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1. A m anu facturer produces sweets of lengthL mm where L has a
continuous uniformdistributionwith range [15, 30].
(a) Find th e probability thata randomly selected sweethas a
length greater than24 mm .(2 )
These sw eets are random ly packed in bags of 20 sweets.
(b) Find the probability that a randomly selected bag will
contain at least 8 sweetslength greater than24 mm.
(3)
(c) Find the probability tha t 2 rando m ly selected bags will
both contain at least 8 swith length greater than 24 mm.
(2)
.
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2. A test statistichas a distributionB(25, p).
Given that
H,:
(a ) find th e critical region for the test statistic such
thatth e probability in each tail is asclose as possible to
2.5%.
(3)
(b ) State th e probability of incorrectly rejectingH using this
critical region.(2)
it" X-
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3. (a) Write downtw o conditions neededto approximateth e
binomial distributionby thePoisson distribution.
(2 )
A machine which manufactures boltsis known to produce 3%
defective bolts.Them achine breaks d ow n and a new m ach ine is
installed. A random sample of 200 btaken fromthose produced by the
newmachineand 12bolts were defective.
(b ) Using a suitable approximation,test at the 5% level of
significance whetheror notthe proportion of defective bolts is
higher with the new machine than with thmachine. State your
hypotheses clearly.
(7)
b).
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4. The numberof houses soldby an estate agentfollows a Poisson
distribution, w itha meanof 2 per week.
(a) Find the probability that in the next 4 weeks the estate
agen t sells,
(i) exactly 3 houses,
(ii) m ore than5 houses.
The estate agent monitors sales in periods of 4 weeks.
(5)
(b) Find the probability that in the next twelve of these 4 week
periods there are exactlynine periods in which more than 5 houses
are sold.
(3)
The estate agent will receivea bonus if he sells more than25
housesin the next 10 weeks.
(c) Use a suitable approximation to estimate the probability
that the estate agent receivesa bonus.
(6)
* n o . o f period5 u/We
= o .
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Question 4 continued
c)
P(X?2S) - -
Jb
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5. The queueing time, X minutes, of a customer at a till of a
supermarket has probabilitydensity function
0 otherwise
(a) Show that the value of & is 4
(b) Write down the value of E(X).
(c) Calculate Var(A).
(4)
(1)
(4)
(d) Find the probability that a randomly chosen customer's
queueing time will differ fromthe mean by at least half a
minute.
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Question 5 continued
r O.S
