10 Question MCQ (including Assertion)

Question 1 of 10

Calculate the standard deviation of the first $10$ odd natural numbers ($1, 3, 5, 7, 9, 11, 13, 15, 17, 19$).

A $5.74$ B $8.25$ C $3.00$ D $9.95$

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Question 2 of 10

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.

A Both (A) and (R) are true and (R) is the correct explanation of (A). B Both (A) and (R) are true but (R) is NOT the correct explanation of (A). C Assertion (A) is true but Reason (R) is false. D Assertion (A) is false but Reason (R) is true.

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Question 3 of 10

Calculate the mean deviation about mean for the continuous distribution: $$\begin{array}{|c|c|c|c|c|} \hline \text{Class} & 0-10 & 10-20 & 20-30 & 30-40 \\ \hline \text{Frequency } f_i & 2 & 3 & 3 & 2 \\ \hline \end{array}$$

A $9.0$ B $7.5$ C $8.0$ D $10.0$

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Question 4 of 10

Investment Fund P has average return $\bar{x}_P = 12\%$ with $\sigma_P = 2\%$. Fund Q has average return $\bar{x}_Q = 15\%$ with $\sigma_Q = 5\%$. Calculate the Coefficients of Variation $C.V._P$ and $C.V._Q$.

A $C.V._P = 16.67\%, C.V._Q = 33.33\%$ B $C.V._P = 20.00\%, C.V._Q = 30.00\%$ C $C.V._P = 12.00\%, C.V._Q = 15.00\%$ D $C.V._P = 25.00\%, C.V._Q = 33.33\%$

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Question 5 of 10

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.

A Both (A) and (R) are true and (R) is the correct explanation of (A). B Both (A) and (R) are true but (R) is NOT the correct explanation of (A). C Assertion (A) is true but Reason (R) is false. D Assertion (A) is false but Reason (R) is true.

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Question 6 of 10

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.

A Both (A) and (R) are true and (R) is the correct explanation of (A). B Both (A) and (R) are true but (R) is NOT the correct explanation of (A). C Assertion (A) is true but Reason (R) is false. D Assertion (A) is false but Reason (R) is true.

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Question 7 of 10

Temperature readings $x_i$ have a standard deviation $\sigma_x = 4.8\,^\circ\text{C}$. The values are transformed using $y_i = \frac{x_i - 10}{2}$. Calculate the standard deviation of $y_i$ ($\sigma_y$).

A $2.4\,^\circ\text{C}$ B $-2.6\,^\circ\text{C}$ C $4.8\,^\circ\text{C}$ D $1.2\,^\circ\text{C}$

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Question 8 of 10

The standard deviation of $x_1, x_2, \dots, x_n$ is $\sigma_x = 6$. Calculate the standard deviation of $y_i = -2x_i + 5$.

A $-12$ B $12$ C $17$ D $36$

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Question 9 of 10

Calculate the mean deviation about the median for the dataset: $3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21$.

A $5.27$ B $4.85$ C $5.80$ D $6.12$

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Question 10 of 10

Assertion (A): The variance of the first $n$ natural numbers is $\frac{n^2 - 1}{12}$.
Reason (R): The sum of the first $n$ natural numbers is given by $\sum_{i=1}^n i = \frac{n(n+1)}{2}$.

A Both (A) and (R) are true and (R) is the correct explanation of (A). B Both (A) and (R) are true but (R) is NOT the correct explanation of (A). C Assertion (A) is true but Reason (R) is false. D Assertion (A) is false but Reason (R) is true.

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10 Question MCQ (including Assertion)