October 2020 Paper 2 Q8

OCR MEICurrent spec12 marksIncludes samplingData Processing Normal Distribution

8 Rosella is carrying out an investigation into the age at which adults retire from work in the city where she lives. She collects a sample of size 50, ensuring this comprises of 25 randomly selected retired men and 25 randomly selected retired women.

(a) State the name of the sampling method she uses. [1]

Fig. 8.1 shows the data she obtains in a frequency table and Fig. 8.2 shows these data displayed in a histogram.

Age in years at retirement	45 –	50 –	55 –	60 –	65 –	70 –	75 – 80
Frequency density	0.4	1.8	2.4	2.2	1.8	1.2	0.2

Fig. 8.1

Fig. 8.2: histogram of frequency density against age in years, bars from 45 to 80 with heights 0.4, 1.8, 2.4, 2.2, 1.8, 1.2, 0.2 — Fig. 8.2

(b) How many people in the sample are aged between 50 and 55? [1]

Rosella obtains a list of the names of all 4960 people who have retired in the city during the previous month.

systematic sampling such that every item on the list could be selected,
simple random sampling.

[4]

Rosella collects two simple random samples, one of size 200 and one of size 500, from her list. The histograms in Fig. 8.3 show the data from the sample of size 200 on the left and the data from the sample of 500 on the right.

Fig. 8.3: two histograms of frequency density against age in years, for the sample of size 200 (ages 45 to 85) and the sample of size 500 (ages 40 to 80); both are roughly symmetrical and bell-shaped, peaking between 55 and 65 — Fig. 8.3

(d) With reference to the histograms shown in Fig. 8.2 and Fig. 8.3, explain why it appears reasonable to model the age of retirement in this city using the Normal distribution. [1]

Summary statistics for the sample of 500 are shown in Fig. 8.4.

Statistics
n	500
Mean	60.0515
σ	6.5717
s	6.5783
Σx	30025.7601
Σx²	1824686.322
Min	36.0793
Q1	55.2573
Median	59.9202
Q3	64.4239
Max	81.742

Fig. 8.4

(e) Use an appropriate Normal model based on the information in Fig. 8.4 to estimate the number of people aged over 65 who retired in the city in the previous month. [4]

(f) Identify a limitation in using this model to predict the number of people aged over 65 retiring in the following month. [1]

Scheme	Marks	AO
Quota sampling	B1	1.2
	[1]

Scheme	Marks	AO
9	B1	1.1
	[1]

Mark scheme (c)

Scheme	Marks	AO
Systematic: select every 24^th number on the list	M1	2.4
start randomly between \(n = 1\) and \(n \geqslant 184\) and stop when 200 have been selected (if \(n > 184\), must cycle through list)	A1	1.1
Simple random sampling: assign each item in the list a unique number (eg from 1 to 4960)	E1	2.4
generate random numbers until a sample of 200 has been selected soi	E1	1.1
	[4]

Notes

M1: alternatively select every 25^th number on the list
alternatively select every 24.8^th value on list, rounding as appropriate

A1: alternatively start randomly between \(n = 1\) and \(n \geqslant 25\), and cycle through the list again, stopping when 200 have been selected
alternatively start randomly with any value on list. Cycle through the list repeatedly until 200 items have been selected

E1: alternatively allow any process where each member of the population has an equal chance of being selected

E1: allow any process where each possible sample has an equal chance of being selected

Mark scheme (d)

Scheme	Marks	AO
as the size of the sample increases, the shape of the distribution appears more and more “Normal” oe	B1	2.4
	[1]

Notes

B1: must refer to shape and closer to Normal shape for larger sample

Mark scheme (e)

Scheme	Marks	AO
use of \(\mathrm{N}(60.0515,\ 6.5783^2)\) to find \(\mathrm{P}(X > 65)\)	M1	3.3
awrt 0.23	A1	3.4
\(4960 \times\) their 0.226	M1	3.1b
1121 or 1120 or 1119	A1	3.5a
	[4]

Notes

M1: condone use of \(6.5717^2\) or parameters rounded to 3 sf
M0 if continuity correction used or eg \(\mathrm{P}(X > 64)\) found

Scheme	Marks	AO
eg there may be seasonal fluctuations such as teachers retiring in August	B1	3.5b
	[1]