Statistics and Econometrics Questions and Answers

Q: Which of the following is not a constituent body of National Statistics Office ?

The body that is not part of the NSO is the Indian Statistical Institute.The National Statistics Office under the Ministry of Statistics and Programme Implementation was formed by merging the Central Statistics Office and the National Sample Survey Office, and its divisions include the Data Informatics and Innovation Division, which handles IT and data processing.The Indian Statistical Institute in Kolkata, founded by P.C. Mahalanobis in 1931, is an autonomous institute of national importance under its own Act of 1959. The ministry funds it, but it is not a constituent of the NSO.

Q: If kurtosis is defined as \(\left(\frac{m_{4}}{\sigma^{4}} - 3\right)\), where \(m_{4}\) is the 4th order central moment and \(\sigma\) is the standard deviation, then its value is positive for

The value is positive for a leptokurtic distribution.The ratio \(m_{4}/\sigma^{4}\) (often written \(\beta_{2}\)) equals 3 for a normal distribution, so subtracting 3 measures excess kurtosis relative to the normal curve. A leptokurtic distribution is more peaked, with heavier tails, so \(\beta_{2} \gt 3\) and the measure is positive.A mesokurtic distribution has \(\beta_{2} = 3\), and the normal distribution is the standard mesokurtic case, so both give exactly zero. A platykurtic distribution is flatter than the normal, with \(\beta_{2} \lt 3\), so its value is negative.

Q: Which one of the followings is a relative measure of dispersion?

The relative measure is the Coefficient of Variation.It divides the standard deviation by the mean, \(CV = \frac{\sigma}{\bar{x}} \times 100\), so the units cancel and the result is a pure number. That is what makes a measure relative: it can compare the spread of series measured in different units or with very different averages.Range, mean absolute deviation about the mean and quartile deviation are all absolute measures, expressed in the same units as the data. Each has a relative counterpart (coefficient of range, coefficient of mean deviation, coefficient of quartile deviation), but the options name the absolute forms.

Q: Let \(Y_{t}\) be the value of \(Y\) in year \(t\). The natural logarithmic value of \(Y_{t}\) is regressed on \(t\). A simple linear regression yields the statistically significant estimates of intercept and slope coefficient as 12 and 0.1512, respectively. The calculated compound annual growth rate of \(Y\) is

The compound annual growth rate is 16.32%.The fitted semi-log trend is an exponential growth path:\[ \begin{aligned} \ln Y_{t} &= 12 + 0.1512\,t \\ Y_{t} &= e^{12}\left(e^{0.1512}\right)^{t} \end{aligned} \]Each year \(Y\) is multiplied by \(e^{0.1512}\), so the compound rate \(r\) satisfies:\[ \begin{aligned} 1 + r &= e^{0.1512} \approx 1.1632 \\ r &\approx 0.1632 = 16.32\% \end{aligned} \]15.12% is the slope itself, which is the instantaneous (continuously compounded) growth rate, not the compound annual rate. 116% treats the growth factor 1.16 as if it were the rate, and 1.16% misplaces the decimal point of that factor.

Q: For a sample of employees participating in an annual appraisal, the effect of the participants' performance on their wages is studied. Information are collected, and variables of interest are defined below:\(Wage_{i}\) = wage of the employee \(i\)\(Pass_{i}\) =1, if the employee \(i\) passes in the annual appraisal0, if the employee \(i\) fails in the annual appraisal\(Fail_{i}\) =1, if the employee \(i\) fails in the annual appraisal0, if the employee \(i\) passes in the annual appraisalBased on the above information, which one of the following regression models cannot be estimated?

The model that cannot be estimated is \(Wage_{i} = \alpha + \beta Pass_{i} + \gamma Fail_{i} + u_{1i}\).Every employee either passes or fails, so \(Pass_{i} + Fail_{i} = 1\) for all \(i\). That sum is exactly the column of ones that carries the intercept \(\alpha\). Including an intercept and both dummies therefore creates perfect multicollinearity, the dummy variable trap: the regressor matrix is not of full rank and least squares cannot separate \(\alpha\), \(\beta\) and \(\gamma\).The other three are estimable. With both dummies and no intercept, \(\beta\) and \(\gamma\) are simply the mean wages of those who pass and those who fail. With an intercept and one dummy, the omitted category becomes the base group: \(\alpha\) is its mean wage and the dummy's coefficient is the difference between the groups.

