The time in hours that Davin plays on his games machine each day is normally distributed with mean $$$3.5$$$ and standard deviation $$$0.9$$$. Calculate an estimate for the number of days in a year (365 days) on which Davin plays on his games machine for between $$$2.8$$$ and $$$4.2$$$ hours. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_w20_qp_51 Year：2020 Question No：5(c)

Answer:

$\mathrm{P}(2.8<\mathrm{X}<4.2)=1-2 \times$ their $\mathbf{5}(\mathbf{a})$
$\equiv 2(1-$ their $5(\mathbf{a}))-1$
$\equiv 2(0 \cdot 5-$ their $5(\mathbf{a}))$
$=0 \cdot 5636$
Number of days $=365 \times 0.5636=205 \cdot 7$
So, 205 (days)
Alternative method for question $5(c)$
$ \mathrm{P}\left(\frac{2.8-3.5}{0.9}\right. $<z<$ \left.\frac{4.2-3.5}{0.9}\right) $
$ \begin{array}{l} =\Phi(0.7778)-(1-\Phi 0.7778) \\ =0.7818-(1-0.7818) \\ =0.5636 \end{array} $
Number of days $ =365 \times 0.5636=205.7 $
So, 205 (days)

Knowledge points:

5.5.1 understand the use of a normal distribution to model a continuous random variable, and use normal distribution tables (Sketches of normal curves to illustrate distributions or probabilities may be required.)

5.5.2.1 finding the value of $P\left(X>x_{1}\right)$, or a related probability, given the values of $x_{1}, \mu, \sigma$.

Answer:

Knowledge points:

Solution:

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