Answer :
To determine which statement is true about the data, we need to calculate the mean and the median of the number of pages in the books for both Box A and Box B step-by-step.
### Step 1: Calculate the Mean
Mean (Average) Calculation:
The mean is calculated using the formula:
[tex]\[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \][/tex]
Box A:
Sum of number of pages in Box A:
[tex]\[ 32 + 32 + 28 + 28 + 28 + 28 + 25 + 25 + 35 = 261 \][/tex]
Number of books in Box A: 9
Mean for Box A:
[tex]\[ \text{Mean}_{\text{Box A}} = \frac{261}{9} \approx 29.0 \][/tex]
Box B:
Sum of number of pages in Box B:
[tex]\[ 48 + 20 + 32 + 20 + 32 + 32 + 40 + 20 + 21 = 265 \][/tex]
Number of books in Box B: 9
Mean for Box B:
[tex]\[ \text{Mean}_{\text{Box B}} = \frac{265}{9} \approx 29.44 \][/tex]
### Step 2: Calculate the Median
Median Calculation:
The median is the middle value when the data points are arranged in ascending order. If the number of data points is odd, the median is the middle number. If the number is even, the median is the average of the two middle numbers.
Box A:
Arranged data in ascending order:
[tex]\[ 25, 25, 28, 28, 28, 28, 32, 32, 35 \][/tex]
Since there are 9 data points (an odd number), the median is the 5th value.
Median for Box A:
[tex]\[ \text{Median}_{\text{Box A}} = 28 \][/tex]
Box B:
Arranged data in ascending order:
[tex]\[ 20, 20, 20, 21, 32, 32, 32, 40, 48 \][/tex]
Since there are 9 data points (an odd number), the median is the 5th value.
Median for Box B:
[tex]\[ \text{Median}_{\text{Box B}} = 32 \][/tex]
### Comparison of Mean and Median:
1. Compare Mean:
- Mean of Box A: 29.0
- Mean of Box B: 29.44
- [tex]\(\therefore\)[/tex] The mean of Box B is greater than the mean of Box A.
2. Compare Median:
- Median of Box A: 28
- Median of Box B: 32
- [tex]\(\therefore\)[/tex] The median of Box B is greater than the median of Box A.
### Conclusion:
Based on the calculations:
- The mean of Box B is greater than the mean of Box A.
- The median of Box B is greater than the median of Box A.
The true statements are:
- "The mean of Box B is greater than the mean of Box A."
- "The median of Box B is greater than the median of Box A."
### Step 1: Calculate the Mean
Mean (Average) Calculation:
The mean is calculated using the formula:
[tex]\[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \][/tex]
Box A:
Sum of number of pages in Box A:
[tex]\[ 32 + 32 + 28 + 28 + 28 + 28 + 25 + 25 + 35 = 261 \][/tex]
Number of books in Box A: 9
Mean for Box A:
[tex]\[ \text{Mean}_{\text{Box A}} = \frac{261}{9} \approx 29.0 \][/tex]
Box B:
Sum of number of pages in Box B:
[tex]\[ 48 + 20 + 32 + 20 + 32 + 32 + 40 + 20 + 21 = 265 \][/tex]
Number of books in Box B: 9
Mean for Box B:
[tex]\[ \text{Mean}_{\text{Box B}} = \frac{265}{9} \approx 29.44 \][/tex]
### Step 2: Calculate the Median
Median Calculation:
The median is the middle value when the data points are arranged in ascending order. If the number of data points is odd, the median is the middle number. If the number is even, the median is the average of the two middle numbers.
Box A:
Arranged data in ascending order:
[tex]\[ 25, 25, 28, 28, 28, 28, 32, 32, 35 \][/tex]
Since there are 9 data points (an odd number), the median is the 5th value.
Median for Box A:
[tex]\[ \text{Median}_{\text{Box A}} = 28 \][/tex]
Box B:
Arranged data in ascending order:
[tex]\[ 20, 20, 20, 21, 32, 32, 32, 40, 48 \][/tex]
Since there are 9 data points (an odd number), the median is the 5th value.
Median for Box B:
[tex]\[ \text{Median}_{\text{Box B}} = 32 \][/tex]
### Comparison of Mean and Median:
1. Compare Mean:
- Mean of Box A: 29.0
- Mean of Box B: 29.44
- [tex]\(\therefore\)[/tex] The mean of Box B is greater than the mean of Box A.
2. Compare Median:
- Median of Box A: 28
- Median of Box B: 32
- [tex]\(\therefore\)[/tex] The median of Box B is greater than the median of Box A.
### Conclusion:
Based on the calculations:
- The mean of Box B is greater than the mean of Box A.
- The median of Box B is greater than the median of Box A.
The true statements are:
- "The mean of Box B is greater than the mean of Box A."
- "The median of Box B is greater than the median of Box A."
