6
A group of 12 students consisting of 6 girls and 6 boys are to be seated at tables A, B and C , each with 4 chairs numbered 1, 2, 3 and 4 as shown below. 1
2
A
1
4
2
3
B
1
4
3
2
C
4
3
Find the number of ways this can be done if
7
(i)
there is no restriction,
(ii)
there are 2 girls and 2 boys at each table,
(iii)
there are 2 girls and 2 boys at each table and students of the same gender are seated opposite each other.
A company has 6 departments and a total of 240 staff. ach department has a different number of staff. !he management, concerned by a large turno"er, decides to sur"ey a sample of 30 staff to see# their opinions on the wor#ing conditions in the company. !wo methods are suggested on how the sample can be selected. Method 1: Use Simple Random Sampling Method 2: Use Stratified Sampling
8
(i)
$escribe how each method can be carried out.
(ii)
%i"e one reason for choosing ðod 1 o"er ðod 2, (a) ðod 2 o"er ðod 1. (b)
!he number of flaws in a roll of a particular design of wallpaper is #nown to follow a 'oisson distribution with mean 0.1(. (i)
Find the probability that, of two randomly chosen rolls of wallpaper, one has e)actly 1 flaw and one has more than 1 flaw.
(ii)
9
*sing a suitable appro)imation, find the probability that in a random sample of (0 rolls of wallpaper, there is an a"erage of more than 0.3 flaw per roll.
From records, a shot putter has an a"erage throwing distance of +.2 metres. -is coach taught him a new techniue and after three months of training, his distance, in metres, on 10 different occasions is as follows. .11
+.+
.02
.0+
.10
+.0
.0(
+.(
+.+/
+.+6
!he coach is interested to find out if the shot putter has impro"ed with the new techniue. (i)
tate, gi"ing your reasons, whether a z test or t test should be used.
(ii)
arry out the test in (i) at the ( significance le"el, stating any assumption necessary for "alidity.
1
10
!he probability that a hoc#ey team wins any match is
2
and the probability that it
1
loses any match is 6 . !hree points are awarded for a win, one point for a draw and no point for a defeat. !he team plays four matches and the outcome of each match is independent of the other matches. (i)
how that the probability that the team has e)actly one draw and e)actly one 1
defeat is
6.
Find the probability that the team
11
(ii)
wins the first match and goes on to win e)actly one other match,
(iii)
wins e)actly one match, gi"en that it obtains four points.
n a certain factory, it is #nown that ( of all articles produced are defecti"e. (i)
how that the probability of a randomly chosen sample of si5e 10 containing e)actly one defecti"e article is 0.31(, correct to 3 significant figures.
(ii)
+0 random samples, each of si5e 10, are drawn. *sing a suitable appro)imation, find the probability that at least 2( but fewer than 3( samples will each contain e)actly one defecti"e article.
12
1
A factory manufactures a large number of containers with presson lids. !he diameter of a randomly chosen lid is X cm, where X follows a normal distribution with mean 12.1 and standard de"iation 0.03. !he diameter of the top of a randomly chosen container into which the lid is pressed is Y cm, where Y follows an independent normal distribution with mean 12.0 and standard de"iation 0.03. (a)
Find the probability that the sum of the diameters of 3 randomly chosen lids and twice the diameter of the top of a randomly chosen container is less than 60.( cm.
(b)
A lid and a container are chosen randomly and paired. A pairing is accepted if 0.02 < X − Y < 0.1/ and discarded otherwise. (i)
Find the probability that a pairing is accepted.
(ii)
stimate the number of trials reuired to obtain 2000 accepted pairings.
A stationery retailer supplies pens to offices. n order to encourage customers to buy in bul#, the stationery retailer came up with the following price scheme. umber of pens ordered, x *nit price in dollar, y
(
10
20
40
+0
160
1.(0
1.20
1.10
0.
0.0
0.+0
(i)
alculate the "alue of the product moment correlation coefficient.
(ii)
%i"e a s#etch of the scatter diagram for the data and hence comment on the "alue of the product moment correlation coefficient found in (i).
(iii)
tate, with a reason, which model of the form y = a + bw is appropriate, where a and b are constants, b > 0 and w is as follows. 2 !7 w = x ,
1 w
"7
=
, #7 w = ln x . x
*sing the appropriate model selected abo"e, find (i$)
the "alue of the product moment correlation coefficient,
($)
the euation of the regression line of y on w , and hence obtain an estimate of the number of pens ordered when a unit price of 81.40 is uoted. omment on the reliability of your answer.
