Answer :
To solve this problem, we need to determine two things:
1. The class interval that contains the median.
2. The estimate for the mean number of hours.
### Step 1: Find the Class Interval that Contains the Median
To find the median class interval, we first need to calculate the cumulative frequencies. We sum the frequencies progressively to get the cumulative frequencies.
Here are the steps to construct the cumulative frequency distribution:
| Number of hours [tex]$(h)$[/tex] | Frequency | Cumulative Frequency |
|------------------------|-----------|-----------------------|
| [tex]$0 \leqslant h < 1$[/tex] | 4 | 4 |
| [tex]$1 \leqslant h < 2$[/tex] | 8 | 4 + 8 = 12 |
| [tex]$2 \leqslant h < 3$[/tex] | 11 | 12 + 11 = 23 |
| [tex]$3 \leqslant h < 4$[/tex] | 7 | 23 + 7 = 30 |
So, the cumulative frequencies are:
[tex]\[ 4, 12, 23, 30 \][/tex]
The total number of children is 30. The median is the value that lies at the [tex]$\frac{30}{2} = 15^{th}$[/tex] position in the ordered dataset. To find which class interval contains this position, we look at the cumulative frequencies:
- The first cumulative frequency is 4, which does not contain the 15th position.
- The second cumulative frequency is 12, which still does not contain the 15th position.
- The third cumulative frequency is 23, which does contain the 15th position (since 12 < 15 ≤ 23).
Therefore, the class interval that contains the median is:
[tex]\[ 2 \leqslant h < 3 \][/tex]
### Step 2: Work Out an Estimate for the Mean Number of Hours
To estimate the mean number of hours, we use the midpoint of each class interval as a representative value for that interval. We will then multiply each midpoint by its corresponding frequency, sum these products, and divide by the total number of children.
Here are the midpoints and their calculations:
- Midpoint of [tex]$0 \leqslant h < 1$[/tex] is 0.5
- Midpoint of [tex]$1 \leqslant h < 2$[/tex] is 1.5
- Midpoint of [tex]$2 \leqslant h < 3$[/tex] is 2.5
- Midpoint of [tex]$3 \leqslant h < 4$[/tex] is 3.5
Next, we calculate the total hours by summing the product of each midpoint and its frequency:
[tex]\[ (4 \times 0.5) + (8 \times 1.5) + (11 \times 2.5) + (7 \times 3.5) \][/tex]
Breaking down the calculations:
[tex]\[ 4 \times 0.5 = 2 \][/tex]
[tex]\[ 8 \times 1.5 = 12 \][/tex]
[tex]\[ 11 \times 2.5 = 27.5 \][/tex]
[tex]\[ 7 \times 3.5 = 24.5 \][/tex]
Adding these together gives the total hours:
[tex]\[ 2 + 12 + 27.5 + 24.5 = 66 \][/tex]
Now, calculate the mean:
[tex]\[ \text{Mean} = \frac{\text{Total Hours}}{\text{Total Number of Children}} = \frac{66}{30} = 2.2 \][/tex]
### Summary of Solutions
1. The class interval that contains the median is:
[tex]\[ 2 \leqslant h < 3 \][/tex]
2. The estimate for the mean number of hours is:
[tex]\[ 2.2 \][/tex]
1. The class interval that contains the median.
2. The estimate for the mean number of hours.
### Step 1: Find the Class Interval that Contains the Median
To find the median class interval, we first need to calculate the cumulative frequencies. We sum the frequencies progressively to get the cumulative frequencies.
Here are the steps to construct the cumulative frequency distribution:
| Number of hours [tex]$(h)$[/tex] | Frequency | Cumulative Frequency |
|------------------------|-----------|-----------------------|
| [tex]$0 \leqslant h < 1$[/tex] | 4 | 4 |
| [tex]$1 \leqslant h < 2$[/tex] | 8 | 4 + 8 = 12 |
| [tex]$2 \leqslant h < 3$[/tex] | 11 | 12 + 11 = 23 |
| [tex]$3 \leqslant h < 4$[/tex] | 7 | 23 + 7 = 30 |
So, the cumulative frequencies are:
[tex]\[ 4, 12, 23, 30 \][/tex]
The total number of children is 30. The median is the value that lies at the [tex]$\frac{30}{2} = 15^{th}$[/tex] position in the ordered dataset. To find which class interval contains this position, we look at the cumulative frequencies:
- The first cumulative frequency is 4, which does not contain the 15th position.
- The second cumulative frequency is 12, which still does not contain the 15th position.
- The third cumulative frequency is 23, which does contain the 15th position (since 12 < 15 ≤ 23).
Therefore, the class interval that contains the median is:
[tex]\[ 2 \leqslant h < 3 \][/tex]
### Step 2: Work Out an Estimate for the Mean Number of Hours
To estimate the mean number of hours, we use the midpoint of each class interval as a representative value for that interval. We will then multiply each midpoint by its corresponding frequency, sum these products, and divide by the total number of children.
Here are the midpoints and their calculations:
- Midpoint of [tex]$0 \leqslant h < 1$[/tex] is 0.5
- Midpoint of [tex]$1 \leqslant h < 2$[/tex] is 1.5
- Midpoint of [tex]$2 \leqslant h < 3$[/tex] is 2.5
- Midpoint of [tex]$3 \leqslant h < 4$[/tex] is 3.5
Next, we calculate the total hours by summing the product of each midpoint and its frequency:
[tex]\[ (4 \times 0.5) + (8 \times 1.5) + (11 \times 2.5) + (7 \times 3.5) \][/tex]
Breaking down the calculations:
[tex]\[ 4 \times 0.5 = 2 \][/tex]
[tex]\[ 8 \times 1.5 = 12 \][/tex]
[tex]\[ 11 \times 2.5 = 27.5 \][/tex]
[tex]\[ 7 \times 3.5 = 24.5 \][/tex]
Adding these together gives the total hours:
[tex]\[ 2 + 12 + 27.5 + 24.5 = 66 \][/tex]
Now, calculate the mean:
[tex]\[ \text{Mean} = \frac{\text{Total Hours}}{\text{Total Number of Children}} = \frac{66}{30} = 2.2 \][/tex]
### Summary of Solutions
1. The class interval that contains the median is:
[tex]\[ 2 \leqslant h < 3 \][/tex]
2. The estimate for the mean number of hours is:
[tex]\[ 2.2 \][/tex]
