IB Maths AI HL Topic 4: Statistics and probability 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

Difficulty

Easy Medium Hard

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]

Question 4

Paper 1Easy6 marks

A student records the distance downstream from a spring and the number of aquatic invertebrate taxa found at each of six fieldwork sites. Figure 1 shows the results.

(a)

Using your GDC, find Spearman’s rank correlation coefficient, $r_s$ for these data. Give your answer to three significant figures.

[2]

(b)

Using your GDC, find Pearson’s product moment correlation coefficient, $r$ for these data. Give your answer to three significant figures.

[2]

(c)

The student wants to describe a monotonic association, and considers the point $(10,5)$ to be unusual compared with the other observations. State which coefficient is more appropriate for this purpose and give a reason.

[2]

Question 5

Paper 1Easy6 marks

A running coach records the weekly training time, $x$ hours, and 5 km race time, $y$ minutes, for six runners.

$x$	2	4	5	7	9	10
$y$	31.8	29.4	28.7	26.3	25.5	23.9

(a)

Use your graphic display calculator to find Pearson’s product-moment correlation coefficient, $r$ Give your answer to three significant figures.

[1]

(b)

Describe the direction and strength of the correlation.

[1]

(c)

Use your graphic display calculator to find the equation of the regression line of $y$ on $x$ in the form $y=ax+b$ Give the values of $a$ and $b$ to three significant figures.

[2]

(d)

Use the regression line to estimate the 5 km race time for a runner who trains for 6.5 hours per week. Give your answer to three significant figures.

[1]

(e)

Interpret the value of $a$ in this context.

[1]

Question 6

Paper 1Hard6 marks

During a greenhouse trial, a horticulturist measures the height of a young plant on days 0 to 5. The measurements, rounded to the nearest centimetre, are shown in Figure 1.

(a)

Use your GDC to fit a quadratic least-squares regression curve to the observations. Give its equation, $R^2$ and $SS_{\mathrm{res}}$

[2]

(b)

Use your GDC to fit a linear least-squares regression curve to the observations. Give its $R^2$ $SS_{\mathrm{res}}$ and Pearson’s product moment correlation coefficient, $r$ Show numerically that $R^2=r^2$ for this linear model.

[2]

(c)

A horticulturist claims that the quadratic model must be used to estimate the plant’s height on day 8 because it has the larger $R^2$ Assess this claim using the graph and your results.

[2]

Question 7

Paper 2Easy5 marks

A school surveys 50 students about how they travelled to school on a particular day. Of these students, 23 travelled by bus, 19 travelled by train, and 8 used both the bus and the train.

One student is selected at random from the 50 students.

(a)

Find the probability that the student travelled by bus or train.

[2]

(b)

Find the probability that the student travelled by neither bus nor train.

[1]

(c)

Given that the student travelled by bus, find the probability that the student also travelled by train.

[2]

Question 8

Paper 2Medium5 marks

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)

Calculate $\mathrm{E}(X)$ and $\mathrm{Var}(X)$

[3]

(b)

For a parcel with excess handling time $X$ the additional handling charge $Y$ in AUD, is modelled by $Y=3X+2$ Hence determine $\mathrm{E}(Y)$ and $\mathrm{Var}(Y)$

[2]

Question 9

Paper 1Hard6 marks

A local emergency department records two types of admissions during each 20-minute interval. The two streams of admissions are independent, and admissions occur independently at a uniform average rate throughout each interval. The mean numbers are 1.4 type A admissions and 0.9 type B admissions per interval.

A planner is considering a normal, binomial or Poisson model for the total number, $N$ of admissions in an interval.

(a)

Show that a suitable model for $N$ is a Poisson distribution with mean 2.3.

[2]

(b)

The department can assess at most $C$ admissions during an interval. The planner requires the probability that the department can assess all admissions to be at least 0.95.

Using your model from part (a), determine the smallest possible integer value of $C$ Show numerical evidence that it is the smallest value satisfying the requirement.

[4]

Question 10

Paper 2Hard6 marks

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.

(a)

Show that $P(X<494)=0.0304$

[2]

(b)

Using your result from part (a), determine the smallest value of $U$ to the nearest $0.1$ g, that meets the company’s requirement. A manager proposes setting $U$ to the 95th percentile of the distribution. Assess whether this proposal meets the requirement.

[4]

Question 11

Paper 1Medium6 marks

A parcel-delivery company defines its daily forecast error as

\text{actual number of parcels}-\text{forecast number of parcels}.

