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We are told that the Taylors Have recorded their weekly grocery expenses for the past 12 weeks and determined that the mean weekly expense was 60.26. Later , Mrs.Taylor discovered that 1 weeks expense of $74 was incorrectly recorded as $47.
We can find the correct mean weekly expenses by calculating the mean of difference between 74 and 47, then adding it to the given mean weekly expenses.
[tex]\frac{74-47}{12}[/tex]
[tex]\frac{27}{12}=2.25[/tex]
Now let us add 2.25 in 60.26 to find the correct mean weekly expenses.
[tex]\text{Correct mean weekly expenses}=60.26+2.25[/tex]
[tex]\text{Correct mean weekly expenses}=62.51[/tex]
Therefore, the correct mean weekly expenses for the Taylors will be 62.51.
The correct mean weekly grocery expense for Taylor is $62.51
How to find mean of a discrete data?
Suppose the data values be [tex]x_1, x_2, \cdots, x_n[/tex].
Then, their mean is evaluated as:
[tex]\overline{x} = \dfrac{x_1 +x_2 + \cdots + x_n}{n}[/tex]
where n = number of observations.
For the given condition, Taylor recorded the weekly expenses for 12 weeks. Therefore, we get n = 12
Let the real weekly expenses for Taylor be [tex]x_1, x_2, \cdots, x_{12}[/tex]
Let [tex]i^{th}[/tex] observation is misinterpreted as T
Then, the mean of that misinterpreted set of data would be
[tex]\overline{x}' = \dfrac{x_1 + x_2 + \cdots + x_{12} + (-x_i + T)}{12}\\\overline{x}' = \overline{x} + \dfrac{T - x_i}{12}\\\overline{x} = \overline{x}' - \dfrac{T - x_i}{12}[/tex]
Since we're given that:
- [tex]T = 47[/tex]
- [tex]x_i = 74[/tex]
- [tex]\overline{x}' = 60.26[/tex]
Therefore, we get the mean of original weekly expenses as:
[tex]\overline{x} = 60.26 - \dfrac{47 - 74}{12} = 60.26 + 2.25\\\\\overline{x} = 62.51[/tex]
Therefore, the correct mean weekly grocery expense for Taylor is $62.51
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