Question 15·Hard·One-Variable Data Distributions; Measures of Center and Spread
The dot plot above represents a data set of the number of deliveries a courier made on each of 12 days.
Later, the courier learns that the tracking app recorded 1 more delivery than the courier actually made on each day.
To correct the data set, 1 is subtracted from each of the 12 values. Then, one additional day is included in the data set, during which the courier made deliveries.
What is the least integer value of such that the mean of the new data set is greater than the mean of the original data set?
For mean-change questions, work with totals. Compute the original sum and mean. A “subtract 1 from each of 12 values” change decreases the sum by 12 without needing to rewrite the whole list. After adding a new value, write the new mean as an expression and solve an inequality to find the least integer that makes the mean larger.
Hints
Use a total (sum), not individual recomputation
Find the original total number of deliveries from the dot plot, then divide by 12 to get the original mean.
Track how the sum changes under the correction
Subtracting 1 from each of 12 values changes the total sum by a fixed amount.
Write the new mean as an expression
After the correction, add the new value and remember the new data set has 13 values.
Use an inequality and then pick the smallest integer
Solve “new mean old mean,” then take the least integer that satisfies it.
Desmos Guide
Enter the original data in a list
In Desmos, enter
L=[2,4,4,4,6,6,6,6,8,8,8,10]
Compute the original mean
Type mean(L) to confirm the original mean is 6.
Represent the corrected sum
The corrected 12-day sum is sum(L)-12. (Subtracting 1 from 12 values lowers the sum by 12.)
Test candidate values
For each answer choice , compute (sum(L)-12+k)/13 and check whether it is greater than 6. Choose the smallest that makes it greater.
Step-by-step Explanation
Compute the original sum and mean from the dot plot
From the dot plot, the 12 values are:
- One
- Three s
- Four s
- Three s
- One
So the original sum is
With 12 values, the original mean is
Write the new mean in terms of
Subtracting 1 from each of the 12 original values decreases the total sum by 12, so the corrected 12-day sum is .
Then one additional day with deliveries is added, so the new sum is and the new number of values is .
Thus the new mean is
Set up and solve the inequality
We want the new mean to be greater than the original mean 6:
Multiply both sides by 13:
So
Choose the least integer that works
Since must be an integer and , the least possible value is .
Therefore, the correct answer is 19.