1. BUSINESS MATHEMATICS
&
STATISTICS

2. Hypothesis testing : Chi-Square Distribution
LECTURE 43
Hypothesis testing : Chi-Square Distribution
Part 1

3. EXAMPLE
An inspector took a sample of 100 tins of beans The sample weight is 225 g
Standard deviation is 5 g
Calculate with 95% confidence the range of the population mean
STEM = s.d/(n )^1/2
s.d is not known
Use s.d of sample as an approximation
STEM = 5/(100)^1/2 = 0.5
95% confidence interval = 225 +/- 2 x 0.5 or
From 224 to 226 g

4. PROBLEM REVISITED
Box of 500 components may have 25 or 5% faulty components
Overall faulty items = 2%
P = 2%; n= 500;
STEP = [(2 x 98)/500]^1/2
= 0.626
To find the probability that the sample percentage is 5% or over
z = (5 – 2)/STEP = 3/0.626 = 4.79
Area against z = 4.79 is negligible
Chance of such a sample is very small

5. FINITE POPULATION CORRECTION FACTOR
If population is very large compared to the sample then multiply STEM and STEP by the Finite Population Correction Factor =
[1 – (n/N)]^1/2
Where
N = Size of the population
n = Size of the sample
n = less than 0.1N

6. TRAINING MANAGER’S PROBLEM
New refresher course for training of workers
Traing Manager would like to assess the effect of retraining if any
Particular questions
Is quality of product better than produced before retraining?
Has the speed of machines increased?
Do some classes of workers respond better to retraining than others?
Training Manager Hopes TO:
Compare the new position with established
Test a theory or hypothesis about the course

7. CASE STUDY Before the course Worker X produced 4% rejects
After the course
Out of 400 items 14 were defective = 3.5%
An improvement?
The 3.5% figure may not demonstrate overall improvement
It does not follow that every single sample of 400 items contains exactly 4% rejects
To draw a sound conclusion
Sampling variations must be taken into account
We do not begin by assuming what we are trying to prove

8. NULL HYPOTHESIS
Must begin with the assumption that there is no change at all
This initial assumption is called
NULL HYPOTHESIS
Implication of Null Hypothesis
That the sample of 400 items taken after the course was drawn from a population in which the percentage of reject items is still 4%

9. NULL HYPOTHESIS EXAMPLE
Data
P = 4%; n = 400
STEP
= [P(100 – P)/n]^1/2
= [4(100 – 4)/400]^1/2 = 0.98%
At 95% confidence limit
Range = 4 +/- 2 x 0.98 = 2.04 to 5.96 %
Conclusion
Sample with 3.5% rejects is not inconsistent
No ground to assume that % rejects has changed at all
On the strength of sample there were no grounds for rejecting Null Hypothesis

10. ANOTHER EXAMPLE Before the course 5% rejects After the course
2.5% rejects (10 out of 400)
P = 5
STEP = [5(100 – 5)/400]^1/2
= [5 x 95/400]^1/2
= 1.09
Range at 95 % Confidence Limits
= 5 +/- 2 x 1.09
= 2.82 % to 7.18 %
Conclusion
Doubt about Null Hypothesis most of the time
Null hypothesis to be rejected

11. PROCEDURE FOR CARRYING OUT HYPOTHESIS TEST
Formulate null hypothesis
Calculate STEP & P +/- 2 x STEP
Compare the sample % with this interval to see whether it is inside or outside
(A) If the sample falls outside the interval, reject the null hypothesis ( sample differs significantly from the population %)
(B) If the sample falls inside the interval,
do not reject the null hypothesis ( sample does not differ significantly from the population % at 5% level)

12. HOW THE RULE WORKS?
Bigger the difference between the sample and population percentages: Less likely that population percentages will be applicable
When the difference is so big that the sample falls outside the 95% interval
Then the population percentages cannot be applied
Null Hypothesis must be rejected
If sample belongs to majority and it falls within 95% interval
Then there are no grounds for doubting the Null Hypothesis

13. FURTHER POINTS ABOUT HYPOTHESIS TESTING
99% interval requires 2.58 x STEP. Interval wider. Less likely to conclude that something is significant
(A) We might conclude there is a significant difference when there is none. Chance of error = 5% (type 1 Error)
(B) We might decide that there is no significant difference when there is one (Type 2 Error)

14. BUSINESS MATHEMATICS
&
STATISTICS