Statistics and Econometrics Questions and Answers

Q: \(\frac{\sum_{i}(P_{i},t)\times(Q_{i},o)}{\sum_{i}(P_{i},o)\times(Q_{i},o)}\) \(\times 100\) is the formula of ........................ .

It is the formula of Laspeyre's Price Index.The numerator values base-year quantities (\(Q_{o}\)) at current prices (\(P_{t}\)), and the denominator values the same base-year quantities at base-year prices (\(P_{o}\)). Using base-year quantities as weights is the defining feature of the Laspeyres index:\( P_{L} = \frac{\sum P_{t}Q_{o}}{\sum P_{o}Q_{o}} \times 100 \)The Paasche index uses current-year quantities, \(\frac{\sum P_{t}Q_{t}}{\sum P_{o}Q_{t}}\). The GDP deflator is a Paasche-type index, and the CPI is a particular application rather than a formula.

Q: Which one of the following is not the test of adequacy of index number ?

The one that is not a test of adequacy is the Bowley Test.Irving Fisher proposed tests that a good index number should satisfy: the time reversal test, the factor reversal test and the circular test (along with the unit test). Fisher's ideal index satisfies the first two.Bowley is known for an index number formula (the arithmetic mean of the Laspeyres and Paasche indices) and for his measure of skewness, not for a test of adequacy.

Q: Which of the following is true in case of geometric mean ?(i) It is used to find the average percent increase in sales, production etc.(ii) Used in construction of index numbers(iii) Most suitable when large weights have to be given to small item and small weights to large items

(i), (ii) and (iii) are true.(i) The geometric mean is the right average for rates of growth, such as average percentage increase in sales or production over several years.(ii) It is used in constructing index numbers, notably Fisher's ideal index, which is the geometric mean of the Laspeyres and Paasche indices.(iii) It gives more weight to small items and less to large ones than the arithmetic mean does, so textbooks describe it as suitable where that weighting is wanted.Since all three statements hold, the options keeping only some of them, or calling them all false, are wrong.

Q: The empirical relationship between mean, median and mode can be expressed as follows :

The relationship is Mode = 3 Median - 2 Mean.For moderately skewed distributions, Karl Pearson observed that the median lies about one-third of the way from the mean to the mode, which gives the empirical formula:\[ \text{Mode} = 3\,\text{Median} - 2\,\text{Mean} \]A check: in a symmetric distribution all three are equal, and \(3m - 2m = m\). The other combinations fail that check (for example \(2m - 3m = -m\)).

Q: If three dice are thrown once, then there will be how many exhaustive number of cases ?

There are 216 exhaustive cases.Each die can show any of 6 faces, independently of the others, so by the multiplication principle:\[ 6 \times 6 \times 6 = 6^{3} = 216 \]18 adds the faces instead of multiplying (\(6 + 6 + 6\)), 108 is half of 216, and 72 corresponds to no standard count. Exhaustive cases list every possible outcome, so all \(6^{3}\) combinations count.

Q: Which of the following is not used to test the stationarity ?

The one not used to test stationarity is the Engle-Granger Test.The Engle-Granger procedure is a test for cointegration: whether a linear combination of non-stationary series is stationary, that is, whether they share a long-run relationship.The Dickey-Fuller and Phillips-Perron tests are unit root tests, the standard tests of stationarity, and the autocorrelation function (correlogram) is also used to judge stationarity: a slowly decaying ACF suggests a non-stationary series.

Q: For which of the following the 2SLS method is most appropriate method of estimation ?

2SLS is most appropriate for an Over Identified Equation.In simultaneous-equation models, an over-identified equation has more excluded exogenous variables than needed, so indirect least squares gives more than one estimate of the structural parameters. Two-stage least squares combines all the instruments into one: it first regresses the endogenous regressors on all exogenous variables, then uses the fitted values in the second stage.An exactly identified equation can be estimated by indirect least squares (2SLS gives the same result), and an under-identified (unidentified) equation cannot be estimated by any method.

