Let's analyze the effect of the [tex]$100 gift card on the measures of center (mean and median) by calculating them step-by-step.
### Calculating the Mean
1. List the prizes and their quantities:
- 44 prizes of \$[/tex]1
- 25 prizes of \[tex]$5 (meal)
- 15 prizes of \$[/tex]5 (gift card)
- 10 prizes of \[tex]$10
- 5 prizes of \$[/tex]20
- 1 prize of \[tex]$100
2. Calculate the total sum of all the prize values:
\[
\text{Total sum} = (44 \times 1) + (25 \times 5) + (15 \times 5) + (10 \times 10) + (5 \times 20) + (1 \times 100)
\]
\[
= 44 + 125 + 75 + 100 + 100 + 100
= 544
\]
3. Calculate the total number of prizes:
\[
\text{Total prizes} = 44 + 25 + 15 + 10 + 5 + 1 = 100
\]
4. Calculate the mean value:
\[
\text{Mean} = \frac{\text{Total sum}}{\text{Total prizes}} = \frac{544}{100} = 5.44
\]
### Calculating the Median
1. Arrange the prizes in ascending order. Since there are 100 prizes, the middle two prizes are the 50th and 51st values.
2. Combine the counts up to find these positions:
- Prizes 1 to 44 are \$[/tex]1
- Prizes 45 to 69 are \[tex]$5 (meal)
- Prizes 70 to 84 are \$[/tex]5 (gift card)
- Prizes 85 to 94 are \[tex]$10
- Prizes 95 to 99 are \$[/tex]20
- Prize 100 is \[tex]$100
3. The 50th and 51st prizes are both among the \$[/tex]5 prizes (specifically, the second set of \[tex]$5 values).
4. Since there are an even number of prizes (100), the median is the average of the 50th and 51st prize values:
\[
\text{Median} = \frac{\$[/tex]5 + \[tex]$5}{2} = \$[/tex]5
\]
### Effect of the \[tex]$100 Gift Card
1. Mean: The \$[/tex]100 gift card increases the mean value of the prizes because it is significantly higher than the other prize values, thereby raising the total sum substantially.
2. Median: The \[tex]$100 gift card does not affect the median value because the median is determined by the middle values. In this case, the middle values are \$[/tex]5, which remains unaffected by adding a single \$100 prize at the end.
Therefore, the correct answer is:
- It increases the mean value of the prizes.
