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

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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, 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 5

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]

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