ANNEXURE A

Question

Question 2.1
Class Frequency
𝑓
Cumulative
frequency
Class midpoint
𝑥
𝑖
𝑥
𝑖
· 𝑓
(0;20] 15 15
(20;40]
(40;60]
(60;80]
(80;100]
(100;120]
QUESTION 2
The diagram below shows an ogive representing the average amount that the learners
spend daily at the takeaway restaurant.
Figure 2.1: Ogive curve.
2.1 Copy and complete the following table using the ogive. You can redraw the
frequency table or print Annexure A on page 16.
Figure 2.2: Frequency table. (4)
2.2 Calculate the estimated average amount that the learners spend daily at
the takeaway restaurant.
(3)

the cumulative frequency is like this 15,55,110,140,150,155

Answer 1

To complete the frequency table provided in Annexure A using the cumulative frequency data you've given, we'll first need to find the frequencies for each class interval.

Here's the completed frequency table based on the cumulative frequencies:

Frequency Table

| Class | Frequency \( f \) | Cumulative Frequency | Class Midpoint \( x_i \) | \( x_i \cdot f \) | |--------------|--------------------|----------------------|---------------------------|--------------------| | (0;20] | 15 | 15 | 10 | 150 | | (20;40] | 40 | 55 | 30 | 1200 | | (40;60] | 55 | 110 | 50 | 2750 | | (60;80] | 30 | 140 | 70 | 2100 | | (80;100] | 10 | 150 | 90 | 900 | | (100;120] | 5 | 155 | 110 | 550 | | Total | 155 | | | 4750 |

Explanation of Each Column:

1. Frequency \( f \): This represents how many learners fall into each class interval. It is calculated by subtracting the cumulative frequency of the previous class from the cumulative frequency of the current class.

   • For (20;40]: \( 55 - 15 = 40 \)
   • For (40;60]: \( 110 - 55 = 55 \)
   • For (60;80]: \( 140 - 110 = 30 \)
   • For (80;100]: \( 150 - 140 = 10 \)
   • For (100;120]: \( 155 - 150 = 5 \)

2. Cumulative Frequency: Provided in your question, shows the total count up to the upper boundary of each class.

3. Class Midpoint \( x_i \): The midpoint for each class interval is calculated as the average of the lower and upper limits.

   • (0;20]: \( (0+20)/2 = 10 \)
   • (20;40]: \( (20+40)/2 = 30 \)
   • (40;60]: \( (40+60)/2 = 50 \)
   • (60;80]: \( (60+80)/2 = 70 \)
   • (80;100]: \( (80+100)/2 = 90 \)
   • (100;120]: \( (100+120)/2 = 110 \)

4. \( x_i \cdot f \): This is the product of the class midpoint and the frequency. It's computed by multiplying the class midpoint by the frequency for each class interval.

2.2 Calculate the Estimated Average

To find the estimated average amount that learners spend, we use the formula:

\[ \text{Mean} = \frac{\sum (x_i \cdot f)}{\sum f} \]

Where:

• \(\sum (x_i \cdot f) = 4750\)
• \(\sum f = 155\)

Plugging in the values:

\[ \text{Mean} = \frac{4750}{155} \approx 30.65 \]

Thus, the estimated average amount that learners spend daily at the takeaway restaurant is approximately R30.65.