Standard +0.3 This is a standard S2 continuous distribution question requiring routine application of pdf/cdf relationships. Part (a) uses the uniform distribution property for immediate answers, part (b) is a guided integration with the integral provided, parts (c) and (d) involve straightforward cdf evaluation and quartile calculation. All techniques are textbook exercises with no novel insight required, making it slightly easier than average.
The random variable \(X\) has probability density function defined by
$$f(x) = \begin{cases} \frac{1}{2} & 0 \leqslant x \leqslant 1 \\ \frac{1}{18}(x - 4)^2 & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$
State values for the median and the lower quartile of \(X\). [2 marks]
Show that, for \(1 \leqslant x \leqslant 4\), the cumulative distribution function, \(\mathrm{F}(x)\), of \(X\) is given by
$$\mathrm{F}(x) = 1 + \frac{1}{54}(x - 4)^3$$
(You may assume that \(\int (x - 4)^2 \, dx = \frac{1}{3}(x - 4)^3 + c\).) [4 marks]
Determine \(\mathrm{P}(2 \leqslant X \leqslant 3)\). [2 marks]
Show that \(q\), the upper quartile of \(X\), satisfies the equation \((q - 4)^3 = -13.5\). [3 marks]
Hence evaluate \(q\) to three decimal places. [1 mark]
Question 7:
7
TOTAL
Answer all questions in the spaces provided.
1 Judith, the village postmistress, believes that, since moving the post office counter
into the local pharmacy, the mean daily number of customers that she serves has
increased from 79.
In order to investigate her belief, she counts the number of customers that she serves
on 12 randomly selected days, with the following results.
88 81 84 89 90 77 72 80 82 81 75 85
Stating a necessary distributional assumption, test Judith’s belief at the 5% level of
significance. (9 marks)
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2 It is claimed that a new drug is effective in the prevention of sickness in
holiday-makers. A sample of 100 holiday-makers was surveyed, with the following
results.
Sickness No sickness Total
Drug taken 24 56 80
No drug taken 11 9 20
Total 35 65 100
Assuming that the 100 holiday-makers are a random sample, use a w2 test, at the
5% level of significance, to investigate the claim. (8 marks)
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Sickness | No sickness | Total
Drug taken | 24 | 56 | 80
No drug taken | 11 | 9 | 20
Total | 35 | 65 | 100
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3 The continuous random variable X has a rectangular distribution defined by
(cid:1)
k (cid:1)3k4x4k
fðxÞ ¼
0 otherwise
(a) (i) Sketch the graph of f. (2 marks)
1
(ii) Hence show that k ¼ . (2 marks)
2
(b) Find the exact numerical values for the mean and the standard deviation of X.
(3 marks)
(cid:2) (cid:3)
1
(c) (i) Find P X 5(cid:1) . (2 marks)
4
(cid:2) (cid:3)
1
(ii) Write down the value of P X 6¼ (cid:1) . (1 mark)
4
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4 The error, X (cid:1)C, made in measuring a patient’s temperature at a local doctors’
surgery may be modelled by a normal distribution with mean m and standard
deviation s.
The errors, x(cid:1)C, made in measuring the temperature of each of a random sample of
10 patients are summarised below.
X X
2
x ¼ 0:35 and ðx(cid:1)(cid:1)xxÞ ¼ 0:12705
Construct a 99% confidence interval for m, giving the limits to three decimal places.
(5 marks)
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5 The number of telephone calls received, during an 8-hour period, by an IT company
that request an urgent visit by an engineer may be modelled by a Poisson distribution
with a mean of 7.
(a) Determine the probability that, during a given 8-hour period, the number of telephone
calls received that request an urgent visit by an engineer is:
(i) at most 5; (1 mark)
(ii) exactly 7; (2 marks)
(iii) at least 5 but fewer than 10. (3 marks)
(b) Write down the distribution for the number of telephone calls received each hour that
request an urgent visit by an engineer. (1 mark)
(c) The IT company has 4 engineers available for urgent visits and it may be assumed
that each of these engineers takes exactly 1 hour for each such visit.
