An investigator has missed a value while collecting data in an experiment. Denoting the missing value by x, the observations are: 10, 4, 11, 6, 17, 15, 9, 8, x. What should be the value of x, if he wants mean = median = mode for this data set?
Step-by-step Solution:
Let the 9 observations be: 4, 6, 8, 9, 10, 11, 15, 17, and \( x \). For the mode to exist and equal the mean and median, \( x \) must duplicate one of the existing values. Thus, \( x \) itself must be the mode. The sum of the 8 known observations is: \[ 4 + 6 + 8 + 9 + 10 + 11 + 15 + 17 = 80 \] The mean of all 9 observations is: \[ \text{Mean} = \frac{80 + x}{9} \] Since Mean = Mode = \( x \): \[ \frac{80 + x}{9} = x \] \[ 80 + x = 9x \] \[ 8x = 80 \implies x = 10 \] Let's verify if the median is also 10 when \( x = 10 \). The sorted sequence is: 4, 6, 8, 9, 10, 10, 11, 15, 17. Since there are 9 total terms, the median is the 5th term, which is indeed 10. All constraints are satisfied when \( x = 10 \).