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प्रश्न
The Mean of the following arrayed data, 5, 8, (3x − 1), (4x + 1), (3x + 7), is equal to its Median. Find the value of x.
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उत्तर
Given:
The arrayed data set is 5, 8, (3x − 1), (4x + 1), and (3x + 7). The mean of this data set is equal to its median.
Formula:
Mean = Sum of observations / Number of observations
Median for odd number of observations = Value of the middle observation
Solution:
Number of observations (n) = 5 [Counting the given terms in the data set]
Median = 3x − 1 [Identifying the third and middle term of the arrayed data]
Sum of observations = 5 + 8 + (3x − 1) + (4x + 1) + (3x + 7) [Adding all the given terms to find the total sum]
Sum of observations = 10x + 20 [Combining like terms to simplify the sum]
\[\therefore \text{Mean} = \frac{10x + 20}{5} \quad [\text{Dividing the total sum by the number of observations}]\]
\[\therefore \text{Mean} = 2x + 4 \quad [\text{Simplifying the fraction}]\]
2x + 4 = 3x − 1 [Equating the mean to the median as stated in the problem]
4 + 1 = 3x − 2x [Transposing variable and constant terms to opposite sides]
x = 5 [Simplifying the expression to isolate x]
∴ x = 5