Q: If Y is dependent and X2 and X3 are independent variables, then r12.3 measures the following :

\(r_{12.3}\) measures the partial correlation between Y and X2.With Y labelled 1, X2 labelled 2 and X3 labelled 3, the subscripts before the dot name the two variables being correlated, and those after the dot are held constant. So \(r_{12.3}\) is the correlation between Y and X2 after removing the linear influence of X3 from both.The simple correlation between Y and X2 is \(r_{12}\), without the dot. The partial correlation between Y and X3 would be \(r_{13.2}\), and X2 with X3 would be \(r_{23}\).

Q: If the value of r (coefficient of correlation) is less than the probable error, it means :

It means the value of r is not at all significant.The probable error of the correlation coefficient is \(PE = 0.6745 \times \frac{1 - r^{2}}{\sqrt{n}}\). The usual rules of interpretation are:If \(r\) is less than PE, the correlation is not significant at all.If \(r\) is more than six times PE, it is definitely significant.In between, the result is not conclusive.So a value of \(r\) below its probable error cannot be called significant, moderately or highly.

41.

\(\frac{\sum_{i}(P_{i},t)\times(Q_{i},o)}{\sum_{i}(P_{i},o)\times(Q_{i},o)}\) \(\times 100\) is the formula of ........................ .

A.Paasche Price Index
B.Laspeyre's Price Index
C.Consumer Price Index
D.GDP Deflator

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: B

It is the formula of Laspeyre's Price Index.

The numerator values base-year quantities (\(Q_{o}\)) at current prices (\(P_{t}\)), and the denominator values the same base-year quantities at base-year prices (\(P_{o}\)). Using base-year quantities as weights is the defining feature of the Laspeyres index:

\( P_{L} = \frac{\sum P_{t}Q_{o}}{\sum P_{o}Q_{o}} \times 100 \)

The Paasche index uses current-year quantities, \(\frac{\sum P_{t}Q_{t}}{\sum P_{o}Q_{t}}\). The GDP deflator is a Paasche-type index, and the CPI is a particular application rather than a formula.

Discuss in Forum

42.

Which one of the following is not the test of adequacy of index number ?

A.Time Reversal Test
B.Factor Reversal Test
C.Circular Test
D.Bowley Test

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: D

The one that is not a test of adequacy is the Bowley Test.

Irving Fisher proposed tests that a good index number should satisfy: the time reversal test, the factor reversal test and the circular test (along with the unit test). Fisher's ideal index satisfies the first two.

Bowley is known for an index number formula (the arithmetic mean of the Laspeyres and Paasche indices) and for his measure of skewness, not for a test of adequacy.

Discuss in Forum

43.

Which of the following is true in case of geometric mean ?

(i) It is used to find the average percent increase in sales, production etc.
(ii) Used in construction of index numbers
(iii) Most suitable when large weights have to be given to small item and small weights to large items

A.Only (i) is true
B.Only (i) and (iii) are true
C.(i), (ii) and (iii) are false
D.(i), (ii) and (iii) are true

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: D

(i), (ii) and (iii) are true.

(i) The geometric mean is the right average for rates of growth, such as average percentage increase in sales or production over several years.
(ii) It is used in constructing index numbers, notably Fisher's ideal index, which is the geometric mean of the Laspeyres and Paasche indices.
(iii) It gives more weight to small items and less to large ones than the arithmetic mean does, so textbooks describe it as suitable where that weighting is wanted.

Since all three statements hold, the options keeping only some of them, or calling them all false, are wrong.

Discuss in Forum

44.

The empirical relationship between mean, median and mode can be expressed as follows :

A.Mode = 3 Median - 2 Mean
B.Mode = 2 Median - 3 Mean
C.Mode = 3 Median - 3 Mean
D.Mode = 6 Median - 5 Mean

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: A

The relationship is Mode = 3 Median - 2 Mean.

For moderately skewed distributions, Karl Pearson observed that the median lies about one-third of the way from the mean to the mode, which gives the empirical formula:

\[ \text{Mode} = 3\,\text{Median} - 2\,\text{Mean} \]

A check: in a symmetric distribution all three are equal, and \(3m - 2m = m\). The other combinations fail that check (for example \(2m - 3m = -m\)).

