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Date	May 2017	Marks available	2	Reference code	17M.2.SL.TZ1.T_5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Calculate	Question number	T_5	Adapted from	N/A

Question

The table below shows the distribution of test grades for 50 IB students at Greendale School.

M17/5/MATSD/SP2/ENG/TZ1/05

A student is chosen at random from these 50 students.

A second student is chosen at random from these 50 students.

The number of minutes that the 50 students spent preparing for the test was normally distributed with a mean of 105 minutes and a standard deviation of 20 minutes.

Calculate the mean test grade of the students;

[2]

a.i.

Calculate the standard deviation.

[1]

a.ii.

Find the median test grade of the students.

[1]

b.

Find the interquartile range.

[2]

c.

Find the probability that this student scored a grade 5 or higher.

[2]

d.

Given that the first student chosen at random scored a grade 5 or higher, find the probability that both students scored a grade 6.

[3]

e.

Calculate the probability that a student chosen at random spent at least 90 minutes preparing for the test.

[2]

f.i.

Calculate the expected number of students that spent at least 90 minutes preparing for the test.

[2]

f.ii.

Markscheme

$\frac{{1(1) + 3(2) + 7(3) + 13(4) + 11(5) + 10(6) + 5(7)}}{{50}} = \frac{{230}}{{50}}$ (M1)

Note: Award (M1) for correct substitution into mean formula.

$= 4.6$ (A1) (G2)

[2 marks]

a.i.

$1.46{\text{ }}(1.45602 \ldots )$ (G1)

[1 mark]

a.ii.

5 (A1)

[1 mark]

b.

$6 - 4$ (M1)

Note: Award (M1) for 6 and 4 seen.

$= 2$ (A1) (G2)

[2 marks]

c.

$\frac{{11 + 10 + 5}}{{50}}$ (M1)

Note: Award (M1) for $11 + 10 + 5$ seen.

$= \frac{{26}}{{50}}{\text{ }}\left( {\frac{{13}}{{25}},{\text{ }}0.52,{\text{ }}52\% } \right)$ (A1) (G2)

[2 marks]

d.

$\frac{{10}}{{{\text{their }}26}} \times \frac{9}{{49}}$ (M1)(M1)

Note: Award (M1) for $\frac{{10}}{{{\text{their }}26}}$ seen, (M1) for multiplying their first probability by $\frac{9}{{49}}$ .

OR

$\frac{{\frac{{10}}{{50}} \times \frac{9}{{49}}}}{{\frac{{26}}{{50}}}}$

Note: Award (M1) for ${\frac{{10}}{{50}} \times \frac{9}{{49}}}$ seen, (M1) for dividing their first probability by $\frac{{{\text{their }}26}}{{50}}$ .

$= \frac{{45}}{{637}}{\text{ (}}0.0706,{\text{ }}0.0706436 \ldots ,{\text{ }}7.06436 \ldots \% )$ (A1)(ft) (G3)

Note: Follow through from part (d).

[3 marks]

e.

${\text{P}}(X \geqslant 90)$ (M1)

OR

M17/5/MATSD/SP2/ENG/TZ1/05.f.i/M (M1)

Note: Award (M1) for a diagram showing the correct shaded region $( > 0.5)$ .

$0.773{\text{ }}(0.773372 \ldots ){\text{ }}0.773{\text{ }}(0.773372 \ldots ,{\text{ }}77.3372 \ldots \% )$ (A1) (G2)

[2 marks]

f.i.

$0.773372 \ldots \times 50$ (M1)

$= 38.7{\text{ }}(38.6686 \ldots )$ (A1)(ft) (G2)

Note: Follow through from part (f)(i).

[2 marks]

f.ii.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.

[N/A]

c.

[N/A]

d.

[N/A]

e.

[N/A]

f.i.

[N/A]

f.ii.