National Sample Surveys came into existence in:

A.1950
B.1951
C.1949
D.1952

Exam Year(s)

DSSSB PGT - 2023,

Answer: A

The National Sample Survey, the large scale survey of households by the Government of India, was set up in 1950 on the advice of P. C. Mahalanobis, and its first round was carried out in 1950. So 1950 is the year in which it came into existence. The years 1949, 1951 and 1952 are close dates that are often confused with it. The answer is 1950, option A. The survey is carried out by the National Sample Survey Office, which collects data on households, consumption and employment.

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Hypothesis can be stated in all ways except as:

A.Directional statement
B.Relation between variables
C.General statements
D.Declarative statements

Exam Year(s)

DSSSB PGT - 2023,

Answer: C

A hypothesis can be stated as a directional statement (predicting the direction of a relation), as a statement of the relation between variables, or as a declarative statement. A general statement is a broad claim and not a testable proposition about variables, so it is not a way of stating a hypothesis. The key gives general statements, option C. A hypothesis is a testable statement and so a general statement is too vague to be tested directly.

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Formulation of hypothesis is based on:

A.Interviews
B.Inferences
C.Observations
D.Conclusion

Exam Year(s)

DSSSB PGT - 2023,

Answer: C

A hypothesis is a tentative statement that is formed on the basis of observations of facts and can be tested. It rests on the researcher's observations of the phenomenon, while inferences and conclusions follow after the testing. The answer is observations, option C.

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Statistics as data does not include:

A.Quantitative data
B.Raw data
C.Descriptive statistics
D.Tools of analysis

Exam Year(s)

DSSSB PGT - 2023,

Answer: D

Statistics in the sense of data means numerical facts, that is quantitative data, and raw data, and also descriptive statistics, the summary figures. The tools of analysis belong to statistics as a method (the second meaning of the word) and are not data. The answer is tools of analysis, option D.

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Match the pair:

1. Collection of data	A. Business Statistic
2. Aggregate of facts	B. No Statistics
3. Analysis of time series	C. Statistics
4. One single data	D. Statistical method

A.1 (B), 2(A), 3(D), 4(C)
B.1 (A), 2(C), 3(D), 4(B)
C.1 (D), 2(B), 3(C), 4(A)
D.1 (D), 2(C), 3(A), 4(B)

Exam Year(s)

DSSSB PGT - 2023,

Answer: D

Statistics as a method, the tools of analysis, is the statistical method, so collection of data goes with D. The word statistics (the plural) as aggregates of facts goes with C. Analysis of a time series is a case of business statistics, A, and a single datum is not statistics, B. So 1 (D), 2 (C), 3 (A), 4 (B), option D.

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Which of the following is not a constituent body of National Statistics Office ?

A.Central Statistical Office
B.Data Informatics and Innovation Division
C.Indian Statistical Institute
D.National Sample Survey Office

Exam Year(s)

HPPSC HPAS - 2025,

Answer: C

The body that is not part of the NSO is the Indian Statistical Institute.

The National Statistics Office under the Ministry of Statistics and Programme Implementation was formed by merging the Central Statistics Office and the National Sample Survey Office, and its divisions include the Data Informatics and Innovation Division, which handles IT and data processing.

The Indian Statistical Institute in Kolkata, founded by P.C. Mahalanobis in 1931, is an autonomous institute of national importance under its own Act of 1959. The ministry funds it, but it is not a constituent of the NSO.

Discuss in Forum

If kurtosis is defined as \(\left(\frac{m_{4}}{\sigma^{4}} - 3\right)\), where \(m_{4}\) is the 4^th order central moment and \(\sigma\) is the standard deviation, then its value is positive for

A.a leptokurtic distribution
B.a mesokurtic distribution
C.a platykurtic distribution
D.a normal distribution

Exam Year(s)

IIT JAM - 2026,

Answer: A

The value is positive for a leptokurtic distribution.

The ratio \(m_{4}/\sigma^{4}\) (often written \(\beta_{2}\)) equals 3 for a normal distribution, so subtracting 3 measures excess kurtosis relative to the normal curve. A leptokurtic distribution is more peaked, with heavier tails, so \(\beta_{2} \gt 3\) and the measure is positive.

A mesokurtic distribution has \(\beta_{2} = 3\), and the normal distribution is the standard mesokurtic case, so both give exactly zero. A platykurtic distribution is flatter than the normal, with \(\beta_{2} \lt 3\), so its value is negative.

