Statistics and Econometrics Questions and Answers

Answer: C

The strength of correlation is measured by the coefficient of determination, the square of the correlation coefficient, and not by the coefficient itself. For X and Y the value is 0.3 squared, which is 0.09, and for P and Q it is 0.6 squared, which is 0.36. The ratio is 0.36 divided by 0.09, which is 4, so the strength of correlation between P and Q is four times that between X and Y, option C. The answer two times comes from comparing 0.6 with 0.3 directly, which treats the coefficient as a linear measure and is wrong. Three and five times have no basis.

Answer: C

The matching is: acceptance of the null hypothesis when it is false with a Type II error (2), a distribution whose mean equals its degrees of freedom with the chi-square distribution (4), a homogeneous population with simple random sampling (1), and mean greater than mode with a positively skewed distribution (3), giving 2 4 1 3, option C. Simple random sampling is the suitable design when the population is homogeneous. Option A swaps the last two items. Options B and D mispair the Type II error and the chi-square.

Answer: B

The test statistic for the null hypothesis that the slope is 2.1 is the difference between the estimate and the hypothesised value divided by the standard error: \[ |t| = \frac{|2 - 2.1|}{0.46} \] \[ |t| = \frac{0.1}{0.46} \approx 0.217. \] So option B, 0.217, is correct. The value 4.34 is close to 2 divided by 0.46, which is the t value for the null hypothesis that the slope is zero, not 2.1. The estimate of 2 is very near the hypothesised 2.1, so the t value must be small and the null is not rejected, which also rules out 4.609 and 4.34.

Answer: C

The model has k explanatory variables and an intercept, so \(k + 1\) parameters are estimated from n observations. The degrees of freedom of the t statistic for testing a single coefficient are the number of observations less the number of estimated parameters, \[ df = n - (k + 1) = n - k - 1. \] So option C is right. The option \(n - k\) would apply if there were no intercept or if k counted the intercept. The options \(n - k + 1\) and \(n + k - 1\) do not correspond to any count of parameters.

Answer: A

The correct match is a-2, b-3, c-4, d-1. A Type II error is the probability of not rejecting the null hypothesis when it is not true, a mean greater than the mode shows a positively skewed distribution, a homogeneous population is the case for which simple random sampling is suited, and the time reversal test requires that the price index for the current year on the base year multiplied by the index for the base on the current year is equal to one. This is the first option. The other options give a Type II error or a skewed distribution the wrong description.

Answer: C

The coefficient of correlation is 0.6, the third option. The slope of the regression of Y on X is \(b_{yx} = 0.4\) and the slope of the regression of X on Y is \(b_{xy} = 0.9\), and the coefficient of correlation is the geometric mean of the two slopes:

\[ r = \sqrt{b_{yx} \, b_{xy}} \]

\[ r = \sqrt{0.4 \times 0.9} = \sqrt{0.36} \]

\[ r = 0.6 \]

The sign is positive because both slopes are positive. The values 0.4 and 0.8 are not the root of the product, and 0.5 would need a product of 0.25.

Answer: D

Of the indices listed, the Marshall-Edgeworth index number satisfies the time reversal test but not the factor reversal test, the fourth option. Fisher's ideal index satisfies both tests, while the Laspeyres and Paasche indices satisfy neither, since each of them uses the weights of only one of the periods. The Marshall-Edgeworth index uses the average of the quantities of the two periods as weights, which makes it reversible in time, but the product of its price and quantity indices does not equal the value ratio.

Answer: A

The degrees of freedom of a chi-square test of association in a contingency table with r rows and c columns are:

\[ (r - 1)(c - 1) \]

\[ (3 - 1)(5 - 1) = 8 \]

The key therefore gives 8, the first option. The tabulated value of chi-square at 8 degrees of freedom and 5 percent is about 15.5, and the computed value of 8.0 is below it, so the hypothesis of no association is not rejected. The other options of 6, 4 and 3 do not match the number of rows and columns of the table.

Answer: C

Quota sampling is a non-probability sampling method, the third option. In it the interviewer fills fixed quotas of people with given characteristics and chooses the individuals, so a person's chance of being selected is not known. Systematic, stratified random and simple random sampling are all probability methods, as each unit has a known chance of selection set by a random procedure.

Answer: A

The coefficient of determination is the square of the correlation coefficient, so it is 0.6 x 0.6 = 0.36. Thus 36 per cent of the variation in Y is explained by X.

60 per cent would be the figure if r were used and not squared, which is a common error, and 6 per cent has no basis in the calculation. So the first option is right. The remaining 64 per cent is explained by other factors and the disturbance term, so the model does not explain the whole variation.