Syllabus sections

Topic 4—Statistics and probability » SL 4.3—Mean, median, mode. Mean of grouped data, standard deviation. Quartiles, IQR

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18N.1.SL.TZ0.T_2c.i:
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19M.2.SL.TZ1.T_3a:
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18M.1.SL.TZ1.T_2b.i:
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19M.1.SL.TZ2.T_6b.i:
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18M.1.SL.TZ1.T_6b:
Complete the table.
18M.1.SL.TZ1.S_2a:
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19M.1.SL.TZ2.S_8b:
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17M.1.SL.TZ2.S_8a.ii:
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19M.1.SL.TZ1.T_2d.ii:
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19M.1.SL.TZ2.S_8a:
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18M.1.SL.TZ2.T_12c:
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18M.2.SL.TZ2.T_2b.iii:
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18M.2.SL.TZ2.T_2d:
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18M.2.SL.TZ2.T_2e:
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18M.2.SL.TZ2.T_2g:
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On graph paper, draw a box-and-whisker diagram for these data. Use a scale of 2 cm to represent 5000 km.
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Write down the median.
18N.2.SL.TZ0.T_3d.i:
Calculate an estimate of the mean examination result.
17M.1.SL.TZ2.S_8a.i:
Find the median number of hours worked by the employees.
19M.2.SL.TZ1.T_3c.iii:
For these data, write down the upper quartile.
18M.1.AHL.TZ2.H_3a:
Find the value of p.
19M.1.SL.TZ2.T_6c:
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State, giving a reason, which school cafeteria has more meals that do not meet the requirement.
18M.1.SL.TZ2.S_3b.i:
Write down the value of the new mean.
17M.1.SL.TZ2.S_8b.i:
Find the amount of money an employee earned for working 40 hours;
19M.2.SL.TZ1.T_3c.ii:
For these data, write down the lower quartile.
18M.2.SL.TZ2.T_2b.ii:
Use the cumulative frequency curve to find the lower quartile.
17M.1.SL.TZ2.S_8b.ii:
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18N.2.SL.TZ0.T_3d.ii:
Calculate an estimate of the standard deviation, giving your answer correct to three decimal places.
18M.2.SL.TZ2.T_2f:
Find the value of m.
18N.1.SL.TZ0.T_2a:
Write down the modal length of the rods.
17M.1.SL.TZ2.S_8d:
Only 10 employees earned more than £ $k$ . Find the value of $k$ .
20N.1.SL.TZ0.T_3b:
Use your graphic display calculator to find an estimate of the standard deviation of the weights of mangoes from this harvest.
19M.1.SL.TZ2.S_8c:
Find the mean number of hours that the girls in this group spent watching television that week.
18M.2.SL.TZ1.T_2g:
Given that this flight was not heavily delayed, find the probability that it travelled between 500 km and 5000 km.
19M.1.SL.TZ1.T_2c:
Write down the range of the animal speeds.
19M.2.SL.TZ1.S_1b:
Find the value of $p$ .
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Write down, for $N$ , the modal class;
17M.2.AHL.TZ2.H_1b:
Calculate the mean score.
18M.2.SL.TZ2.T_2c:
Hence write down the interquartile range.
22M.1.SL.TZ1.3b.i:
One of the adults surveyed is $42$ years old. Estimate the age of their eldest child.
17M.2.AHL.TZ2.H_1a:
One of the players is chosen at random. Find the probability that this player’s score was 5 or more.
19M.1.SL.TZ2.S_8d.i:
Find the total number of hours the group of boys spent watching television that week.
17M.2.SL.TZ1.S_1a.ii:
Find the value of the range.
20N.1.SL.TZ0.S_8d:
Find $m$ .
17M.1.SL.TZ2.S_8c:
Find the number of employees who earned £200 or less.
20N.1.SL.TZ0.S_8b:
Write down the value of the median distance in kilometres (km).
20N.2.SL.TZ0.S_9c:
Find the probability that the bus journey takes less than $45$ minutes.
20N.2.SL.TZ0.S_9d:
Find the probability that Fiona will arrive on time.
20N.2.SL.TZ0.S_9e:
This year, Fiona will go to school on $183$ days.

Calculate the number of days Fiona is expected to arrive on time.
20N.1.SL.TZ0.T_7d.ii:
Using your answers to parts (b) and (c), find the height of Flower $D$ .
19M.1.SL.TZ2.S_8e.i:
the mean number of hours that the group of boys spent watching television.
17M.2.SL.TZ1.S_1a.i:
Write down the mode.
18M.1.SL.TZ1.S_2b:
One student sent k text messages, where k > 11 . Given that k is an outlier, find the least value of k.
20N.2.SL.TZ0.S_9b:
Find $σ$ .
17M.1.SL.TZ2.T_7c:
Find the value of $x$ and of $y$ .
18M.1.AHL.TZ2.H_3b.i:
Find μ, the expected value of X.
20N.1.SL.TZ0.S_8c:
Find the value of $b$ .
20N.1.SL.TZ0.T_7b:
Using this information, write down an equation in $p$ and $q$ .
20N.1.SL.TZ0.T_7c:
Write down a second equation in $p$ and $q$ .
18M.1.AHL.TZ2.H_3b.ii:
Find P(X > μ).
16N.2.SL.TZ0.S_8c:
During week 3 each student spent 5% less time browsing the Internet than during week 1. For week 3, find

