QT Notes for Ph.D. Scholars

1. QT NOTES
MAHESH KUMAR JOSHI
20PHEN017
Ph. D. Scholar
JECRC UNIVERSITY, JAIPUR
09928138739

2. Population:-
Population is an aggregate of objects, animate or inanimate under study. The population
may be finite or infinite.
Sampling:-
A finite subset of statistical individuals in a population is called a Sample and the number
of individuals in a sample is called the sample size.
For the purpose of determining population characteristics, instead of enumerating the
entire population, the individuals in the sample only are observed. Then the sample
characteristics are utilized to approximately determine or estimate the population.
The error involved in such approximation is known as sampling error. But sampling
results in considerable gains, especially in time and cost not only in respect of making
observations of characteristics but also in the subsequent handling of the data.
Parameter and Statistic:-
In order to avoid verbal confusion with the statistical constants of the population. viz.,
mean (μ), variance ( σ2). etc., which are usually referred to as parameters, statistical
measures computed from the sample observations alone, e.g., mean ( x_bar ). variance (
s2 ). etc ., have been termed by Professor R.A. Fisher as statistics.

3. Population Mean
Population Standard Deviation
The term population mean, which is the average score of the population on a given variable,
is represented by:
μ = ( Σ Xi ) / N
The symbol ‘μ’ represents the population mean. The symbol ‘Σ Xi’ represents the sum of
all scores present in the population (say, in this case) X1 X2 X3 and so on. The symbol
‘N’ represents the total number of individuals or cases in the population.
The population standard deviation is a measure of the spread (variability) of the scores on a
given variable and is represented by:
σ = sqrt[ Σ ( Xi – μ )2 / N ]
The symbol ‘σ’ represents the population standard deviation. The term ‘sqrt’ used in this
statistical formula denotes square root. The term ‘Σ ( Xi – μ )2’ used in the statistical
formula represents the sum of the squared deviations of the scores from their population
mean.

4. Population Variance
The population variance is the square of the population standard deviation and is represented
by:
σ2 = Σ ( Xi – μ )2 / N
The symbol ‘σ2’ represents the population variance.
Sample Mean
The sample mean is the average score of a sample on a given variable and is represented by:
x_bar = ( Σ xi ) / n
The term “x_bar” represents the sample mean. The symbol ‘Σ xi’ used in this formula
represents the represents the sum of all scores present in the sample (say, in this case)
x1 x2 x3 and so on. The symbol ‘n,’ represents the total number of individuals or
observations in the sample.

5. Sample Standard Deviation
The statistic called sample standard deviation, is a measure of the spread (variability) of
the scores in the sample on a given variable and is represented by:
s = sqrt [ Σ ( xi – x_bar )2 / ( n – 1 ) ] for large sample
OR
s = sqrt [ Σ ( xi – x_bar )2 / n ] (for sample size < 30 for T – test in stats.)
The term ‘Σ ( xi – x_bar )2’ represents the sum of the squared deviations of the scores
from the sample mean.
Sample Variance
The sample variance is the square of the sample standard deviation and is represented by:
s2 = Σ ( xi – x_bar )2 / ( n – 1 ) for large sample
OR
s2 = Σ ( xi – x_bar )2 / n (for sample size < 30 for T – test in stats.)
The symbol ‘s2’ represents the sample variance.

6. HYPOTHESIS:-
An idea that is suggested as the possible explanation for something but
has not yet been found to be true or correct.
What is Hypothesis?
Hypothesis is an assumption that is made on the basis of some evidence. This
is the initial point of any investigation that translates the research questions
into a prediction. It includes components like variables, population and the
relation between the variables. A research hypothesis is a hypothesis that is
used to test the relationship between two or more variables.
Define a Problem into Statement is called a Hypothesis.

7. Null Hypothesis
In probability and statistics, the null hypothesis is a comprehensive statement or default
status that there is zero happening or nothing happening. For example, there is no
connection among groups or no association between two measured events. It is
generally assumed here that the hypothesis is true until any other proof has been brought
into the light to deny the hypothesis.
In statistics, the null hypothesis is usually denoted by letter H with subscript ‘0’ (zero),
such that H0. It is pronounced as H-null or H-zero or H-nought
The formula for the null hypothesis is:
H0: p = p0
In Statistics, a null hypothesis is a type of hypothesis which explains the population
parameter whose purpose is to test the validity of the given experimental data.
Such a hypothesis, which is usually a hypothesis of no difference, is called null
hypothesis and is usually denoted by Ho. According to Prof. RA. Fisher, null hypothesis
is the hypothesis which is tested for possible rejection under the assumption that it is
true.

