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To find the estimated mean age of attendees at the snooker club using the provided frequency table, we will follow these steps:
### Step 1: List the Age Intervals and Frequencies
Given age intervals and their corresponding frequencies are:
- [tex]\( 14-18 \)[/tex]: 9 people
- [tex]\( 19-23 \)[/tex]: 3 people
- [tex]\( 24-28 \)[/tex]: 2 people
- [tex]\( 29-33 \)[/tex]: 1 person
- [tex]\( 34-38 \)[/tex]: 5 people
### Step 2: Calculate the Mid-Point of Each Age Interval
To find the mid-point of each age interval, we use the formula:
[tex]\[ \text{Mid-point} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} \][/tex]
Thus, the mid-points for each interval are:
- For [tex]\( 14-18 \)[/tex]: [tex]\( \frac{14 + 18}{2} = 16 \)[/tex]
- For [tex]\( 19-23 \)[/tex]: [tex]\( \frac{19 + 23}{2} = 21 \)[/tex]
- For [tex]\( 24-28 \)[/tex]: [tex]\( \frac{24 + 28}{2} = 26 \)[/tex]
- For [tex]\( 29-33 \)[/tex]: [tex]\( \frac{29 + 33}{2} = 31 \)[/tex]
- For [tex]\( 34-38 \)[/tex]: [tex]\( \frac{34 + 38}{2} = 36 \)[/tex]
### Step 3: Multiply Each Mid-Point by the Corresponding Frequency
Next, we multiply the mid-point of each interval by the corresponding frequency:
- [tex]\( 16 \times 9 = 144 \)[/tex]
- [tex]\( 21 \times 3 = 63 \)[/tex]
- [tex]\( 26 \times 2 = 52 \)[/tex]
- [tex]\( 31 \times 1 = 31 \)[/tex]
- [tex]\( 36 \times 5 = 180 \)[/tex]
### Step 4: Sum Up the Products
Now, we sum up these products to find the total sum of mid-points times frequencies:
[tex]\[ 144 + 63 + 52 + 31 + 180 = 470 \][/tex]
### Step 5: Sum the Frequencies
Sum the total frequencies to find the total number of people:
[tex]\[ 9 + 3 + 2 + 1 + 5 = 20 \][/tex]
### Step 6: Calculate the Estimated Mean Age
The estimated mean age is calculated by dividing the total sum of the mid-points times frequencies by the total number of people:
[tex]\[ \text{Estimated Mean} = \frac{\text{Total Mid-Point Frequency}}{\text{Total Frequency}} = \frac{470}{20} = 23.5 \][/tex]
### Step 7: Round to the Nearest Whole Number
Finally, we round 23.5 to the nearest whole number, which gives us:
[tex]\[ \text{Estimated Mean Age} \approx 24 \][/tex]
Therefore, the estimated mean age of the people attending the snooker club, rounded to the nearest whole number, is [tex]\( \boxed{24} \)[/tex].
### Step 1: List the Age Intervals and Frequencies
Given age intervals and their corresponding frequencies are:
- [tex]\( 14-18 \)[/tex]: 9 people
- [tex]\( 19-23 \)[/tex]: 3 people
- [tex]\( 24-28 \)[/tex]: 2 people
- [tex]\( 29-33 \)[/tex]: 1 person
- [tex]\( 34-38 \)[/tex]: 5 people
### Step 2: Calculate the Mid-Point of Each Age Interval
To find the mid-point of each age interval, we use the formula:
[tex]\[ \text{Mid-point} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} \][/tex]
Thus, the mid-points for each interval are:
- For [tex]\( 14-18 \)[/tex]: [tex]\( \frac{14 + 18}{2} = 16 \)[/tex]
- For [tex]\( 19-23 \)[/tex]: [tex]\( \frac{19 + 23}{2} = 21 \)[/tex]
- For [tex]\( 24-28 \)[/tex]: [tex]\( \frac{24 + 28}{2} = 26 \)[/tex]
- For [tex]\( 29-33 \)[/tex]: [tex]\( \frac{29 + 33}{2} = 31 \)[/tex]
- For [tex]\( 34-38 \)[/tex]: [tex]\( \frac{34 + 38}{2} = 36 \)[/tex]
### Step 3: Multiply Each Mid-Point by the Corresponding Frequency
Next, we multiply the mid-point of each interval by the corresponding frequency:
- [tex]\( 16 \times 9 = 144 \)[/tex]
- [tex]\( 21 \times 3 = 63 \)[/tex]
- [tex]\( 26 \times 2 = 52 \)[/tex]
- [tex]\( 31 \times 1 = 31 \)[/tex]
- [tex]\( 36 \times 5 = 180 \)[/tex]
### Step 4: Sum Up the Products
Now, we sum up these products to find the total sum of mid-points times frequencies:
[tex]\[ 144 + 63 + 52 + 31 + 180 = 470 \][/tex]
### Step 5: Sum the Frequencies
Sum the total frequencies to find the total number of people:
[tex]\[ 9 + 3 + 2 + 1 + 5 = 20 \][/tex]
### Step 6: Calculate the Estimated Mean Age
The estimated mean age is calculated by dividing the total sum of the mid-points times frequencies by the total number of people:
[tex]\[ \text{Estimated Mean} = \frac{\text{Total Mid-Point Frequency}}{\text{Total Frequency}} = \frac{470}{20} = 23.5 \][/tex]
### Step 7: Round to the Nearest Whole Number
Finally, we round 23.5 to the nearest whole number, which gives us:
[tex]\[ \text{Estimated Mean Age} \approx 24 \][/tex]
Therefore, the estimated mean age of the people attending the snooker club, rounded to the nearest whole number, is [tex]\( \boxed{24} \)[/tex].
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