(i) the median;

(ii) the variance.
20N.2.SL.TZ0.S_9a:
Find the probability that it will take Fiona between $15$ minutes and $30$ minutes to walk to the bus stop.
19M.2.AHL.TZ1.H_3c:
A ninth student also takes the test.

Explain why the median is unchanged.
EXN.2.SL.TZ0.1:
A data set consisting of $16$ test scores has mean $14.5$ . One test score of $9$ requires a second marking and is removed from the data set.

Find the mean of the remaining $15$ test scores.
22M.2.SL.TZ1.2a:
Find the value of $x$ .
19M.1.SL.TZ2.T_14a.ii:
Find the price that is two standard deviations above the mean price.
19M.1.SL.TZ1.T_2d.i:
For these eight animals find the mean speed.
17N.1.SL.TZ0.T_1a.ii:
For the students in this group write down the median age.
19M.2.SL.TZ1.T_3f:
Write down the number of degrees of freedom for this test.
16N.1.SL.TZ0.T_2b:
Draw a box-and-whisker diagram on the grid below to represent the Vitamin C content, in milligrams, for this sample.
19M.1.SL.TZ2.T_6a:
State what 13 represents in the given diagram.
17N.2.SL.TZ0.T_1c.i:
Use your graphic display calculator to estimate the mean of $N$ ;
18M.2.SL.TZ1.T_2e:
State, with a reason, whether you would reject the null hypothesis.
16N.2.SL.TZ0.T_1c:
Plot the point $(\bar x,{\text{ }}\bar y)$ on your scatter diagram and label this point M.
17M.1.SL.TZ2.T_7b:
Using the mean score, write down a second equation in terms of $x$ and $y$ .
20N.1.SL.TZ0.T_3c:
On the grid below, draw a histogram for the data in the table.
16N.2.SL.TZ0.T_1e:
Write down the equation of the regression line $y$ on $x$ for these eight male students.
17N.1.SL.TZ0.T_1b:
Draw a box-and-whisker diagram, for these students’ ages, on the following grid.
17M.1.SL.TZ2.T_7a:
Write down an equation, in terms of $x$ and $y$ , for the total number of times the die was rolled.
18M.1.SL.TZ2.S_3a:
Find n.
19M.2.SL.TZ1.S_1a:
For these data, find the mean distance from a student’s home to school.
17M.2.SL.TZ1.S_1b.ii:
Find the variance.
19M.1.SL.TZ2.S_8d.ii:
Find the mean number of hours that all 30 girls and boys spent watching television that week.
18M.2.SL.TZ1.T_2h:
Two flights are chosen at random from those which were slightly delayed.

Find the probability that each of these flights travelled at least 5000 km.
17N.2.SL.TZ0.T_1g:
State the result of the test. Give a reason for your answer.
16N.2.SL.TZ0.S_8b:
During week 2, the students worked on a major project and they each spent an additional five hours browsing the Internet. For week 2, write down

(i) the mean;

(ii) the standard deviation.
18M.1.SL.TZ1.T_6c.ii:
Using the table, calculate an estimate for the mean number of people being followed on the social media website by these 160 students.
18M.1.SL.TZ2.S_3b.ii:
Find the value of the new variance.
20N.1.SL.TZ0.S_8a:
Find the value of $a$ .
20N.1.SL.TZ0.S_8e:
The first $150$ athletes that completed the race won a prize.

