S3 June 2015 Q3

EdexcelOld spec11 marksConfidence Intervals

3. A nursery has 16 staff and 40 children on its records. In preparation for an outing the manager needs an estimate of the mean weight of the people on its records and decides to take a stratified sample of size 14.

(a) Describe how this stratified sample should be taken. (3)

The weights, \(x\) kg, of each of the 14 people selected are summarised as

\[\sum x = 437 \text{ and } \sum x^2 = 26983\]

(b) Find unbiased estimates of the mean and the variance of the weights of all the people on the nursery’s records. (4)

The estimates of the standard error of the mean for the staff and for the children are 5.11 and 1.10 respectively.

(d) Comment on these values with reference to your answer to part (c) and give a reason for any differences. (2)

Mark scheme (a)

Scheme	Marks
Label staff (from 1 – 16) and children (from 1 – 40)	B1
Use random numbers to select	B1
4 staff and 10 children	B1
	(3)

Notes

1st B1 for labelling/numbering/listing staff and children

2nd B1 for use of random numbers or “randomly select” in each group (may be implied)

3rd B1 for selecting the correct number of staff and children
e.g. randomly select 4 staff and 10 children scores 2nd and 3rd B marks since randomly selecting and the “each group” is implied,

Mark scheme (b)

Scheme	Marks
\(\bar{x} = \hat{\mu} = 31.2142\ldots\) awrt 31.2	B1
\(s^2 = \dfrac{26983 - 14 \times \text{"}31.2\ldots\text{"}^2}{13}\)	M1 A1ft
\(= 1026.33\ldots\) awrt 1030	A1
	(4)

Notes

B1 for awrt 31.2

M1 for a correct expression ft their \(\bar{x}\) and allow transcription error in \(\sum x^2\) e.g. 29683

1st A1ft for a fully correct expression ft their \(\bar{x}\) only

2nd A1 for awrt 1030

Mark scheme (c)

Scheme	Marks
\(\dfrac{\text{"}\sqrt{1026.33\ldots}\text{"}}{\sqrt{14}}\), \(= 8.562..\) awrt 8.56	M1, A1
	(2)

Notes

M1 for attempting \(\dfrac{\text{"their } s\text{"}}{\sqrt{14}}\) (must have 14)

A1 for awrt 8.56

Mark scheme (d)

Scheme	Marks
The variation within each stratum is quite small (o.e.)	B1
The difference in the means will be quite large, (so variations from the overall mean will be large giving a larger overall s.e.)	B1
	(2)
	(11 marks)

Notes

1st B1 for a suitable comment about variation (se) suggesting that variation (se) within strata is less than that overall

2nd B1 for a suitable reason about means, pointing out that the individuals’ weights will vary a lot from the overall mean and so overall s.e. will be higher.