Problem Solving: Quantitative Analysis For Business

Questions:

1. Clearly distinguish in your own words among the features of the three approaches to Under what circumstances would each type be more appropriate than the others?
 
2. A recent road safety study found that in 77% of all accidents the driver was wearing a Accident reports indicated that 92% of those drivers escaped serious injury (defined as hospitalisation or death), but only 63% of the non-belted drivers were so fortunate. What is the probability that a driver who was seriously injured was not wearing seatbelt? (Use the decision tree method to obtain your answer.)
 
3. The aptitude test scores of applicants to a university graduate program are normally distributed with mean 500 and standard deviation If the university wishes to set the cut-off score for graduate admission so that only the top 10 percent of applicants qualify for admission, what is the required cut-off score? What percentage of applicants have test scores within two standard deviations of the mean?
 
4. For the following discuss whether a sample or a census would be Indicate any assumptions you make:

‘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:

 
a. ‘The Central Limit Theorem is the cornerstone of statistical estimation.’
 
b. ‘The overriding factor in determining sample size is the requirement for precision in estimates of population ’
 
6. Explain the difference between sampling error and non-sampling Briefly describe three types of non-sampling error.
 
7. How large a sample is needed so that a 95% confidence interval for the mean number of years of attained education has a margin of error equal to one year?

 

 

Answers:

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

 posterior probability

 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.б + 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).

 

 

5) a. The Central Limit theorem forms the basis of Statistical Inference. The main basis of the central limit theorem is when the sample size is large then a set of i.i.d r.v’s with mean and standard deviation can be approximated by the normal distribution. Although the original distribution may vary from normal But by CLT it assumes Normality. The normal distribution is one of the most important distributions in statistics (especially for continuous data) and because of its extensive usage many statistical inference is dependent on the normal assumption. The central limit theorem makes the way for different data sets to approximately assume normal distribution as the properties of normal distribution are very useful for further inferencial refernce.  If the central limit theorem was not formulated then many hypothesis problems would have never existed (Lewis-Beck, 1993). Therefore the contribution of the central limit theorem cannot be denied for statistics and therefore it is termed as the cornerstone of statistical estimation.
 
b. By precision we measure how much the original value of the statistic is approximated by the estimated statistic. Calculation of precision is very important to judge how much error is shown by the statistic. Precision can be the value of 1 divided by the standard error. Therefore if precision of the estimates needs to be improved by a certain rate then the sample size must also be increased by that rate. Now the overiding factor for estimatimg sample size includes the standard error and the margin of error and on the basis of these factors the required sample size is generated, therefore these factors are required for the calculation of sample size with proper precision.

 

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

References

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.

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