A random sample of 16 independent days has a mean forecast error of 18.4 parcels and a sample standard deviation of 12.0 parcels. The daily forecast errors may be modelled by a normal population, whose standard deviation is unknown.

(a)

State which distribution should be used to construct a confidence interval for the population mean forecast error, $\mu$

[1]

(b)

Calculate a 95% confidence interval for $\mu$ Give both endpoints to one decimal place.

[3]

(c)

The planning director claims that the forecasts are unbiased, so that $\mu=0$ Use your interval to comment on this claim and state the direction of any bias indicated.

[2]

Question 12

Paper 1Hard6 marks

A field team measures the nitrate concentration in water samples from a catchment. The eight samples are selected independently at random. Assume that the population of nitrate concentrations is normally distributed, with known standard deviation $1.2\text{ mg L}^{-1}$

Figure 1 shows the measured concentrations. The team will test whether the population mean concentration exceeds $5.0\text{ mg L}^{-1}$ using

H_0:\mu=5.0,\qquad H_1:\mu>5.0,

where $\mu$ is measured in $\text{mg L}^{-1}$

(a)

Use Figure 1 to determine the sample mean concentration.

[1]

(b)

Using your answer to part (a), carry out the stated test at the 5% significance level. State your conclusion in context.

[2]

(c)

Suppose that the true population mean concentration is $5.8\text{ mg L}^{-1}$ Determine the probability of a Type II error for the test in part (b). Give your answer to three significant figures.

[3]

Question 13

Paper 1Hard6 marks

During fieldwork, students randomly selected 50 plots in each of three habitat types and recorded whether a particular plant species was present.

Habitat type	Plant present	Plant absent
Woodland interior	22	28
Woodland edge	35	15
Open scrub	23	27

(a)

State appropriate null and alternative hypotheses for investigating whether plant presence is related to habitat type.

[2]

(b)

Using an appropriate statistical test and your GDC, determine the test statistic, its degrees of freedom and the p-value.

[2]

(c)

At the 5% significance level, state what the data indicate. A volunteer claims that the test proves woodland edge causes a greater probability of plant presence than open scrub. Assess this claim.

[2]

Question 14

Paper 1Easy6 marks

A campus club records which of two cafés its members visit each Friday. The transition diagram in Figure 1 gives the probabilities that a member visits each café on the next Friday, based on the café visited this Friday.

Let $s_n$ be the column state matrix after $n$ weekly transitions, using the state order North, South. On the Friday of the survey, the state matrix is

s_0=\begin{pmatrix}0.75\\0.25\end{pmatrix}.

(a)

Write down the transition matrix $T$ where $T_{ij}$ is the probability of moving from state $j$ to state $i$

[2]

(b)

Calculate $s_1=Ts_0$

[2]

(c)

Hence, using your result from part (b), calculate $s_2$

[2]

Question 15

Paper 1Hard6 marks

A rail authority models the monthly change in commuters’ usual station between North (N) and South (S). The state matrix is a column matrix, with entries giving the proportions using N and S, in that order. The transition matrix is

T=\begin{pmatrix}0.8&0.3\\0.2&0.7\end{pmatrix},

where rows represent the station used in the next month and columns represent the station used in the current month. Initially,

s_0=\begin{pmatrix}0.10\\0.90\end{pmatrix}.

(a)

Calculate $s_1$

[1]

(b)

Find the least whole number of months $n\geq 1$ for which the proportion using N is within $0.01$ of its long-term proportion. Justify why this condition holds for every subsequent month.

[5]

Question 16

Paper 1Hard6 marks

A university models changes in students’ preferred meal-collection hub (A, B or C) from one month to the next. The columns of the transition matrix $T$ represent the current hub, and the rows represent the hub in the following month. The state matrices are column matrices.

T= \begin{pmatrix} 0.6&0.2&0.1\\ x&0.5&0.2\\ 0.4-x&0.3&0.7 \end{pmatrix}

The initial state matrix and the state matrix one month later are

s_0= \begin{pmatrix} 0.50\\ 0.30\\ 0.20 \end{pmatrix}, \qquad s_1= \begin{pmatrix} 0.38\\ 0.34\\ 0.28 \end{pmatrix}.

The same transition matrix applies each month, and the resulting Markov chain is regular.

(a)

Determine the value of $x$

[2]

(b)

A manager claims that more than half of the students will use hub C in the long term because $0.7$ of the students currently using C remain there the following month.

Determine the long-term proportions of students using each hub and decide whether the manager’s claim is correct. Give the proportions to three significant figures.

[4]

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