16N.2.SL.TZ0.S_8

pestleMathematicsAASLPaper 216N· sl-4-3-mean-median-mode-mean-of-grouped-data-standard-deviation-quartiles-iqrsource ↗

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.

$\sum\limits_{i = 1}^{10} {{x_i} = 252,{\text{ }}\sigma = 5{\text{ and median}} = 27.}$

Find the mean number of hours spent browsing the Internet.

[2]

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]

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]

Markscheme / solution

* 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 $\,\,\,\,\,$ $\frac{{\Sigma x}}{{10}},{\text{ }}\frac{{252}}{n},{\text{ }}\frac{{252}}{{10}}$

mean $= 25.2{\text{ (hours)}}$ A1 N2

[2 marks]

(i) mean $= 30.2{\text{ (hours)}}$ A1 N1

(ii) $\sigma = 5{\text{ (hours)}}$ A1 N1

[2 marks]

(i) valid approach (M1)

eg $\,\,\,\,\,$ 95%, 5% of 27

correct working (A1)

eg $\,\,\,\,\,$ $0.95 \times 27,{\text{ }}27 - (5\% {\text{ of }}27)$

median $= 25.65{\text{ (exact), }}25.7{\text{ (hours)}}$ A1 N2

(ii) METHOD 1

variance $= {({\text{standard deviation}})^2}$ (seen anywhere) (A1)

valid attempt to find new standard deviation (M1)

eg $\,\,\,\,\,$ ${\sigma _{new}} = 0.95 \times 5,{\text{ }}4.75$

variance $= 22.5625{\text{ }}({\text{exact}}),{\text{ }}22.6$ A1 N2

METHOD 2

variance $= {({\text{standard deviation}})^2}$ (seen anywhere) (A1)

valid attempt to find new variance (M1)

eg $\,\,\,\,\,$ ${0.95^2}{\text{ }},{\text{ }}0.9025 \times {\sigma ^2}$

new variance $= 22.5625{\text{ }}({\text{exact}}),{\text{ }}22.6$ A1 N2

[6 marks]

Examiners’ report

[N/A]