At 10am on a particular day, all 4 engineers are available for urgent visits.
(i) State the maximum possible number of telephone calls received between 10am and
11am that request an urgent visit and for which an engineer is immediately available.
(1 mark)
(ii) Calculate the probability that at 11am an engineer will not be immediately available
to make an urgent visit. (4 marks)
(d) Give a reason why a Poisson distribution may not be a suitable model for the number
of telephone calls per hour received by the IT company that request an urgent visit
by an engineer. (1 mark)
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6 (a) The number of strokes, R, taken by the members of Duffers Golf Club to complete
the first hole may be modelled by the following discrete probability distribution.
r 42 3 4 5 6 7 8 59
PðR ¼ rÞ 0 0.1 0.2 0.3 0.25 0.1 0.05 0
(i) Determine the probability that a member, selected at random, takes at least 5 strokes
to complete the first hole. (1 mark)
(ii) Calculate EðRÞ. (2 marks)
(iii) Show that VarðRÞ ¼ 1:66. (4 marks)
(b) The number of strokes, S, taken by the members of Duffers Golf Club to complete
the second hole may be modelled by the following discrete probability distribution.
s 42 3 4 5 6 7 8 59
PðS ¼ sÞ 0 0.15 0.4 0.3 0.1 0.03 0.02 0
Assuming that R and S are independent:
(i) show that PðRþS48Þ ¼ 0:24; (5 marks)
(ii) calculate the probability that, when 5 members are selected at random, at least 4 of
them complete the first two holes in fewer than 9 strokes; (3 marks)
(iii) calculate PðR ¼ 4jRþS48Þ. (3 marks)
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r | 42 | 3 | 4 | 5 | 6 | 7 | 8 | 59
PðR ¼ rÞ | 0 | 0.1 | 0.2 | 0.3 | 0.25 | 0.1 | 0.05 | 0
s | 42 | 3 | 4 | 5 | 6 | 7 | 8 | 59
PðS ¼ sÞ | 0 | 0.15 | 0.4 | 0.3 | 0.1 | 0.03 | 0.02 | 0
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7 The random variable X has probability density function defined by
8 1
04x41
>
> 2
>
>
<
fðxÞ ¼ 1 2
ðx(cid:1)4Þ 14x44
>18
>
>
>
:
0 otherwise
(a) State values for the median and the lower quartile of X. (2 marks)
(b) Show that, for 14x44, the cumulative distribution function, FðxÞ, ofX is given
by
1 3
FðxÞ ¼ 1þ ðx(cid:1)4Þ
54
ð
2 1 3
ðYou may assume that ðx(cid:1)4Þ dx ¼ ðx(cid:1)4Þ þc:Þ (4 marks)
3
(c) Determine Pð24X 43Þ. (2 marks)
3
(d)(i) Show that q, the upper quartile of X, satisfies the equation ðq(cid:1)4Þ ¼ (cid:1)13:5.
(3 marks)
(ii) Hence evaluate q to three decimal places. (1 mark)
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The random variable $X$ has probability density function defined by
$$f(x) = \begin{cases} \frac{1}{2} & 0 \leqslant x \leqslant 1 \\ \frac{1}{18}(x - 4)^2 & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item State values for the median and the lower quartile of $X$. [2 marks]
\item Show that, for $1 \leqslant x \leqslant 4$, the cumulative distribution function, $\mathrm{F}(x)$, of $X$ is given by
$$\mathrm{F}(x) = 1 + \frac{1}{54}(x - 4)^3$$
(You may assume that $\int (x - 4)^2 \, dx = \frac{1}{3}(x - 4)^3 + c$.) [4 marks]
\item Determine $\mathrm{P}(2 \leqslant X \leqslant 3)$. [2 marks]
\item
\begin{enumerate}[label=(\roman*)]
\item Show that $q$, the upper quartile of $X$, satisfies the equation $(q - 4)^3 = -13.5$. [3 marks]
\item Hence evaluate $q$ to three decimal places. [1 mark]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2010 Q7 [12]}}