If the mean of a set of observations is , then the mean of the observations where is:
The mean of the observations , can be calculated as follows:
First, let's find the number of terms in the new set of observations. For each i, we have one term, so the number of terms is .
Next, we calculate the sum of the new observations:
To simplify this expression, we can rewrite it as:
Now, let's express the indices in terms of i:
For ,the index is 3 .
For , the index is 6 .
For , the index is 11 .
And so on.
We can observe that the indices follow a pattern given by the expression .
Therefore, the sum of the new observations can be written as:
Simplifying further, we have:
Now, let's consider the mean of the new observations:
Therefore, the correct answer is .
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