Section B ( 52 marks )
10 . A machine manufactures a large number of nails . The length , L mm , of a nail is normally distributed , where
2
L ~ N ( � , 3.5 ) .
There are 12 % of the nails which are longer than 59.2 mm , and they are classified as long nails .
( a ) Find the value of � .
[ 4 ]
There are three different types of classifications for the nails , based on their lengths . The following table shows the percentages and the weights of each classification :
Classification |
Short |
Medium |
Long |
Percentage |
55 % |
33 % |
12 % |
Weight of a nail 0.8 g 1.1 g 1.4 g
A nail is classified as short if it is shorter than q cm.
( b ) Find the value of q . [ 3 ]
( c ) 10 nails are randomly selected . Let X be the number of short nails in this sample .
( i ) Find E ( X ) .
( ii ) Find the probability that more than 5 of these nails are short .
All short and medium nails , in total m nails , are selected . It is given that the total weight of the nail is 949 kg .
[ 6 ]
( d ) Find the value of m . [ 4 ]
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