The City Sleep Study

Stand Up & Math · Descriptive statistics · SL & HL 4.1–4.3
📋 The Scenario

Background — read before you begin

A public health researcher surveyed 80 residents aged 13–25 about nightly sleep hours. The results are below. Keep this card — you will use this data for all four phases.

Hours (h)4≤h<55≤h<66≤h<77≤h<88≤h<99≤h<1010≤h<11
Frequency3718241693
1
Phase 1
Sampling
Task 1a
Identify the sampling method.
She stands at a mall entrance on Saturday and asks everyone until she has 80. Name the method. State one source of bias.
Task 1b
Compare three sampling methods.
Describe: (i) simple random, (ii) systematic, (iii) stratified by age group (13–17 and 18–25). Which is most appropriate? Justify.
Task 1c
Stratified sample sizes.
City has 3 200 teenagers and 4 800 young adults. How many from each group in a stratified sample of 80? Show full working.
2
Phase 2
Cumulative Frequency
Task 2a
Build the cumulative frequency table.
Add a cumulative frequency row. Use upper class boundaries: 5, 6, 7, 8, 9, 10, 11.
Task 2b
Sketch the ogive. Read off Q₁, Q₂, Q₃.
Draw a clearly labelled S-shaped curve. Use dotted lines to mark and record the three quartiles.
Task 2c
Find the 20th and 90th percentiles.
Read both from your graph. Explain in context what the 90th percentile means for this study.
Task 2d
Calculate IQR and range.
Find IQR = Q₃ − Q₁. State the range. What does IQR tell us that range does not?
3
Phase 3
Mean, σ & Box Plot
Task 3a
Estimate the mean.
Calculate an estimate for the mean using the frequency table. Show full working on the board.
Task 3b
Find the standard deviation.
Your GDC gives two values. Choose the appropriate one for this study. Justify your choice and interpret it in context.
Task 3c
Check for outliers.
Lower fence = Q₁ − 1.5 × IQR. Upper fence = Q₃ + 1.5 × IQR. Are there outliers? What would one mean in context?
Task 3d
Draw the box and whisker diagram.
Use min, Q₁, Q₂, Q₃, max. Mark outliers with ×. Compare the position of the mean and median.
4
Phase 4
Comparing Two Cities
Task 4a
Draw a second box plot.
Neighbouring city: mean = 6.8 h, σ = 1.1 h, Q₁ = 6.1 h, Q₂ = 6.7 h, Q₃ = 7.4 h. Plot on the same scale as Phase 3.
Task 4b
Compare the two distributions.
Write four statements comparing the two cities using: centre (mean and median), spread (IQR and σ), skew, and a public health implication. Always compare in context.
Task 4c
Effect of a linear transformation.
Data is converted from hours to minutes (× 60). Without recalculating, state the new values of mean, σ, Q₁, Q₃, IQR, and range. What changes? What stays the same?