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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

1 ( 1 ) + 3 ( 2 ) + 7 ( 3 ) + 13 ( 4 ) + 11 ( 5 ) + 10 ( 6 ) + 5 ( 7 ) 50 = 230 50     (M1)

 

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

 

= 4.6     (A1)     (G2)

[2 marks]

a.i.

1.46   ( 1.45602 )     (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.

11 + 10 + 5 50     (M1)

 

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

 

= 26 50   ( 13 25 ,   0.52 ,   52 % )     (A1)     (G2)

[2 marks]

d.

10 their  26 × 9 49     (M1)(M1)

 

Note:     Award (M1) for 10 their  26 seen, (M1) for multiplying their first probability by 9 49 .

 

OR

10 50 × 9 49 26 50

 

Note:     Award (M1) for 10 50 × 9 49 seen, (M1) for dividing their first probability by their  26 50 .

 

= 45 637  ( 0.0706 ,   0.0706436 ,   7.06436 % )     (A1)(ft)     (G3)

 

Note:     Follow through from part (d).

 

[3 marks]

e.

P ( X 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   ( 0.773372 )   0.773   ( 0.773372 ,   77.3372 % )     (A1)     (G2)

[2 marks]

f.i.

0.773372 × 50     (M1)

= 38.7   ( 38.6686 )     (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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Topic 4—Statistics and probability » SL 4.9—Normal distribution and calculations
Topic 4—Statistics and probability

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