Assertion Reasoning Quiz

Assertion (A): If the mean of \(100\) observations was initially calculated as \(40\), but one value recorded as \(50\) was corrected to \(40\), the correct mean becomes \(39.9\).
Reason (R): The correct sum of observations is obtained by subtracting the incorrect value (\(50\)) and adding the correct value (\(40\)) to the initial total (\(4000\)).

Assertion (A): Two groups of 50 students each have mean scores \(\bar{x}_1 = 60\) and \(\bar{x}_2 = 60\) with variances \(\sigma_1^2 = 4\) and \(\sigma_2^2 = 9\). The combined variance of all 100 students is \(6.5\).
Reason (R): When combining two datasets of equal size \(n_1 = n_2\), the combined arithmetic mean is equal to the average of their individual means, \(\bar{x} = \frac{\bar{x}_1 + \bar{x}_2}{2}\).

Assertion (A): The variance of the first \(8\) even natural numbers (\(2, 4, 6, 8, 10, 12, 14, 16\)) is equal to \(21\).
Reason (R): The variance of the first \(n\) even natural numbers is given by the formula \(\sigma^2 = \frac{n^2 - 1}{12}\).

Assertion (A): A standardized variable \(z_i = \frac{x_i - \bar{x}}{\sigma_x}\) created from dataset \(X\) has mean \(\bar{z} = 0\) and variance \(\sigma_z^2 = 1\).
Reason (R): Standardizing a variable converts raw observations into standard deviation units measured relative to the mean.

Assertion (A): The mean deviation about median for the dataset \(3, 3, 4, 5, 7, 9, 10, 12, 18, 19, 21\) is equal to \(5.27\).
Reason (R): For an odd number of observations \(n\), the median is given by the \((\frac{n+1}{2})^{\text{th}}\) observation when ordered.

Assertion (A): If each observation in a dataset \(x_1, x_2, \dots, x_n\) is increased by a constant \(k = 5\), the variance of the resulting new dataset remains equal to the variance of the original dataset.
Reason (R): Adding a constant \(k\) to each observation shifts the mean by \(k\), so the individual deviations \((y_i - \bar{y}) = (x_i + k) - (\bar{x} + k) = x_i - \bar{x}\) remain completely unchanged.