Question 495·Hard·Math

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The dot plot above represents a data set of the number of cups of soup sold at a café on 30 different days.

The café removes the record for the day with the smallest number of cups sold and replaces it with a new day on which 7 cups were sold, creating a new data set of 30 days.

I. The median number of cups sold

II. The mean number of cups sold

III. The range of the numbers of cups sold

Which choice states the quantities above that are greater for the new data set than for the original data set?

A dot plot shows each value and how many times it occurs. For a replacement, check three things separately: the ordered middle for the median, the total and count for the mean, and the endpoints for the range. A value that crosses the middle can change the median even when the number of observations stays the same. A higher maximum alone doesn't guarantee a wider range.

Hints

Hint 1
For an even number of days, the median comes from the two middle positions. With $30$ days, those are positions $15$ and $16$ . Which stacks contain those positions before and after the replacement?
Hint 2
The mean is the total cups sold divided by the number of days. A replacement keeps the count at $30$ , so compare the cups sold on the removed day and the new day.
Hint 3
The range is the largest value minus the smallest. Here, both endpoints can change: the smallest day is removed, and the new day may become the largest. Check both endpoints before comparing ranges.

Step-by-step

Compare the middle, total, and endpoints
Step 1
Find the original middle
No Desmos needed. The dot plot shows the ordered positions, and the replacement lets you compare totals without adding all $30$ values. A median is the middle of the ordered data. With $30$ days, use the 15th and 16th values. The one $2$ is first, the five $3$ s fill positions $2$ through $6$ , and the ten $4$ s fill positions $7$ through $16$ . Both middle values are $4$ , so the original median is $4$ .
Step 2
Recount the middle after replacement
Remove the lone $2$ and put the new $7$ at the far right. The five $3$ s now fill positions $1$ through $5$ , the ten $4$ s fill positions $6$ through $15$ , and the $5$ s begin at position $16$ . The middle pair changes from $4,4$ to $4,5$ . Its average is now above $4$ , so I increases. Recount the middle when a replacement crosses it.
Step 3
Compare the totals for the mean
The mean is the total cups sold divided by the number of days. Find how much replacing $2$ with $7$ raises the total:
$7-2=5$
There are still $30$ days, so dividing a larger total by the same count gives a larger mean. II increases.
Step 4
Check both ends for the range
The range is the largest value minus the smallest. Originally, the endpoints are $2$ and $6$ . Subtract them:
$6-2=4$
After the replacement, the endpoints are $3$ and $7$ . Subtract them:
$7-3=4$
The range stays the same, so III does not increase. The median and mean are greater for the new data set. Choice A.

Question 495·Hard·Math

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