Discuss in Forum

Which one of the followings is a relative measure of dispersion?

A.Range
B.Mean Absolute Deviation about Mean
C.Coefficient of Variation
D.Quartile Deviation

Exam Year(s)

IIT JAM - 2026,

Answer: C

The relative measure is the Coefficient of Variation.

It divides the standard deviation by the mean, \(CV = \frac{\sigma}{\bar{x}} \times 100\), so the units cancel and the result is a pure number. That is what makes a measure relative: it can compare the spread of series measured in different units or with very different averages.

Range, mean absolute deviation about the mean and quartile deviation are all absolute measures, expressed in the same units as the data. Each has a relative counterpart (coefficient of range, coefficient of mean deviation, coefficient of quartile deviation), but the options name the absolute forms.

Discuss in Forum

Let \(Y_{t}\) be the value of \(Y\) in year \(t\). The natural logarithmic value of \(Y_{t}\) is regressed on \(t\). A simple linear regression yields the statistically significant estimates of intercept and slope coefficient as 12 and 0.1512, respectively. The calculated compound annual growth rate of \(Y\) is

A.15.12%
B.1.16%
C.16.32%
D.116%

Exam Year(s)

IIT JAM - 2026,

Answer: C

The compound annual growth rate is 16.32%.

The fitted semi-log trend is an exponential growth path:

\[ \begin{aligned} \ln Y_{t} &= 12 + 0.1512\,t \\ Y_{t} &= e^{12}\left(e^{0.1512}\right)^{t} \end{aligned} \]

Each year \(Y\) is multiplied by \(e^{0.1512}\), so the compound rate \(r\) satisfies:

\[ \begin{aligned} 1 + r &= e^{0.1512} \approx 1.1632 \\ r &\approx 0.1632 = 16.32\% \end{aligned} \]

15.12% is the slope itself, which is the instantaneous (continuously compounded) growth rate, not the compound annual rate. 116% treats the growth factor 1.16 as if it were the rate, and 1.16% misplaces the decimal point of that factor.

Discuss in Forum

10.

For a sample of employees participating in an annual appraisal, the effect of the participants' performance on their wages is studied. Information are collected, and variables of interest are defined below:

\(Wage_{i}\) = wage of the employee \(i\)

\(Pass_{i}\) =
1, if the employee \(i\) passes in the annual appraisal
0, if the employee \(i\) fails in the annual appraisal

\(Fail_{i}\) =
1, if the employee \(i\) fails in the annual appraisal
0, if the employee \(i\) passes in the annual appraisal

Based on the above information, which one of the following regression models cannot be estimated?

A.\(Wage_{i} = \alpha + \beta Pass_{i} + \gamma Fail_{i} + u_{1i}\) ; where, \(\alpha\), \(\beta\), and \(\gamma\) are the model parameters and \(u_{1i}\) is the error term
B.\(Wage_{i} = \beta Pass_{i} + \gamma Fail_{i} + u_{2i}\) ; where, \(\beta\) and \(\gamma\) are the model parameters and \(u_{2i}\) is the error term
C.\(Wage_{i} = \alpha + \gamma Fail_{i} + u_{3i}\) ; where, \(\alpha\) and \(\gamma\) are the model parameters and \(u_{3i}\) is the error term
D.\(Wage_{i} = \alpha + \beta Pass_{i} + u_{4i}\) ; where, \(\alpha\) and \(\beta\) are the model parameters and \(u_{4i}\) is the error term

Exam Year(s)

IIT JAM - 2026,

Answer: A

The model that cannot be estimated is \(Wage_{i} = \alpha + \beta Pass_{i} + \gamma Fail_{i} + u_{1i}\).

Every employee either passes or fails, so \(Pass_{i} + Fail_{i} = 1\) for all \(i\). That sum is exactly the column of ones that carries the intercept \(\alpha\). Including an intercept and both dummies therefore creates perfect multicollinearity, the dummy variable trap: the regressor matrix is not of full rank and least squares cannot separate \(\alpha\), \(\beta\) and \(\gamma\).

The other three are estimable. With both dummies and no intercept, \(\beta\) and \(\gamma\) are simply the mean wages of those who pass and those who fail. With an intercept and one dummy, the omitted category becomes the base group: \(\alpha\) is its mean wage and the dummy's coefficient is the difference between the groups.

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