Multiple choice

Let $\overline x$, M and $\sigma^2$ be respectively the mean mode and variance of n observations $x_1, x_2, ....., x_n$ and $d_i=-x_1-a, i=1, 2, ....., n,$ where a is any number. Statement I : Variance of $d_1, d_1, ....., d_n$ is $\sigma^2$ Statement II : Mean and mode of $d_1, d_2, ....., d_n$ are $-\overline x-a$ and $-M-a$, respectively

  1. Statement I and Statement II are both false

  2. Statement I and Statement II are both true

  3. Statement I is true and Statement II is false

  4. Statement I is false and Statement II is true
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B Correct answer
AI explanation

The linear transformation for the new data is given as d_i equals negative x_i minus a, meaning the observations are reflected across the y-axis and then shifted by negative a. Variance is invariant under changes in location and is unaffected by reflections, so the variance of the d values remains identical to the original variance, making Statement I true. Furthermore, the mean and mode of the transformed data undergo the exact same linear transformation, becoming negative x-bar minus a and negative M minus a respectively, confirming that Statement II is also true. Therefore, both statements are true.

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