‘An examination of dwellings in a large city to obtain information about the extent of present home insulation and the costs to bring substandard dwellings up to minimum insulation standards.’
5. Comment on the veracity of the following .statements:
1. The three different probability approaches are
The Classical Approach
The Frequency Approach
The Axiomatic Approach
Probability theory is based on the concept of randomness, i.e. the random experiment.
In random experiment the toatal number of all possible cases is defined as the sample space.
and the total number of events favourable to be be counted before calculating probability.
Therefore the required Classical definition of probability
= Total number of all possible cases favourable to event E / Total number of all possible cases.
The Frequency Approach to probability is a simpler approach. The probability is to be calculated for the event A (Barnett, 1999). Suppose the whole population cannot be the sample space to calculate this probability. Then from a specified sample we calculate the probability fuction
Therefore by the frequency approach
Probability function of the event E
= Total number of cases that have that particular criterion(Here E)/ Total number of sample cases
The Axiomatic Approch uses axioms to define probability. The Axiomatic probability has mainly three axioms. Any function satisfying the three axioms is called the probability function if
- P(A) >0, considering any event A
- P(Ω) = 1, this will be the case for all sure events where Ω is a sure event.
- Suppose A1 and A2 are two events which are mutually exclusive then
P(A1 U A2) =P(A1)+P(A2) , this will be the case for all mutually exclusive events.
Generally the classical definition is used in games of cards, dice and coins which are not very related to day to day life. Moreover classical definition cannot be applied if the events are not equally likely and the number of events are not finite.
The Frequency approach to probability has many practical uses and can be used in studies where data is collected by sampling.
The Axiomatic approach is combination of both the approaches, better to say both the classical and frequency approaches can be fitted in the axiomatic approach. The axiomatic approach helps probability theory to grow further because from the formulas based on the axioms probaility theory has invented new ideas.
2. The given probabilities are placed in the decision tree
Here we have to calculate the posterior probability
=P(driver not wearing seatbelt/ Driver was seriously injured) (From the bayes theorem )
= .37*.23 = .0851
3. Here X is the marks of an aptitude test for applicants in a university.
Given X ~ N(500,60)
The university wants only the top 10 percent pf the applicants for selection. From the table of the Z score it is seen that at 10% i.e. at .01 the Z score is 1.28 Since we calculated it for standard normal, to return to the normal , the equation used is X = Z.Ð± + µ = 500 + 1.28*60 = 576.80
4. From the given information about the population it is seen that the sample units are dwellings of a large city. Therefore the population size is big and also it quite a labourious job to inspect all the dwellings individually. Therefore a sample from the population can be more useful because it would reduce the cost of the survey and also reduce the time of the survey. If a proper sampling plan is selected it would further improve the accuracy of the results because cencus being a lenghty processcan loose track and therfore provide faulty results. A stratified random sampling can be used in this case, the whole city must be divided into stratas for example simple stratas like north, south , east and west and then simple random sampling can be conducted from those stratas (Blank, 1968).
6. Sampling error occurs only in case of sampling and non sampling method occurs both in case of sampling and census. Sampling is a probabilistic method and therefore the problems are related to those methods like wrong sample selection and wrong sample selection whereas non sampling errors include major and basic problems of sampling like design of data, data collection , therefore nonsampling error being both common to sampling and census.
The first kind of nonsampling errors occur due to improper planning like lack of eligible investigators, improper data collection methods and faulty data designing. The second kind of nonsampling error occur due to faulty response like respondent manipulating answers, lack of memory, misunderstanding in part of the respondant and prestige issues e.t.c. and also due to non response. The third kind of nonsampling error occur due to errors in coverage, error in compilation of data and improper data placement by the investigator.
7. Here X = mean number of years of attained education
We want to estimate the sample size for the mean number of years.
Let us assume that the standard deviation be s. The margin of error is given as 1. Here confidence level is 95% and critical value for 95% is 1.96 and by the help of this critical value the standard deviation is to be calculated.
Finally the required sample size = 4*s2/12
Barnett, V. (1999). Comparative statistical inference. Chichester: Wiley.
Blank, S. (1968). Descriptive statistics. New York: Appleton-Century-Crofts.
Lewis-Beck, M. (1993). Regression analysis. London: Sage Publications.