8. Alternative Hypothesis
Alternative hypothesis defines there is a statistically important relationship between two
variables.
The alternative hypothesis is a statement used in statistical inference experiment. It is
contradictory to the null hypothesis and denoted by Ha or H1.
In this hypothesis, the difference between two or more variables is predicted by the
researchers, such that the pattern of data observed in the test is not due to chance.
Any hypothesis which is complementary to the null hypothesis is called an alternative
hypothesis, usually denoted by H1 .For example, if we want to test the null hypothesis that
the population has a specified mean μ0, (say), i.e., Ho: μ = μ0, then the alternative
hypothesis could be
(i) H1 : μ ≠ μ0 (i.e . μ > μ0 or μ < μ0)
(ii) H1 : μ > μ0
(iii) H1 : μ < μ0
The alternative hypothesis in (i) is known as a two-tailed alternative and the alternatives in
(ii) and (iii) are known as right-tailed and left-tailed alternatives respectively. The setting
of alternative hypothesis is very important since it enables us to decide whether we have to
use a single-tailed (right or left) or two tailed Test.

9. Left-Tailed: Here, it is expected that the sample proportion (π) is less than a specified
value which is denoted by π0, such that;
H1 : π < π0
Right-Tailed: It represents that the sample proportion (π) is greater than some value,
denoted by π0.
H1 : π > π0
Two-Tailed: According to this hypothesis, the sample proportion (denoted by π) is not
equal to a specific value which is represented by π0.
H1 : π ≠ π0
Note: The null hypothesis for all the three alternative hypotheses, would be H1 : π = π0.
A test of any statistical hypothesis where the alternative hypothesis is one tailed
(right tailed or left tailed) is called a one tailed test.

10. Difference between Null & Alternative Hypothesis
Single Tailed Test

11. FOUR STEPS TO HYPOTHESIS TESTING
The goal of hypothesis testing is to determine the likelihood that a population
parameter, such as the mean, is likely to be true:
Step 1: State the hypotheses.
Step 2: Set the criteria for a decision.
Step 3: Compute the test statistic.
Step 4: Make a decision.

12. A summary of hypothesis testing.

13. When Calculated value < Tabulated Value (Significant Value)
Then we say that Null Hypothesis is accepted.
When Calculated value > Tabulated Value (Significant Value)
Then we say that Null Hypothesis is rejected & Alternative Hypothesis is Accepted.
Calculated Value is found through Different Different Test
But Tabulated value found From degree of freedom (From Different Test).

14. Types of Test
1. T test (Used For Small Sample Size Less than 30 & Unknown Population Variance)
1.1 T-Test for Sample Mean
1.2 T-Test for Difference of Mean (For Independent Population)
1.3 Paired T-Test for Difference of Mean (Same Set Population)
1.4 T-Test for testing the significance of an Observed Sample Correlation Coefficient
2. Chi- Square Test
2.1 To Test If the Hypothetical Value of the population variance
2.2 To Test the Goodness of Fit
2.3 To test the independence of attributes
3. F-Distribution Test (Mean but According to variance)
4. Z- Test (Used For Large Sample)
*Note:- In T & Z Test Due to Symmetric, So we calculate (From Table) Different Value for 1
Tail & 2 Tail.
But For F-Test & Chi-Square Test There is no symmetrical So No difference between 1 Tail
& 2 Tail.

16. 1. T test (Used For Small Sample Size Less than 30 & Unknown Population Variance)
1.1 T-Test for Sample Mean
Under the Null Hypothesis H0 :
i) There is no significance difference between the sample mean x_bar & the population
mean µ,
ii) The sample has been drawn from the population mean µ.
Here S2 is Mean Square Error.

17. Relation Between Sample Variance s2 and Mean Square Error S2 :
Hence, for numerical problem the formula for the T- Test will be:-

22. Some Problems based on T-test Single Mean:

25. II Application of T- Test:- When we need to compare 2 Independent Population like 2
Section of a class, etc.

33. In ii case we apply T- Paired Test.

36. III Application of T- Test:- For Same Set Population

40. IV Application of T- Test:- For Correlation Coefficient

43. F- Distribution Test (Mean but
According to variance):-

47. Chi- Square Test
To Test If the Hypothetical Value of the population variance
To Test the Goodness of Fit
To test the independence of attributes

48. To Test If the Hypothetical Value of the population
variance

51. To Test the Goodness of Fit

58. To test the independence of attributes