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Question
If each observation of the data is increased by 5, then their mean ______.
Options
remains the same
becomes 5 times the original mean
is decreased by 5
is increased by 5
Solution
If each observation of the data is increased by 5, then their mean is increased by 5.
Explanation:
Let x1, x2, ..., xn be the n observations.
Then, old mean `barx_("old") = (sum_(i = 1)^n x_i)/n` ...(i)
Now, adding 5 in each observation, the new mean becomes
`barx_("new") = ((x_1 + 5) + (x_2 + 5) + ... + (x_n + 5))/n`
⇒ `barx_("new") = ((x_1 + x_2 + ... + x_n) + 5n)/n`
⇒ `barx_("new") = (sum_(i = 1)^n x_i)/n + 5 = barx_("old") + 5` ...[From equation (i)]
⇒ `barx_("new") = barx_("old") + 5`
Hence, the new mean is increased by 5.
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