To complete the table and calculate the estimated average amount spent by learners daily at the takeaway restaurant, we need to extract the necessary information from the ogive curve mentioned. Since I can't see the actual ogive, I'll provide a template for how to complete the frequency table and how to calculate the average given appropriate data.
Completing the Frequency Table
You would follow these steps to complete the table:
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Extract cumulative frequencies from the ogive. For each class from the ogive, you note down the cumulative frequency at the upper boundary of each interval.
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Calculate class frequencies. The frequency for each class can be calculated by taking the difference between the cumulative frequencies of consecutive classes.
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Determine the class midpoints. The midpoint \( x_i \) for each class can be calculated as follows:
\[ x_i = \frac{\text{Lower boundary} + \text{Upper boundary}}{2} \]
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Calculate \( x_i \cdot f \). Multiply the midpoint by the class frequency.
Example Frequency Table
Based on the instructions, here's how you could structure it. You would need to fill in the frequencies based on the ogive:
| Class | Frequency \(f\) | Cumulative Frequency | Class Midpoint \(x_i\) | \(x_i \cdot f\) | |------------|------------------|----------------------|-------------------------|------------------| | (0; 20] | 15 | 15 | 10 | 150 | | (20; 40] | x1 | 15 + x1 | 30 | 30 * x1 | | (40; 60] | x2 | 15 + x1 + x2 | 50 | 50 * x2 | | (60; 80] | x3 | 15 + x1 + x2 + x3 | 70 | 70 * x3 | | (80; 100] | x4 | 15 + x1 + x2 + x3 + x4 | 90 | 90 * x4 | | (100; 120] | x5 | 15 + x1 + x2 + x3 + x4 + x5 | 110 | 110 * x5 |
Calculation of Estimated Average
Once you have populated the frequency table, you can calculate the estimated average amount \( \bar{x} \) using the formula:
\[ \bar{x} = \frac{\sum (x_i \cdot f)}{\sum f} \]
Where \( \sum (x_i \cdot f) \) is the sum of the products of midpoints and their respective frequencies, and \( \sum f \) is the total frequency.
Example Calculation
As an illustrative example, suppose you found the frequencies to be:
- (0; 20]: 15
- (20; 40]: 25
- (40; 60]: 20
- (60; 80]: 10
- (80; 100]: 5
- (100; 120]: 5
You would have:
| Class | Frequency \(f\) | Cumulative Frequency | Class Midpoint \(x_i\) | \(x_i \cdot f\) | |------------|------------------|----------------------|-------------------------|------------------| | (0; 20] | 15 | 15 | 10 | 150 | | (20; 40] | 25 | 40 | 30 | 750 | | (40; 60] | 20 | 60 | 50 | 1000 | | (60; 80] | 10 | 70 | 70 | 700 | | (80; 100] | 5 | 75 | 90 | 450 | | (100; 120] | 5 | 80 | 110 | 550 |
Now sum these up:
\[ \sum (x_i \cdot f) = 150 + 750 + 1000 + 700 + 450 + 550 = 3000 \]
\[ \sum f = 15 + 25 + 20 + 10 + 5 + 5 = 80 \]
Finally, calculate the average:
\[ \bar{x} = \frac{3000}{80} = 37.5 \]
So, the average amount that learners spend daily at the takeaway restaurant would be 37.5.
Note
Please insert the specific values extracted from the ogive as needed to get accurate results.