Question

The least and greatest numbers in a list of 7 real numbers are 4 and 16, respectively. The median of the list is 8, and the number 11 occurs most often in the list. Which of the following could be the average (arithmetic mean) of the numbers in the list?

I. 9
II. 10
III. 10.5

Answer 1

If we take the information and try to recreate the list in order as much as possible maybe we can find the answer. So we know the smallest and biggest numbers are 4 and 16, so it looks like this:
\[ 4,X_2,X_3,X_4,X_5,X_6,16 \]
The median should be 8, and the median is the middlemost number of a list, so the middle number must be 8:
\[ 4,X_2,X_3,8,X_5,X_6,16 \]
Now the text says 11 should occur most often, since every other number occurs at least once we need to put in two 11's:
\[ 4,X_2,X_3,8,11,11,16 \]
So the equation for the arithmetic mean is now
\[ avg = \frac{1}{7}(4+X_2+X_3+8+11+11+16) = \frac{50+X_2+X_3}{7} \]
Now X2 and X3 must be bigger than 4 and smaller than 8. If both are very close to 4 we get the average:
\[ \frac{50+4+4}{7} \approx 8.3 \]
And if X2 and X3 are close to 8 we get
\[ \frac{50+8+8}{7} \approx 9.4 \]
So the mean is approximately somewhere between 8.3 and 9.4. The answer must be I.

Answer 2

gee thanks!