Given that an athlete took between $22$ and $m$ minutes to complete the $5 km$ race, calculate the probability that they won a prize.
18M.2.SL.TZ1.T_2b:
Calculate the expected frequency of flights travelling at most 500 km and arriving slightly delayed.
16N.2.SL.TZ0.S_8a:
Find the mean number of hours spent browsing the Internet.
18M.1.SL.TZ1.T_2b.ii:
Find the interquartile range.
18M.1.SL.TZ2.T_12b.ii:
Calculate the value of k.
17N.2.SL.TZ0.T_1b.ii:
Write down, for $N$ , the mid-interval value of the modal class.
19M.2.SL.TZ1.T_3e.ii:
State the alternative hypothesis.
18M.1.SL.TZ1.T_2a:
Find the value of x.
18M.2.SL.TZ1.T_2c:
Write down the number of degrees of freedom.
17N.2.SL.TZ0.T_1a:
State whether $N$ is a discrete or a continuous variable.
17N.2.SL.TZ0.T_1f.ii:
Write down the ${\chi ^2}$ statistic.
16N.2.SL.TZ0.T_1d:
Write down the value of $r$ , the Pearson’s product–moment correlation coefficient, for these data.
19M.2.SL.TZ1.T_3d:
Write down the ratio of teenagers to non-teenagers in its simplest form.
17N.1.SL.TZ0.T_1a.i:
For the students in this group find the mean age;
20N.1.SL.TZ0.T_7d.i:
Using your answers to parts (b) and (c), find the height of Flower $A$ .
19M.2.SL.TZ1.T_3g:
Calculate the expected number of teenagers that prefer cats.
17M.2.SL.TZ1.T_5a.i:
Calculate the mean test grade of the students;
20N.1.SL.TZ0.T_7a:
Find the height of Flower null.
16N.1.SL.TZ0.T_2a:
Giving your answer to one decimal place, write down the value of

(i) the median level of Vitamin C content of the oranges in the sample;

(ii) the lower quartile;

(iii) the upper quartile.
17M.2.SL.TZ1.T_5e:
Given that the first student chosen at random scored a grade 5 or higher, find the probability that both students scored a grade 6.
20N.1.SL.TZ0.T_3a:
Write down the modal group for these data.
17M.2.SL.TZ1.T_5d:
Find the probability that this student scored a grade 5 or higher.
18M.2.SL.TZ1.T_2f:
Write down the probability that this flight arrived on time.
17N.2.SL.TZ0.T_1c.ii:
Use your graphic display calculator to estimate the standard deviation of $N$ .
17N.2.SL.TZ0.T_1d:
Find the expected frequency of students choosing the Science category and obtaining 31 to 40 correct answers.
17N.2.SL.TZ0.T_1e.i:
Write down the null hypothesis for this test;
17N.2.SL.TZ0.T_1e.ii:
Write down the number of degrees of freedom.
19M.2.SL.TZ1.T_3h:
Use your graphic display calculator to find the $p$ -value for this test.
18N.2.SL.TZ0.T_3a.ii:
Find the interquartile range.
18N.2.SL.TZ0.T_3c.i:
Write down the modal class.
18N.2.SL.TZ0.T_3c.ii:
Write down the mid-interval value of the modal class.
18N.2.SL.TZ0.T_3e:
The teacher sets a grade boundary that is one standard deviation below the mean.

Use the cumulative frequency graph to estimate the number of students whose final examination result was below this grade boundary.
18M.2.SL.TZ2.T_2a:
Find the number of buses that travelled a distance between 15000 and 20000 kilometres.
16N.2.SL.TZ0.T_1a:
On graph paper, draw a scatter diagram for these data. Use a scale of 2 cm to represent 5 hours on the $x$ -axis and 2 cm to represent 10 points on the $y$ -axis.
16N.2.SL.TZ0.T_1b:
(i) ${\bar x}$ , the mean number of hours spent on social media;

(ii) ${\bar y}$ , the mean number of IB Diploma points.
16N.2.SL.TZ0.T_1f:
Draw the regression line, from part (e), on your scatter diagram.
16N.2.SL.TZ0.T_1g:
Use the given equation of the regression line to estimate the number of IB Diploma points that this girl obtained.
17M.2.SL.TZ1.T_5a.ii:
Calculate the standard deviation.
17M.2.SL.TZ1.T_5c:
Find the interquartile range.
17M.2.SL.TZ1.T_5f.i:
Calculate the probability that a student chosen at random spent at least 90 minutes preparing for the test.
19M.2.SL.TZ1.T_3i:
State the conclusion for this test. Give a reason for your answer.

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Topic 4—Statistics and probability » SL 4.9—Normal distribution and calculations

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