Answer :
### Step-by-Step Solution:
1. Tabulating the Data:
- The prizes and their respective counts are given as follows:
- \[tex]$1 drink: 44 prizes - \$[/tex]5 meal: 25 prizes
- \[tex]$10 gift card: 15 prizes - \$[/tex]20 gift card: 10 prizes
- \[tex]$100 gift card: 5 prizes 2. Total Number of Prizes: - Total number of prizes: \( 44 + 25 + 15 + 10 + 5 = 100 \) 3. Computing the Mean Value of Prizes: - The mean value of prizes is calculated by dividing the total value of prizes by the total number of prizes. - Total value of the prizes: \( 44 \times 1 + 25 \times 5 + 15 \times 10 + 10 \times 20 + 5 \times 100 \) - Total value: \( 44 + 125 + 150 + 200 + 500 = 1019 \) - Mean value: \( \frac{1019}{100} \approx 10.29 \) 4. Computing the Median Value of Prizes: - To find the median, we need to list all the prizes and find the middle value. - Sorted prizes: - \$[/tex]1 (44 times), \[tex]$5 (25 times), \$[/tex]10 (15 times), \[tex]$20 (10 times), \$[/tex]100 (5 times)
- Since the total number of prizes is 100 (an even number), the median is the average of the 50th and 51st values.
- Both the 50th and 51st values are \[tex]$5 (since the first 44 are \$[/tex]1 prizes and the next 25 are \[tex]$5 prizes). - Median value: \( \$[/tex]5 \)
5. Recomputing Without the \[tex]$100 Gift Card: - If the \$[/tex]100 gift card is removed, the counts change as follows:
- \[tex]$1 drink: 44 prizes - \$[/tex]5 meal: 25 prizes
- \[tex]$10 gift card: 15 prizes - \$[/tex]20 gift card: 10 prizes
- \[tex]$100 gift card: 0 prizes - New total number of prizes: \( 44 + 25 + 15 + 10 + 0 = 94 \) - Total value of the prizes without \$[/tex]100 gift card: [tex]\( 44 \times 1 + 25 \times 5 + 15 \times 10 + 10 \times 20 + 0 \times 100 \)[/tex]
- New total value: [tex]\( 44 + 125 + 150 + 200 + 0 = 519 \)[/tex]
- New mean value: [tex]\( \frac{519}{94} \approx 5.52 \)[/tex]
6. Recomputing Median Without the \[tex]$100 Gift Card: - Sorted prizes without \$[/tex]100:
- \[tex]$1 (44 times), \$[/tex]5 (25 times), \[tex]$10 (15 times), \$[/tex]20 (10 times)
- Since the total number of prizes is 94 (an even number), the median is the average of the 47th and 48th values.
- Both the 47th and 48th values are \[tex]$5 (since the first 44 are \$[/tex]1 prizes and the next 25 are \[tex]$5 prizes). - Median value remains: \( \$[/tex]5 \)
7. Effect of \[tex]$100 Gift Card: - Mean Value Comparison: - Mean with \$[/tex]100 gift card: [tex]\( 10.29 \)[/tex]
- Mean without \[tex]$100 gift card: \( 5.52 \) - Difference in mean: \( 10.29 - 5.52 = 4.77 \) - So, the \$[/tex]100 gift card increases the mean value of the prizes.
- Median Value Comparison:
- Median with \[tex]$100 gift card: \( 5 \) - Median without \$[/tex]100 gift card: [tex]\( 5 \)[/tex]
- Difference in median: [tex]\( 5 - 5 = 0 \)[/tex]
- So, the \[tex]$100 gift card does not change the median value of the prizes. ### Conclusion: - The \$[/tex]100 gift card increases the mean value of the prizes.
- The \[tex]$100 gift card does not affect the median value of the prizes. Hence, the correct interpretation of the \$[/tex]100 gift card's effect on the measure of center of the data is:
- It increases the mean value of the prizes.
