Find the mean number of hours spent browsing the Internet.

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.

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.

(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 $k$.

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]

(i)     both correct frequencies     (A1)

eg$\,\,\,\,\,$80, 150

subtracting their frequencies in either order     (M1)

eg$\,\,\,\,\,$$150 - 80,{\text{ }}80 - 150$

70 (students)     A1     N2

(ii)     evidence of a valid approach     (M1)

eg$\,\,\,\,\,$10% of 200, 90%

correct working     (A1)

eg$\,\,\,\,\,$$0.90 \times 200,{\text{ }}200 - 20$, 180 students

$k = 35$     A1     N3

[6 marks]