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A company recorded the number of messages sent by e-mail over a period of 60 working days - NSC Mathematics - Question 2 - 2016 - Paper 2
Question 2
A company recorded the number of messages sent by e-mail over a period of 60 working days. The data is shown in the table below. NUMBER OF MESSAGES NUMBER OF DAYS ... show full transcript
Worked Solution & Example Answer:A company recorded the number of messages sent by e-mail over a period of 60 working days - NSC Mathematics - Question 2 - 2016 - Paper 2
Step 1
Estimate the mean number of messages sent per day, rounded off to TWO decimal places.
Answer
To estimate the mean number of messages per day, we will calculate the weighted average using the midpoints of each interval.
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Calculate midpoints:
- For 10 < x ≤ 20: midpoint = 15
- For 20 < x ≤ 30: midpoint = 25
- For 30 < x ≤ 40: midpoint = 35
- For 40 < x ≤ 50: midpoint = 45
- For 50 < x ≤ 60: midpoint = 55
- For 60 < x ≤ 70: midpoint = 65
- For 70 < x ≤ 80: midpoint = 75
- For 80 < x ≤ 90: midpoint = 85
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Multiply midpoints by the corresponding number of days:
- 15 x 2 = 30
- 25 x 8 = 200
- 35 x 5 = 175
- 45 x 6 = 270
- 55 x 12 = 660
- 65 x 18 = 1170
- 75 x 9 = 675
- 85 x 2 = 170
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Sum of all products = 30 + 200 + 175 + 270 + 660 + 1170 + 675 + 170 = 3080.
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Total days = 60.
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Mean = ( \frac{3080}{60} \approx 51.33 ) messages per day.
Step 2
Draw a cumulative frequency graph (ogive) of the data on the grid provided in the ANSWER BOOK.
Answer
To draw a cumulative frequency graph:
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Calculate cumulative frequencies for each interval:
- 10 < x ≤ 20: 2
- 20 < x ≤ 30: 2 + 8 = 10
- 30 < x ≤ 40: 10 + 5 = 15
- 40 < x ≤ 50: 15 + 6 = 21
- 50 < x ≤ 60: 21 + 12 = 33
- 60 < x ≤ 70: 33 + 18 = 51
- 70 < x ≤ 80: 51 + 9 = 60
- 80 < x ≤ 90: 60 + 2 = 62
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Plot the points on the graph at the upper limits of each interval against the cumulative frequencies.
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Connect the points using a smooth curve to create the ogive.
Step 3
Hence, estimate the number of days on which 65 or more messages were sent.
Answer
To estimate the number of days with 65 or more messages:
- From the cumulative frequency graph (ogive) plotted:
- Locate the point corresponding to 65 messages.
- Draw a horizontal line to intersect with the ogive.
- Drop a vertical line down to the x-axis to read the cumulative frequency corresponding to 65 messages.
- The approximate value from the graph shows around 14 days on which 65 or more messages were sent.