Question 1: A cumulative frequency table for student ages is given below: | Age | Cumulative Frequency | |-------|------------------------| | 10 | 5 | | 12 | 15 | | 14 | 30 | | 16 | 45 | | 18 | 50 | Plot the cumulative frequency curve on the grid provided and describe the shape of the curve.

Answer: Students should plot the points (10,5), (12,15), (14,30), (16,45), (18,50), and draw a smooth S-curve. The shape should start shallow, steepen in the middle, then level off at the end.

Question 2: Given the data points (x=4, y=10), (x=6, y=50), (x=8, y=80), (x=10, y=95), draw the S-curve and identify the approximate point where the curve begins to flatten.

Answer: Curve starts to flatten around x=8, y≈80, indicating the saturation point.

Question 3: Explain why an S-curve in a cumulative frequency graph typically has a sigmoidal shape. Include at least two reasons.

Answer: An S-curve is sigmoidal because it reflects slow initial growth, rapid middle growth, and slowing towards saturation. This shape occurs due to factors like limited population or maximum capacity constraints.

Question 4: A data set shows that 20 students completed a survey in 10 minutes, and 50 students in 20 minutes. Plot an approximate S-curve for the cumulative number of students over time.

Answer: Students should plot (10,20), (20,50), and draw a smooth S-shaped curve showing rapid growth in the middle phase.

Question 5: Identify and explain two common mistakes made when plotting an S-curve from data.

Answer: Common mistakes include: (1) plotting points incorrectly, leading to a jagged or non-sigmoidal curve; (2) drawing a straight line instead of a smooth curve, missing the characteristic S-shape.

Question 6: Construct an S-curve showing the cumulative number of books read over a month, where reading starts slowly, accelerates mid-month, then slows down towards the end. Use the points: (Day 1, 2 books), (Day 10, 15 books), (Day 20, 25 books), (Day 30, 28 books).

Answer: Students plot the points and draw a smooth curve with a slow start, steep middle, and flattening at the end.

Question 7: In the context of a population growth study, explain how the S-curve can indicate the carrying capacity of the environment.

Answer: The flattening of the S-curve at the upper end indicates the maximum population the environment can sustain, known as the carrying capacity.

Question 8: A cumulative frequency graph for test scores shows an S-curve. If a student’s score is at the inflection point, what does this represent in terms of their performance?

Answer: The inflection point indicates the score where the number of students achieving that score or less is increasing most rapidly, often near the median.

Question 9: Using the data: (x=0, y=0), (x=50, y=40), (x=100, y=80), (x=150, y=95), draw and interpret the S-curve.

Answer: The curve starts flat, steepens around middle, then levels off near the maximum, showing saturation at y≈100.

Question 10: What is the significance of the point where the S-curve begins to level off? How is this useful in real-world data analysis?

Answer: This point indicates the approaching maximum capacity or saturation level, useful for predicting limits and making decisions based on data saturation.

Question 11: Identify one typical error when students interpret the shape of an S-curve and suggest how to correct it.

Answer: A common error is assuming the curve is linear throughout; students should recognize the sigmoidal shape and focus on the inflection points and leveling off.