Let

*a*be the unknown eighth number.

__Numbers in ascending order__

1, 2, 3, 4, 9, 12, 13

4 is the median of the seven numbers.

4.5 =

__4 +__

*a*2

4.5 x 2 = 4 +

*a*

9 = 4 +

*a*

*a*= 5

The eighth number is 5.

__4. A box contained five cards numbered 1, 2, 3, 4 and 5. A card was drawn from the box, its number noted and then replaced. The process was repeated 100 times and the table below shows the resulting frequency distribution.__

Numbers 1 2 3 4 5

Frequency 21

*x*

*y*18 17

(a) Show that

*x*+

*y*= 44

(b) If the mean of the distribution is 2.9, show that 2

*x*+ 3

*y*= 112

(c) Find the value of

*x*and

*y*

(d) State the mode and the median of the distribution.

a)

The total frequency is 100 as the process was repeated 100 times.

100 = 21 + 18 + 17 +

*x*+

*y*

100 = 56 +

*x*+

*y*

44 =

*x*+

*y*

b)

2.9 x 100 = 290

21 x 1 +

*x*x 2 +

*y*x 3 + 18 x 4 + 17 x 5 = 178 + 2

*x*+ 3

*y*

= 290

290 = 178 + 2

*x*+ 3

*y*

112 = 2

*x*+ 3

*y*

*c)*

Since 44 =

*x*+

*y*and 112 = 2

*x*+ 3

*y*,

*y*= 112 - 44 x 2

= 112 - 88

= 24

*x*= 44 - 24

= 20

*x*= 20 and

*y*= 24

d)

Numbers 1 2 3 4 5

Frequency 21 20 24 18 17

From this diagram we can tell that the mode is 3.

The median number will be the mean of the 50th and the 51st number.

Numbers which are 1 are the numbers from 1 to 21.

Numbers which are 2 are the numbers from 22 to 42.

Numbers which are 3 are the numbers from 43 to 66.

As 50 and 51 are in the range of numbers which are 3,

the median is 3.

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