Subjective / Written Questions (25)
1. Explain the scope, importance and principal limitations of Statistics. Discuss common ways in which statistical methods may be misused.
2. Distinguish between a statistical population and a sample. Explain the circumstances in which sampling is preferable to a census.
3. Explain nominal, ordinal, interval and ratio scales of measurement, giving one suitable statistical example for each.
4. Differentiate quantitative and qualitative data. Explain primary and secondary data with suitable examples.
5. Distinguish time-series, cross-sectional, longitudinal and multivariate data. State an appropriate example of each.
6. Explain the construction of a frequency distribution from raw observations. Discuss the choice of class intervals and class boundaries.
7. Explain cumulative frequency distributions of the less-than and more-than types and describe their graphical representation by ogives.
8. Discuss the construction and interpretation of bar charts, pie charts, histograms and frequency polygons. Distinguish a histogram from a bar chart.
9. Explain box plots and state how they help in identifying the centre, spread and possible outliers of a dataset.
10. Explain one-dimensional, two-dimensional and multidimensional data visualisation and discuss their appropriate uses in statistical analysis.
11. Define arithmetic mean, geometric mean and harmonic mean. Derive the relation among them for positive observations and discuss their relative suitability.
12. Obtain the arithmetic mean from grouped data and discuss its important merits and limitations.
13. Explain the median and mode for grouped data. Discuss their advantages over the arithmetic mean in skewed distributions.
14. Define quartiles and explain their role in describing the position of observations within a distribution.
15. Compare arithmetic mean, median and mode with respect to algebraic treatment, stability, extreme observations and practical applications.
16. Define a random experiment, sample space and event. Explain the algebra of events with suitable examples.
17. State and explain the axiomatic definition of probability. Show how the basic properties of a probability measure follow from the axioms.
18. Explain the relationship between classical probability and the frequency interpretation of probability. Discuss the role of frequency ratios.
19. Prove or explain Boole's inequality and illustrate its use in bounding the probability of a union of events.
20. Derive the addition law of probability for two events and extend it to three events.
21. Explain conditional probability and the associated conditional probability space. Derive the multiplication rule for two events.
22. State and prove Bayes' theorem for a finite partition of the sample space. Explain its interpretation in terms of revised probabilities.
23. Define independence of events and distinguish pairwise independence from mutual independence, using suitable examples.
24. Define a random variable and explain how a probability distribution is induced by a mapping from the sample space.
25. Distinguish between a discrete and a continuous random variable. Define the probability mass function, probability density function and distribution function, and state their basic properties.