Discuss in Forum

45.

If three dice are thrown once, then there will be how many exhaustive number of cases ?

A.18
B.108
C.216
D.72

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: C

There are 216 exhaustive cases.

Each die can show any of 6 faces, independently of the others, so by the multiplication principle:

\[ 6 \times 6 \times 6 = 6^{3} = 216 \]

18 adds the faces instead of multiplying (\(6 + 6 + 6\)), 108 is half of 216, and 72 corresponds to no standard count. Exhaustive cases list every possible outcome, so all \(6^{3}\) combinations count.

Discuss in Forum

46.

Which of the following is not a non-parametric sampling ?

A.Judgmental sampling
B.Convenience sampling
C.Stratified sampling
D.Quota sampling

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: C

The one that is not non-parametric (non-probability) sampling is Stratified sampling.

Stratified sampling is a probability method: the population is divided into strata and a random sample is drawn from each, so every unit has a known chance of selection.

Judgmental (purposive), convenience and quota sampling choose units by the researcher's judgment or by ease of access, without random selection, so they are non-probability methods. Quota sampling looks like stratified sampling but fills the quotas non-randomly.

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47.

Which of the following is not used to test the stationarity ?

A.Dickey-Fuller Test
B.Autocorrelation Function Test
C.Engle-Granger Test
D.Phillips-Perron Test

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: C

The one not used to test stationarity is the Engle-Granger Test.

The Engle-Granger procedure is a test for cointegration: whether a linear combination of non-stationary series is stationary, that is, whether they share a long-run relationship.

The Dickey-Fuller and Phillips-Perron tests are unit root tests, the standard tests of stationarity, and the autocorrelation function (correlogram) is also used to judge stationarity: a slowly decaying ACF suggests a non-stationary series.

Discuss in Forum

48.

For which of the following the 2SLS method is most appropriate method of estimation ?

A.Under Identified Equation
B.Exactly Identified Equation
C.Exactly Unidentified Equation
D.Over Identified Equation

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: D

2SLS is most appropriate for an Over Identified Equation.

In simultaneous-equation models, an over-identified equation has more excluded exogenous variables than needed, so indirect least squares gives more than one estimate of the structural parameters. Two-stage least squares combines all the instruments into one: it first regresses the endogenous regressors on all exogenous variables, then uses the fitted values in the second stage.

An exactly identified equation can be estimated by indirect least squares (2SLS gives the same result), and an under-identified (unidentified) equation cannot be estimated by any method.

Discuss in Forum

49.

If Y is dependent and X₂ and X₃ are independent variables, then r_12.3 measures the following :

A.Correlation between Y and X\(_{2}\)
B.Correlation between X\(_{2}\) and X\(_{3}\)
C.Partial Correlation between Y and X\(_{3}\)
D.Partial Correlation between Y and X\(_{2}\)

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: D

\(r_{12.3}\) measures the partial correlation between Y and X₂.

With Y labelled 1, X₂ labelled 2 and X₃ labelled 3, the subscripts before the dot name the two variables being correlated, and those after the dot are held constant. So \(r_{12.3}\) is the correlation between Y and X₂ after removing the linear influence of X₃ from both.

The simple correlation between Y and X₂ is \(r_{12}\), without the dot. The partial correlation between Y and X₃ would be \(r_{13.2}\), and X₂ with X₃ would be \(r_{23}\).

Discuss in Forum

50.

If the value of r (coefficient of correlation) is less than the probable error, it means :

A.The value of r is not at all significant
B.The value of r is significant
C.The value of r is moderately significant
D.The value of r is very highly significant

Exam Year(s)

MH SET - 2025, SET Exam - 2025,

Answer: A

It means the value of r is not at all significant.

The probable error of the correlation coefficient is \(PE = 0.6745 \times \frac{1 - r^{2}}{\sqrt{n}}\). The usual rules of interpretation are:

If \(r\) is less than PE, the correlation is not significant at all.
If \(r\) is more than six times PE, it is definitely significant.
In between, the result is not conclusive.

So a value of \(r\) below its probable error cannot be called significant, moderately or highly.

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