𝜇2 against 𝐻1: 𝜇1 ≠ 𝜇2
– 𝛼 = 0.05
– Critical regeion: 𝑡 > 2.048 or 𝑡 < −2.048 Wehere t =
𝑥1− 𝑥2
𝑠 𝑝
1
𝑛1
+
1
𝑛2
– 𝑡 =
70.2−62.7
8.08
1
16
+
1
14
= 2.54
– The difference is ststistically significant