Question 675·Hard·Math

0 / 2000

Two data sets of 24 integers each are summarized in the histograms shown.

For each histogram, each interval represents the frequency of integers greater than or equal to the left endpoint of the interval but less than the right endpoint. For example, in the interval 10–20, the integers are greater than or equal to $10$ but less than $20$ .

Let $\bar{x}$ be the mean of data set X and let $\bar{y}$ be the mean of data set Y. Which choice is the smallest possible value of $\bar{x}-\bar{y}$ ?

A histogram gives counts in intervals, not the exact values, so a mean can vary even when the bars stay fixed. For the smallest possible difference of two means, push each set in the direction that makes the subtraction smaller. Then multiply each chosen value by its bar height to find the totals. Watch out: a bin's right endpoint is excluded.

Hints

Hint 1
A mean is the total divided by the number of values. Both sets have $24$ values, so lowering X's total and raising Y's total makes the difference of their means smaller. Which end of each bar should you use?
Hint 2
The right endpoint is excluded from each interval. Since the values are integers, the greatest value in the interval from $20$ to less than $30$ is $29$ , not $30$ . Apply that to Y's other bars.
Hint 3
A bar's frequency tells you how many values it contains. Multiply each chosen value by its bar height and add within each set. Then divide the difference of the totals by $24$ .

Step-by-step

Push the two means in opposite directions
Step 1
Choose the direction for each set
A mean is the total divided by the number of values. Both sets have $24$ values, so to make $\bar{x}-\bar{y}$ smallest, make X's total as small as possible and Y's total as large as possible. Each value can move within its own bar without changing the histogram.
Step 2
Pick values the bars allow
Use X's left endpoints: $10,20,30,40$ . Use Y's largest included integers: $29,39,49,59$ . For example, the interval from $20$ to less than $30$ includes $29$ but not $30$ . Putting $30$ there would cross into the next bar.
Step 3
Find X's smallest total
The X bars have frequencies $2,5,9,8$ , so count each chosen value that many times. Type $2(10)+5(20)+9(30)+8(40)$ into Desmos. It shows $710$ , the smallest possible total for X.
Step 4
Find Y's largest total
The Y bars have frequencies $4,6,10,4$ . Add $4(29)+6(39)+10(49)+4(59)$ in Desmos. It shows $1076$ , the largest possible total for Y.
Step 5
Subtract the means
Each mean is its total divided by $24$ , so type $\frac{710-1076}{24}$ . Desmos shows $-15.25$ ; its fraction button gives $-\frac{61}{4}$ . Both totals have been pushed as far as the bars allow, so the smallest possible value of $\bar{x}-\bar{y}$ is $-\frac{61}{4}$ . Choice A.

Question 675·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this