A survey was conducted among 100 people about the amount that they paid on a monthly basis for their cellphone contracts - NSC Mathematics - Question 2 - 2019 - Paper 2

We know that the total number of people surveyed is 100. This gives us the equation:

$7 + 12 + a + 35 + b + 6 = 100$

Simplifying this, we have:

$a + b + 60 = 100$

Thus,

$a + b = 40 \quad (1)$

Next, using the mean calculated as R309:

$309 = \frac{(50 \times 7) + (150 \times 12) + (250 \times a) + (350 \times 35) + (450 \times b) + (550 \times 6)}{100}$

Multiplying through by 100:

$30900 = (50 \times 7) + (150 \times 12) + (250 \times a) + (350 \times 35) + (450 \times b) + (550 \times 6$

Substituting known values:

$30900 = 350 + 1800 + 250a + 12250 + 450b + 3300$

Which simplifies to:

$250a + 450b = 30900 - 42350$

This gives:

$250a + 450b = -11450 \quad (2)$

Now, from equation (1) we can express b as:

$b = 40 - a$

Substituting this into equation (2):

$250a + 450(40 - a) = -11450$

Simplifying:

$250a + 18000 - 450a = -11450$

Combining like terms gives:

$-200a = -29450$

Solving for a:

$a = 24$

Now substituting back to find b:

$b = 40 - 24 = 16$

Thus, we have shown that a = 24 and b = 16.

The modal class is the class interval with the highest frequency. In this case, the interval 300 < x ≤ 400 has the highest frequency of 35.

Therefore, the modal class is 300 < x ≤ 400.

To draw an ogive:

1. Calculate the cumulative frequencies for each interval:

   • For 0 < x ≤ 100: 7
   • For 100 < x ≤ 200: 7 + 12 = 19
   • For 200 < x ≤ 300: 19 + 24 = 43
   • For 300 < x ≤ 400: 43 + 35 = 78
   • For 400 < x ≤ 500: 78 + 16 = 94
   • For 500 < x ≤ 600: 94 + 6 = 100

2. Plot these cumulative frequencies against the upper boundaries of each class:

   • (100, 7)
   • (200, 19)
   • (300, 43)
   • (400, 78)
   • (500, 94)
   • (600, 100)

3. Connect the points with a smooth curve to create the ogive. Make sure to label the axes appropriately. This visually represents the cumulative frequency distribution of the data.

From the cumulative frequency calculation, we know that:

• Total number of people surveyed = 100
• Cumulative frequency for interval 500 < x ≤ 600 = 94

Thus, those who paid more than R420 can be calculated:

$100 - 82 = 18$

Therefore, 18 people paid more than R420 per month.