IB Maths AI HL AI 4.1 Data and descriptive statistics questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1 Paper 2

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Question 1

Paper 1Easy6 marks

A transport operator records the journey times, in minutes, for 40 journeys on each of two bus services. The five-number summaries are shown below. No outliers were identified in either sample.

Service	Minimum	Lower quartile	Median	Upper quartile	Maximum
A	12	20	28	36	44
B	8	16	24	32	56

(a)

On a common horizontal scale from 0 to 60 minutes, construct box-and-whisker diagrams for the two services.

[5]

(b)

State which service’s journey-time distribution may be normally distributed, giving a reason based on the symmetry of its box and whiskers.

[1]

Question 2

Paper 2Hard6 marks

A city council is estimating the number of potholes on a typical 1 km road segment. It has two samples, each containing 40 segments.

The app sample consists of segments for which residents submitted reports during one month. Reports were grouped by segment, but the counts were not independently checked. The inspection sample was selected by stratified random sampling, in proportion to the number of segments in each road class. Inspectors counted and checked the potholes.

Figure 1 shows box-and-whisker plots of the recorded number of potholes per segment. Inspection confirmed that the segment represented by the value 14 had 14 distinct potholes. The value 18 in the app sample has not been checked.

For this question, a value is an outlier if it is more than $1.5\times\mathrm{IQR}$ beyond the nearest quartile.

The council proposes deleting both marked high values as recording errors and using the remaining app sample to estimate the typical number of potholes per city segment.

(a)

For each sample, calculate the upper outlier fence and determine whether its marked high value is an outlier.

[2]

(b)

Using your result from part (a), evaluate the council’s proposal. Recommend a defensible way to estimate the typical number of potholes, using evidence from Figure 1 and the sampling information.

[4]

Question 3

Paper 2Hard6 marks

A parcel depot records the dispatch-to-arrival times, in minutes, for equal samples of 60 deliveries on each of two routes. The summaries below give the whisker endpoints, quartiles and any outliers. The outlier is marked with a cross; there are no other outliers.

Route	Lower whisker	$Q_1$	Median	$Q_3$	Upper whisker	Outlier
A	12.4	18.1	22.0	26.0	29.9	54.2 ×
B	15.2	19.1	23.0	27.1	31.0	—

(a)

Calculate the interquartile range for each route. Give your answers in minutes.

[2]

(b)

A supervisor concludes that Route A has more variable delivery times and is less likely to have a normally distributed set of times than Route B. Evaluate this conclusion using the summaries.

[4]

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