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Date November 2016 Marks available 6 Reference code 16N.2.SL.TZ0.S_8
Level Standard Level Paper Paper 2 Time zone Time zone 0
Command term Find Question number S_8 Adapted from N/A

Question

Ten students were surveyed about the number of hours, x , they spent browsing the Internet during week 1 of the school year. The results of the survey are given below.

i = 1 10 x i = 252 ,   σ = 5  and median = 27.

During week 4, the survey was extended to all 200 students in the school. The results are shown in the cumulative frequency graph:

N16/5/MATME/SP2/ENG/TZ0/08.d

Find the mean number of hours spent browsing the Internet.

[2]
a.

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.

[2]
b.

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.

[6]
c.

(i)     Find the number of students who spent between 25 and 30 hours browsing the Internet.

(ii)     Given that 10% of the students spent more than k hours browsing the Internet, find the maximum value of k .

[6]
d.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

attempt to substitute into formula for mean     (M1)

eg Σ x 10 ,   252 n ,   252 10

mean = 25.2  (hours)      A1     N2

[2 marks]

a.

(i)     mean = 30.2  (hours)      A1 N1

(ii)     σ = 5  (hours)      A1     N1

[2 marks]

b.

(i)     valid approach     (M1)

eg 95%, 5% of 27

correct working     (A1)

eg 0.95 × 27 ,   27 ( 5 %  of  27 )

median = 25.65  (exact),  25.7  (hours)      A1     N2

(ii)     METHOD 1

variance = ( standard deviation ) 2 (seen anywhere)     (A1)

valid attempt to find new standard deviation     (M1)

eg σ n e w = 0.95 × 5 ,   4.75

variance = 22.5625   ( exact ) ,   22.6      A1     N2

METHOD 2

variance = ( standard deviation ) 2 (seen anywhere)     (A1)

valid attempt to find new variance     (M1)

eg 0.95 2   ,   0.9025 × σ 2

new variance = 22.5625   ( exact ) ,   22.6      A1     N2

[6 marks]

c.

(i)     both correct frequencies     (A1)

eg 80, 150

subtracting their frequencies in either order     (M1)

eg 150 80 ,   80 150

70 (students)     A1     N2

(ii)     evidence of a valid approach     (M1)

eg 10% of 200, 90%

correct working     (A1)

eg 0.90 × 200 ,   200 20 , 180 students

k = 35      A1     N3

[6 marks]

d.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.
[N/A]
d.

Syllabus sections

Topic 4—Statistics and probability » SL 4.2—Histograms, CF graphs, box plots
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Topic 4—Statistics and probability » SL 4.3—Mean, median, mode. Mean of grouped data, standard deviation. Quartiles, IQR
Topic 4—Statistics and probability

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