This ppt is presentation given by me in my college on the occasion of annual presentation test. My team's topic was Test of Significance of Large Samples for Mean = µ from Engineering Mathematics - 4 of Mumbai University Syllabus for Maths in Semester 4 of Engineering. All the credits to me, my team and https://www.faxscience.co.in/

1. ENGINEERING MATHEMATICS -4
MINI PROJECT
Test of Significance of Large Samples for
Mean = µ
Presented by Group no. 4 [A.I.M.L.]

2. Meet The
Team
Hardi
Patel
Harshit
Patil
Prathamesh
Patil
Tanmay
Patil
Siddhi
patil
Roll No- 17 Roll No- 18 Roll No- 19 Roll No- 20 Roll No- 21

3. Introduction
1
2
4
3
5
TABLES OF
CONTENTS History
How to solve them and Formulaes
Applications
Problems and Solutions

4. Parameter and Statistics
•A measure calculated from
populated data is called
Parameter.
•A measure calculated from
sample data is called Statistics.
Parameter Statistics
Size N n
Mean μ x̄
Standard deviation s
Proportion P p
Correlation
coefficient
r
σ
ρ

5. What is meant by Hypothesis ?
Hypothesis:
A hypothesis is a statement or assumption about a population parameter that can be tested using
statistical methods. It is a tentative explanation for an observed phenomenon and serves as the
basis for conducting statistical tests.
Example of a Hypothesis:
Suppose you are interested in investigating whether a new teaching method improves students'
test scores. Your hypothesis might be: "The average test scores of students using the new
teaching method are different from those of students using the traditional teaching method."

6. Types of Hypothesis ?
Null Hypothesis (H₀):
The null hypothesis is a statement that there is
no significant difference, effect, or relationship in
a population. It represents the default or status
quo assumption. In statistical testing, the null
hypothesis is what researchers aim to test
against and potentially reject.
Example of a Null Hypothesis:
Consider a scenario where a manufacturing
process is being assessed for its consistency in
producing a certain product. The null hypothesis
(H₀) in this case might be: "The mean product
dimensions produced by the manufacturing
process are equal to the target specifications."
Alternative Hypothesis (H₁ or Hα):
The alternative hypothesis is a statement that
contradicts the null hypothesis. It suggests that
there is a significant difference, effect, or
relationship in the population. In essence, it is
what researchers hope to support if there is
enough evidence to reject the null hypothesis.
Example of an Alternative Hypothesis:
Building on the manufacturing process example,
the alternative hypothesis (H₁) could be: "The
mean product dimensions produced by the
manufacturing process are different from the
target specifications."

7. Test of significance for difference for
Large Samples (z test)

8. Test of significance for difference for
Large Samples (z test)

9. Test of significance for difference for
Large Samples (z test)

10. Formulaes

11. Formulaes

12. Historical Context :
Early Statistics and the Foundation for Hypothesis Testing
Early Developments :
Ancient civilizations used simple methods like tallying on cave walls or clay tablets for basic data collection.
In the 17th and 18th centuries:
Governments began censuses for understanding population and wealth.
John Graunt analyzed London's mortality data, setting the stage for demography.
The Big Three :
Sir Ronald A. Fisher (1890-1962):
Known for ANOVA and null hypothesis significance testing (NHST).
Karl Pearson (1857-1936):
Developed correlation analysis and the chi-square test.
Jerzy Neyman (1894-1981):
Worked with Egon Pearson on formalizing null hypothesis testing and defining Type I and Type II errors.

13. Historical Context :
Foundation of Hypothesis Testing
Basic Idea:
Formulate Null Hypothesis (H₀):
Assume no difference exists (e.g., in a drug trial, H₀ might be that the new drug has no effect).
Set Significance Level (α):
Decide how confident you want to be before rejecting H₀ (common levels are 0.05 or 0.01).
Collect Data and Conduct Test:
Analyze data to see if it's unlikely to occur under the assumption of H₀.
Interpret Results:
If the chance of observing the data is very low (p-value < α), we reject H₀.
If the chance is high (p-value > α), we can't conclude much.

14. Quality
Control
Market
Research
Medical
Research
Applications
For determining production
process is meeting quality
standards or not by by
comparing sample means
to specified target values
Use to analyze survey data
to determine if there is
a significant difference
between the means of
different groups or
populations
Use to compare the
effectiveness of different
treatments in clinical trials,
analyzing whether there is
a significant difference in
mean outcomes between
treatment and control
groups.

15. Educational
Assesment
Financial
Analysis
Epidemiology
Applications
Educators may use the Z-test
to evaluate the effectiveness of
teaching methods by comparing
mean test scores of students
before and after implementing
a new instructional approach.
Use to assess whether there is
a significant difference in the
mean returns of different
investment portfolios or to
evaluate the performance
of a stock against a benchmark
index.
Public health researchers may
use the Z-test to compare the
mean incidence rates of diseases
between different populations or
regions.

16. Steps To Solve Hypothesis
Testing:
Step 1
Null Hypothesis
H0 and Alternative
Hypothesis Ha.
Step 2
Test of statistics
Step 3
Fix the level of
significance
Step 4
Determine the critical
region based on 2 tailed or
non- tailed test
Step 5
Descisio
n

17. Q1) Can it be concluded that the average life-span of an Indian is more than 70 years, if a random
samples of 100 Indians has an average life span of 71.8 years with standard deviations of 8.9 years?
Problems and solutions
Solution:
Example: Test of
hypothesis

18. Problems and solutions
Q2) A tyre company claims that the lives of tyres have been mean 42,000 kms with S.D of 4000 kms.A change in the production
process is believed to result in better product.A test sample of 81 new tyres has a mean life of 42,500 kms.Test at 5% level of
significance that the new product is significantly better than the old one.
Solution:

19. THANK YOU!