Let's analyze the impact of the [tex]$100 gift card on the measures of center, which are the mean and the median.
### Mean Calculation
1. Without the $[/tex]100 Gift Card:
- Total Number of Prizes: [tex]\(44 + 25 + 15 + 10 + 5 = 99\)[/tex]
- Total Value of Prizes: [tex]\((44 \times 1) + (25 \times 5) + (15 \times 5) + (10 \times 10) + (5 \times 20) = 44 + 125 + 75 + 100 + 100 = 444$
- Mean Value: \(\frac{444}{99} = 4.484848484848484\)[/tex]
2. With the [tex]$100 Gift Card:
- Total Number of Prizes: \(100\)
- Total Value of Prizes: \(444 + 100 = 544\)
- Mean Value: \(\frac{544}{100} = 5.44\)
### Median Calculation
The median depends on the middle values when the data is sorted in increasing order.
1. Without the $[/tex]100 Gift Card:
- Sorted Prizes list (in terms of number): [tex]\([1, 1, 1, ..., 5, 5, ..., 10, 10, ..., 20, 20]\)[/tex]
- Middle Index: [tex]\(\frac{99 + 1}{2} = 50\)[/tex]
- The median prize is the middle one, which includes prizes at the position [tex]\(50\)[/tex]:
- Position [tex]\(50\)[/tex] falls within the range of [tex]$5 prizes.
- Median Value: \(5\)
2. With the $[/tex]100 Gift Card:
- Sorted Prizes list (in terms of number): [tex]\([1, 1, 1, ..., 5, 5, ..., 10, 10, ..., 20, 20, 100]\)[/tex]
- Middle Index: [tex]\(\frac{100 + 1}{2} = 50.5\)[/tex]
- The median is the average of the prizes at the positions [tex]\(50\)[/tex] and [tex]\(51\)[/tex]:
- Both positions fall within the range of [tex]$5 prizes.
- Median Value: \((5+5)/2 = 5.0\)
### Conclusion
- The mean value of the prizes increases from \(4.484848484848484\) to \(5.44\).
- The median value of the prizes remains the same, at \(5.0\).
Thus, the $[/tex]100 gift card increases the mean value of the prizes while the median value remains unchanged.
