Certainly! Let's work through this step-by-step to determine which increases more when including the new package weight: the median or the mean.
### Step 1: Determine the original mean and median
We have the original weights of the six packages:
[tex]\[ 35, 25, 75, 30, 70, 65 \][/tex]
First, we need to find the original mean (average):
[tex]\[ \text{Mean} = \frac{35 + 25 + 75 + 30 + 70 + 65}{6} = \frac{300}{6} = 50 \][/tex]
Next, we find the original median (middle value in the ordered list):
First, order the weights:
[tex]\[ 25, 30, 35, 65, 70, 75 \][/tex]
Since we have an even number of weights, the median is the average of the two middle numbers:
[tex]\[ \text{Median} = \frac{35 + 65}{2} = \frac{100}{2} = 50 \][/tex]
Original mean and median:
[tex]\[ \text{Mean} = 50 \][/tex]
[tex]\[ \text{Median} = 50 \][/tex]
### Step 2: Adding the new package weight
The new package weighs 155 ounces. Including this, the updated list of weights is:
[tex]\[ 35, 25, 75, 30, 70, 65, 155 \][/tex]
### Step 3: Calculate the new mean and median
Start by finding the new mean:
[tex]\[ \text{New Mean} = \frac{35 + 25 + 75 + 30 + 70 + 65 + 155}{7} = \frac{455}{7} = 65 \][/tex]
Now, the new median (middle value in the ordered list):
First, order the weights:
[tex]\[ 25, 30, 35, 65, 70, 75, 155 \][/tex]
Since we now have an odd number of weights, the median is the middle number:
[tex]\[ \text{Median} = 65 \][/tex]
New mean and median:
[tex]\[ \text{New Mean} = 65 \][/tex]
[tex]\[ \text{New Median} = 65 \][/tex]
### Step 4: Determine the increase
Calculate the increase in the mean and the median:
[tex]\[ \text{Mean Increase} = \text{New Mean} - \text{Original Mean} = 65 - 50 = 15 \][/tex]
[tex]\[ \text{Median Increase} = \text{New Median} - \text{Original Median} = 65 - 50 = 15 \][/tex]
### Step 5: Compare the increases
Both the mean and the median increased by 15. Therefore, they are affected the same amount.
### Conclusion
Given that both the mean and the median increase by the same amount, the correct answer is:
[tex]\[ \boxed{B} \text{ The median and mean are affected the same amount.} \][/tex]
