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Distinguish among classical or a priori probability

(a) Distinguish among classical or a priori probability, relative frequency or a posteriori probability, axiomatic Probability and subjective or personal speak probability, what is the disadvantage of each? Why do we study probability theory?

(b) Describe the general procedure to testing for testing a hypothesis about a population parameter

(c) Differentiate between regression and correlation by giving example also describe the property of correlation coefficient

Answer:

(a)

  • Classic or prior probability:

Classical and prior probability means the number of favorable cases divided by a total number of equally likely mutually exclusive and exhaustive cases. It bases on theory, not on Experiment

P(A)= No.of favorable cases / Total  number of equally likely, mutually exclusive, and exhaustive case

Example:

Through a dice

S=1,2,3,4,5,6

P(1)=1/6

Disadvantage:

  • the event should be equally likely
  • If the total number of cases becomes infinite then it cannot be applied
  • Relative frequency or posterior probability

The relative frequency of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, not in a theoretical sample space but in an actual experiment. In a general sense, the empirical probability estimates probabilities from observation and experience.

Formula:

If all events are likely:

P(A)= No.Of successful trail  /Total no of trail

Disadvantage/limitation:

  • No experiment can perform infinite no of time therefore P(A) remain unknown
  • Axiomatic probability:

Let S be a sample space, let C be the class of all events, and let P be a real-valued function defined on C. Then P is called a probability function, and P(A) is called the probability of event A when the following axioms hold:

Axiom 1:  0<P(A) ≤ 1 for each event A in S.

Axiom 2: P(S) = 1.

Axiom 3: If A and B are mutually exclusive events in S, then P(AUB) = P(A) + P(B).

  • Personal & subjective probability:

as its name describe a subjective or personal probability means someone’s personal judgment about whether a specific outcome is likely to occur

for example a forecaster forecast about the weather that tomorrow has a 60% chance of rain. Subjective Probability is not dependent on past experience it only depends on our personal judgment

Disadvantage:

  • Less Appropriation: It does not base on the experiment or observation its just and guess or an idea so it may be wrong

(b)

  • Hypothesis testing is a scientific process of testing whether or not the hypothesis is plausible.  The following steps are involved in hypothesis testing:
  • The first steps first step is for the analyst to state the two hypotheses so that only one can be right.
  • The second step next step is to formulate an analysis plan, which outlines how the data will be evaluated.
  • The third step is to carry out the plan and physically analyze the sample data.
  • The fourth step The fourth and final step is to analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.
  • The fifth step is to draw a conclusion about the data and interpret the results obtained from There are basically three approaches to hypothesis testing. The researcher should note that all three approaches require different subject criteria and objective statistics, but all three approaches give the same conclusion.

(c)

COMPARISON CORRELATION REGRESSION
definition the degree and direction of the relationship between variables is called correlation It is the technique to investigate the dependencies of a variable on one or more variables for prediction and estimation
Usage To represent the linear relationship between two variables. To fit the best line and estimate one variable on the basis of another variable.
Dependent and Independent variables Both variables are the same Both variables are different.
Indicates The correlation coefficient shows the extent to which two variables move together. Regression indicates the impact of a unit change in the known variable (x) on the estimated variable (y).
Objective To find a numerical value expressing the relationship between variables. To estimate values of random variables on the basis of the values of the fixed variables.
Data representation In single point Represented by line

 

Property of Regression coefficient:

  • the geometric mean of two regression coefficients is equal to correlation Coefficient
  • two regression coefficients have the same sign
  • regression Coefficient is independent of origin but not scale
  • if one of regression Coefficients is greater than one then the other must be less than 1

Property of correlation coefficient

For variables x and y:

  • r is symmetrical with respect to the variable
  • r is a covariance of the value of the two variables measured in standard units
  • the magnitude of R is independent of change of origin and scale
  • r always lie between -1 and + 1
  • r is the charge metric mean of two regression coefficient

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