IB Maths AI HL Topic 4: Statistics and probability questions

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 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.

At a parcel centre, $X$ is the excess handling time, in minutes, for a randomly selected parcel. Figure 1 shows the probability distribution of $X$

A snack company models the mass, $X$ g, of a packet as normally distributed with mean $503$ g and standard deviation $4.8$ g. Packets with mass below $494$ g do not meet the lower specification. The company wants at least $94\%$ of packets to meet the specification $494\leq X\leq U$ where $U$ is the upper specification.

Use your GDC for normal probability and inverse normal calculations.