Let's analyze the problem step by step to understand the changes in the mean and median if Jerry wins [tex]$10,000 in Week 8.
### Step 1: Initial Calculations
Winnings for the first seven weeks:
\[
684, 770, 481, 647, 277, 853, 712
\]
Calculate the initial mean:
\[
\text{Mean}_\text{initial} = \frac{684 + 770 + 481 + 647 + 277 + 853 + 712}{7}
\]
\[
\text{Mean}_\text{initial} = \frac{4424}{7} = 632.0
\]
Calculate the initial median:
To find the median, we first sort the winnings:
\[
277, 481, 647, 684, 712, 770, 853
\]
Since there are 7 numbers, the median is the 4th number in the sorted list:
\[
\text{Median}_\text{initial} = 684
\]
### Step 2: Add Week 8 Winnings
Adding Jerry's win in Week 8:
\[
winnings = [684, 770, 481, 647, 277, 853, 712, 10000]
\]
### Step 3: New Calculations
Calculate the new mean:
\[
\text{Mean}_\text{new} = \frac{684 + 770 + 481 + 647 + 277 + 853 + 712 + 10000}{8}
\]
\[
\text{Mean}_\text{new} = \frac{14424}{8} = 1803.0
\]
Calculate the new median:
To find the median of the updated list, we sort it:
\[
277, 481, 647, 684, 712, 770, 853, 10000
\]
Since there are now 8 numbers, the median is the average of the 4th and 5th numbers:
\[
\text{Median}_\text{new} = \frac{684 + 712}{2} = 698.0
\]
### Step 4: Changes in the Mean and Median
Change in mean:
\[
\text{Mean Increase} = \text{Mean}_\text{new} - \text{Mean}_\text{initial}
\]
\[
\text{Mean Increase} = 1803.0 - 632.0 = 1171.0
\]
Change in median:
\[
\text{Median Increase} = \text{Median}_\text{new} - \text{Median}_\text{initial}
\]
\[
\text{Median Increase} = 698.0 - 684.0 = 14.0
\]
### Step 5: Conclusion
Given the changes calculated:
- The mean increases by \$[/tex]1171.0.
- The median increases by \[tex]$14.0.
The best answer among the provided options is:
\[
d. \text{The mean increases by \$[/tex]1,171, the median increases by \$14.}
\]
