SPS SPS SM Statistics (SPS SM Statistics) 2021 September

1. A random sample of distances travelled to work for 120 commuters from a train station in Devon is recorded. The distances travelled, to the nearest mile, are summarised below.

For this distribution:
a estimate the median. The mid-point of each class was represented by \(x\) and its corresponding frequency by \(f\). The mid-point of the lowest class was taken to be 4.75 giving: $$\Sigma f x = 3552.5 \text { and } \Sigma f x ^ { 2 } = 138043.125$$ b Estimate the mean and the standard deviation of this distribution.
c Explain why the median is less than the mean for these data.
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2. Data relating to the lifetimes (to the nearest hour) of a random sample of 200 light bulbs from the production line of a manufacturer were summarised in a grouped frequency table. The mid-point of each class in the table was represented by \(x\) and the corresponding frequency for that class by \(f\). The data were then coded using: $$y = \frac { ( x - 755.0 ) } { 2.5 }$$ and summarised as follows: $$\sum f y = - 467 , \sum f y ^ { 2 } = 9179$$ Calculate estimates of the mean and the standard deviation of the lifetimes of this sample of bulbs.
(4 marks)
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3. There are 180 students at a college following a general course in computing. Students on this course can choose to take up to three extra options. \begin{displayquote} 112 take systems support,
70 take developing software,
81 take networking,
35 take developing software and systems support,
28 take networking and developing software,
40 take systems support and networking,
4 take all three extra options. \end{displayquote} a Draw a Venn diagram to represent this information. A student from the course is chosen at random.
b Find the probability that this student takes
i none of the three extra options
ii networking only. Students who take systems support and networking are eligible to become technicians.
c Given that the randomly chosen student is eligible to become a technician, find the probability that this student takes all three extra options.
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4. A company assembles drills using components from two sources. Goodbuy supplies the components for \(85 \%\) of the drills whilst Amart supplies the components for the rest.
It is known that \(3 \%\) of the components supplied by Goodbuy are faulty and \(6 \%\) of those supplied by Amart are faulty.
a Represent this information on a tree diagram. An assembled drill is selected at random.
b Find the probability that the drill is not faulty.

5. Figure 2 is a histogram showing the distribution of the time taken in minutes, \(t\), by a group of people to swim 500 m . \begin{figure}[h] \end{figure} a Find the probability that a person chosen at random from the group takes longer than 18 minutes.

6. The discrete random variable \(X\) has probability function
\(\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) & x = 0,1,2
k ( x - 2 ) & x = 3
0 & \text { otherwise } \end{cases}\)
where \(k\) is a positive constant.
a Show that \(k = 0.25\) Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
b Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0\)
c Find the complete probability function for \(X _ { 1 } + X _ { 2 }\).
d Find \(\mathrm { P } \left( 1.3 \leqslant X _ { 1 } + X _ { 2 } \leqslant 3.2 \right)\)

7. Emma throws a fair coin 15 times and records the number of times it shows a head.
a State the appropriate distribution to model the number of times the coin shows a head giving any relevant parameter values.
b Find the probability that Emma records:
i exactly 8 heads
ii at least 4 heads.