IBDP Mathematics AA · SL/HL 4.1–4.3 · Cut along dashed lines
A public health researcher in a large city wants to understand sleep patterns among teenagers and young adults. She surveyed 80 residents aged 13–25 about the number of hours of sleep they get on a typical school/work night. The results are below. You will use this table for all four phases — keep this card throughout.
| Hours of sleep h | 4 ≤ h < 5 | 5 ≤ h < 6 | 6 ≤ h < 7 | 7 ≤ h < 8 | 8 ≤ h < 9 | 9 ≤ h < 10 | 10 ≤ h < 11 |
|---|---|---|---|---|---|---|---|
| Frequency | 3 | 7 | 18 | 24 | 16 | 9 | 3 |
She stands at a shopping mall entrance on Saturday and asks every person who walks in until she has 80. Name this method. State one potential source of bias.
Describe how she could use: (i) simple random, (ii) systematic, (iii) stratified sampling (age groups 13–17 and 18–25). Which is most appropriate? Justify.
The city has 3 200 teenagers (13–17) and 4 800 young adults (18–25). How many from each group in a stratified sample of 80? Show full working.
Add a cumulative frequency row to the data table. Use upper class boundaries as x-values: 5, 6, 7, 8, 9, 10, 11.
Draw a clearly labelled S-shaped curve, then use dotted horizontal and vertical lines to mark and read off Q₁, the median (Q₂), and Q₃ — record their values.
Estimate (i) the 20th percentile and (ii) the 90th percentile. Explain in context what the 90th percentile means.
Find IQR = Q₃ − Q₁. State the range. What does IQR tell us that range does not?
Calculate an estimate for the mean of this dataset using the frequency table. Show your full working.
Your calculator gives you two different standard deviations. Decide which one is appropriate for this study, justify your choice, and explain in a sentence what it tells us about the data.
Lower fence = Q₁ − 1.5 × IQR. Upper fence = Q₃ + 1.5 × IQR. Are there outliers? What would one mean in context?
Use min, Q₁, Q₂, Q₃, max. Mark outliers with ×. Compare mean and median. What does this suggest about the shape?
Mean = 7.0 h, σ = 1.45 h, Q₁ = 6.1 h, median = 6.9 h, Q₃ = 8.0 h, min = 4 h, max = 10 h.
On the same scale as Phase 3, draw a second box and whisker diagram for the teenager sub-group directly below.
Write at least 3 comparative statements using median, IQR, range, and symmetry. Always compare in context — not just numbers.
All values are multiplied by 60. Without recalculating, state the new: mean, σ, Q₁, Q₃, IQR, range. What stays the same? What changes? Why?