1. Tabulating the Data:
- The prizes and their respective counts are given as follows:
- \[tex]$1 drink: 44 prizes - \$[/tex]5 meal: 25 prizes
- \[tex]$10 gift card: 15 prizes - \$[/tex]20 gift card: 10 prizes
- \[tex]$100 gift card: 5 prizes 2. Total Number of Prizes: - Total number of prizes: \( 44 + 25 + 15 + 10 + 5 = 100 \) 3. Computing the Mean Value of Prizes: - The mean value of prizes is calculated by dividing the total value of prizes by the total number of prizes. - Total value of the prizes: \( 44 \times 1 + 25 \times 5 + 15 \times 10 + 10 \times 20 + 5 \times 100 \) - Total value: \( 44 + 125 + 150 + 200 + 500 = 1019 \) - Mean value: \( \frac{1019}{100} \approx 10.29 \) 4. Computing the Median Value of Prizes: - To find the median, we need to list all the prizes and find the middle value. - Sorted prizes: - \$[/tex]1 (44 times), \[tex]$5 (25 times), \$[/tex]10 (15 times), \[tex]$20 (10 times), \$[/tex]100 (5 times)
- Since the total number of prizes is 100 (an even number), the median is the average of the 50th and 51st values.
- Both the 50th and 51st values are \[tex]$5 (since the first 44 are \$[/tex]1 prizes and the next 25 are \[tex]$5 prizes). - Median value: \( \$[/tex]5 \)
5. Recomputing Without the \[tex]$100 Gift Card: - If the \$[/tex]100 gift card is removed, the counts change as follows:
- \[tex]$1 drink: 44 prizes - \$[/tex]5 meal: 25 prizes
- \[tex]$10 gift card: 15 prizes - \$[/tex]20 gift card: 10 prizes
- \[tex]$100 gift card: 0 prizes - New total number of prizes: \( 44 + 25 + 15 + 10 + 0 = 94 \) - Total value of the prizes without \$[/tex]100 gift card: [tex]\( 44 \times 1 + 25 \times 5 + 15 \times 10 + 10 \times 20 + 0 \times 100 \)[/tex]
- New total value: [tex]\( 44 + 125 + 150 + 200 + 0 = 519 \)[/tex]
- New mean value: [tex]\( \frac{519}{94} \approx 5.52 \)[/tex]
6. Recomputing Median Without the \[tex]$100 Gift Card: - Sorted prizes without \$[/tex]100:
- \[tex]$1 (44 times), \$[/tex]5 (25 times), \[tex]$10 (15 times), \$[/tex]20 (10 times)
- Since the total number of prizes is 94 (an even number), the median is the average of the 47th and 48th values.
- Both the 47th and 48th values are \[tex]$5 (since the first 44 are \$[/tex]1 prizes and the next 25 are \[tex]$5 prizes). - Median value remains: \( \$[/tex]5 \)
7. Effect of \[tex]$100 Gift Card: - Mean Value Comparison: - Mean with \$[/tex]100 gift card: [tex]\( 10.29 \)[/tex]
- Mean without \[tex]$100 gift card: \( 5.52 \) - Difference in mean: \( 10.29 - 5.52 = 4.77 \) - So, the \$[/tex]100 gift card increases the mean value of the prizes.
- Median Value Comparison:
- Median with \[tex]$100 gift card: \( 5 \) - Median without \$[/tex]100 gift card: [tex]\( 5 \)[/tex]
- Difference in median: [tex]\( 5 - 5 = 0 \)[/tex]
- So, the \[tex]$100 gift card does not change the median value of the prizes. ### Conclusion: - The \$[/tex]100 gift card increases the mean value of the prizes.
- The \[tex]$100 gift card does not affect the median value of the prizes. Hence, the correct interpretation of the \$[/tex]100 gift card's effect on the measure of center of the data is:
- It increases the mean value of the prizes.
