IB Maths AA SL 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, no calculator Paper 2, calculator

Difficulty

Easy Medium Hard

Question 1

Paper 2, calculatorMedium6 marks

The waiting times, in minutes, for a sample of eight customers at a repair desk were

12, 14, 14, 16, 18, 19, 21, 26.

(a)

Use your GDC to determine the sample mean and sample standard deviation, correct to three significant figures.

[2]

(b)

Hence find the sample variance, correct to three significant figures.

[1]

(c)

For a later report, 3 minutes is subtracted from every waiting time in the sample. A supervisor claims that the standard deviation will therefore also decrease by 3 minutes.

Determine the new sample mean and standard deviation, and comment on the supervisor’s claim.

[3]

A courier company records the delivery times for 40 trips on each of two routes. Figure 1 shows box-and-whisker diagrams for the two sets of times. For this comparison, use the median to represent typical delivery time and the interquartile range (IQR) to represent consistency.

(a)

State the median delivery time for each route.

[2]

(b)

Calculate the IQR for each route.

[2]

(c)

Recommend a route if the company wants both a shorter typical delivery time and more consistent delivery times. Justify your recommendation using the medians and IQRs.

[2]

Question 3

Paper 1, no calculatorEasy6 marks

A student records the average daily sunlight exposure, $x$ hours, and the height after four weeks, $y$ cm, of five seedlings of the same species.

$x$	2	3	4	5	6
$y$	11	12	17	17	18

For these data, the Pearson correlation coefficient is $r=0.93$ and the regression line of $y$ on $x$ is $y=1.9x+7.4$

(a)

State the strength and direction of the correlation. State whether this result alone shows that sunlight causes greater height.

[2]

(b)

Use the regression line to estimate the height of a seedling receiving 5 hours of sunlight per day.

[2]

(c)

Interpret the gradient of the regression line in this context.

[1]

(d)

The observed sunlight exposures range from 2 to 6 hours per day. Explain why using the regression line to estimate the height at 10 hours per day may be unreliable.

[1]

Question 4

Paper 2, calculatorHard6 marks

A small business records its advertising expenditure, $x$ in hundreds of dollars, and the number of enquiries received in a week, $y$ for five weeks.

$x$	1	3	5	7	9
$y$	3	8	4	10	6

(a)

Determine the regression line of $x$ on $y$ giving its coefficients to three significant figures.

[2]

(b)

Use your regression line from part (a) to estimate the advertising expenditure when $y=7$ Give your answer in hundreds of dollars.

[1]

(c)

A manager wishes to estimate $y$ when $x=9$ She rearranges the line from part (a) and obtains $y=14.4$

Determine a regression estimate you would use instead, and explain why the manager’s value should not be treated as the regression estimate for predicting $y$ from $x$

[3]

Question 5

Paper 2, calculatorEasy5 marks

A student uses the spinner shown in Figure 1. Each of its six sectors is equally likely to be selected. Let $A$ be the event that the spinner lands on a number greater than $4$

(a)

List the outcomes in $A$

[1]

(b)

Determine $P(A)$

[1]

(c)

State $P(A')$ where $A'$ is the complement of $A$

[1]

(d)

Determine the expected number of times the spinner lands on a number greater than $4$ in $30$ spins.

[1]

(e)

In a separate trial of $30$ spins, the spinner lands on a number greater than $4$ eight times. Calculate the relative frequency of $A$

[1]

Question 6

Paper 2, calculatorHard6 marks

A box contains 12 counters, each of which is red or blue. The number of red counters is $r$ where $r\in\mathbb{Z}$ and $1\leq r\leq 11$ Two counters are selected successively without replacement.

The probability that the two counters are the same colour is $\frac{17}{33}$

(a)

Determine the possible values of $r$

[2]

(b)

It is also known that the probability that the first counter is red, given that the second counter is red, is $\frac{3}{11}$

(i)

Determine $r$

[2]

(ii)

Hence determine the probability that exactly one of the two counters is red.

[2]

Question 7

Paper 1, no calculatorMedium8 marks

A machine produces metal pins whose lengths are modelled by a normal distribution with mean $42$ mm and standard deviation $3$ mm.

For this question, use the facts that approximately $68\%$ $95\%$ and $99.7\%$ of values lie within one, two and three standard deviations of the mean, respectively.

(a)

Estimate the percentage of pins with lengths between $39$ mm and $45$ mm.

[1]

(b)

Hence, estimate the percentage of pins with lengths between $39$ mm and $48$ mm.

[3]

(c)

A batch contains $800$ pins. Hence, estimate the number of pins in the batch with lengths between $39$ mm and $48$ mm.

[2]

(d)

A pin has length $51$ mm. Estimate the percentage of pins with lengths greater than $51$ mm, giving your reasoning.

[2]

Question 8

Paper 1, no calculatorHard6 marks

In a production run, the number of packets with faulty seals is modelled by $X\sim B(n,p)$ where $n$ and $p$ are unknown. The mean of $X$ is $3$ and its variance is $2.1$

(a)

Determine $n$ and $p$

[4]

(b)

In a separate check, an inspector selects 10 packets without replacement from a lot of 20 packets, of which exactly 6 have faulty seals. A supervisor uses the values of $n$ and $p$ from part (a) to model the number of faulty packets in the check. Comment on whether this is appropriate.

[2]

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