Question

Chucky grabbed 12 items in the grocery store that each had a different price and had a mean cost of about $7.41. One of the items was an entire wheel of cheese that cost $39.99. Chucky then decided to put the wheel of cheese back and only buy the other 11 items. How will removing the wheel of cheese affect the mean and median?

A. Both the mean and median will decrease, but the median will decrease by more than the mean.
B. Both the mean and median will decrease, but the mean will decrease by more than the median.
C. Both the mean and median will increase, but the median will increase by more than the mean.
D. Both the mean and median will increase, but the mean will increase by more than the median.

Answer 1

B

Explanation:

Create a function to help determine his total cost.

12 x $7.41 = $88.92

After putting the cheese back he has 11 items and his total cost went down.

$88.92-$39.99 = $48.93

To find the average cost per item divide by 11.

$48.93/11 = $4.45 per item

The mean (average) has gone down.

The median would be the middle number, so you would need to line the cost of the items up from least to greatest. If you have an even number of items, you take the mean of the middle 2 numbers. The block of cheese would have have given Chucky 12 items so the 2 middle numbers would have been averaged.

A hypothetical 12 and 11 items:

$1 $2 $2 $3 $4 $4 $5 $6 $7 $7 $8 $39.99

Mean $7.42 Median $4.5

$1 $2 $2 $3 $4 $4 $5 $6 $7 $7 $8

Mean $4.45 Median $4

Taking the cheese away would move the middle down 1 number so it would also go down, but not as much as the mean of cost per item.

B. Both the mean and median will decrease, but the mean will decrease by more than the median.

Answer 2

The mean will be (significantly) less.

Depending on the other items, the median is either affected only slightly or not at all.

Of all the answer choices, B is the correct one.

Explanation:

So, we know that initially, Chucky had a total of 12 items. Each of the items has a different price, but the average cost of them is $7.41.

One item of the twelve was a wheel of cheese that costs $39.99.

And we want to know how the average cost and median cost will be affected if we remove this wheel of cheese.

The important thing to note here is that the average cost is $7.41. The wheel of cheese, however, costs $39.99. Thus, it is an outlier. It is significantly more expensive than the other items.

Initially, to find the mean, we had to add the 12 items together and then divide by 12. However, now that we removed the expensive outlier, the average cost will now be less.

Algebraically, we can see this. Let t be the total cost of the items excluding the cheese. Thus, the mean initially is:

`7.41=(t+39.99)/(12)`

Find t. Multiply both sides by 12:

`88.92=t+39.99\\t=48.93`

Therefore, all the other items excluding the cheese totals $48.93.

If we remove the cheese, the total is now $48.93 and with only 11 items. Thus, the mean is now:

`48.93/11\approx\$4.45`

This is almost $3 less than the original mean. Thus, the mean has decreased significantly.

For the median, we really don't know. Recall that the median is simply the "middle" quantity. Assuming all the other products are similarly priced, the median may only change slightly or it may not at all. However, we can only guess for the median as it will depend on the prices of the other products. Nevertheless, it mostly likely decrease.

Edit: Added Letter Answer