Question 5 - A-Level Maths

OCR MEI Further Statistics A AS 2018 June — Question 5 13 marks

Exam Board	OCR MEI
Module	Further Statistics A AS (Further Statistics A AS)
Year	2018
Session	June
Marks	13
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Chi-squared goodness of fit
Type	Spreadsheet-based chi-squared test
Difficulty	Standard +0.3 This is a straightforward chi-squared test question with spreadsheet context. Students must identify sample size (trivial), state standard hypotheses (recall), calculate missing expected frequencies and contributions using given formulas (routine application), and complete the test using tables. The spreadsheet format adds mild complexity, but all calculations follow standard procedures with no novel problem-solving required. Slightly easier than average due to most values being provided.
Spec	5.06a Chi-squared: contingency tables

5 A random sample of workers for a large company were asked whether they are smokers, ex-smokers or have never smoked. The responses were classified by the type of worker: Managerial, Production line or Administrative. Fig. 5 is a screenshot showing part of the spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]

	A	B	C	D	E	F
1	Observed frequencies
2		Smoker	Ex-smoker	Never smoked	Totals
3	Managerial	2	10	5	17
4	Production line	18	15	21	54
5	Administrative	13	6	14	33
6	Totals	33	31	40	104
7
8		Expected frequencies
9		5.3942	5.0673	6.5385
10		17.1346		20.7692
11		10.4712	9.8365	12.6923
12
13		Contributions to the test statistic
14		2.1358	4.8017	0.3620
15		0.0437		0.0026
16			1.4964	0.1347
17				Test statistic	9.66
18

\captionsetup{labelformat=empty} \caption{Fig. 5}

\end{table}

(A) State the sample size.
(B) State the null and alternative hypotheses for a test to investigate whether there is any association between type of worker and smoking status.
Showing your calculations, find the missing values in each of the following cells.

Show mark scheme Show mark scheme source

Question 5:

Answer	Marks	Guidance
5	(i)	A
[1]	2.2a
B	H : No association between smoking status and type

0 of worker.

H : Some association between smoking status and

1

Answer	Marks	Guidance
type of worker.	B1
[1]	1.2	Hypotheses in context – may be

written in terms of independence

Answer	Marks
(ii)	C11 =

C15 =

Answer	Marks
B16 =	M1

A1

M1

A1

Answer	Marks
[4]	1.1a

1.1 1.1a

Answer	Marks
1.1	(NB This can be calculated by

subtraction)

M1 for applied at least once.

Answer	Marks
A1 for both correct.	Accept awrt 16.1 www

For C15 accept answers

between 0.0746 and 0.0752

For B16 accept awrt 0.611

Answer	Marks
(iii)	Degrees of freedom = 4 seen

Critical value = 7.779

Test statistic = 9.66

9.66 > 7.779 significant/reject H

0 The evidence suggests that there is some association

Answer	Marks
between smoking status and type of worker.	B1

B1

M1

A1

Answer	Marks
[4]	3.3

1.1 2.2b

Answer	Marks
3.5a	No further marks from here if wrong.

M1 for sensible comparison with

7.779, leading to a conclusion

For non-assertive correct conclusion

in terms of H .

Answer	Marks
1	See additional notes.
(iv)	For ‘Managerial’ (large) contributions (of 2.1358

and 4.8017) show that fewer than expected are

smokers and more are ex-smokers (and numbers of

never smoked are as expected).

For ‘Production line’ (small) contributions show the

numbers are as expected.

For ‘Administrative’ the (large) contribution (of

1.4964) suggests that fewer than expected are ex-

smokers. (The numbers of smokers and never

Answer	Marks
smoked are as expected).	E1

E1

Answer	Marks
[3]	3.5a

3.5a

Answer	Marks
3.5a	For correct comments relating to both

of the large contributions.

For correct comment relating to the

Answer	Marks
large contribution.	Must compare observed and

expected frequencies. No

marks for general comments

or for statements about

likelihood.

Max 1/3 if no mention of

contributions.

No ISW for additional

incorrect comments

Question 5:
5 | (i) | A | Sample size = 104 | B1
[1] | 2.2a
B | H : No association between smoking status and type
0
of worker.
H : Some association between smoking status and
1
type of worker. | B1
[1] | 1.2 | Hypotheses in context – may be
written in terms of independence
(ii) | C11 =
C15 =
B16 = | M1
A1
M1
A1
[4] | 1.1a
1.1
1.1a
1.1 | (NB This can be calculated by
subtraction)
M1 for applied at least once.
A1 for both correct. | Accept awrt 16.1 www
For C15 accept answers
between 0.0746 and 0.0752
For B16 accept awrt 0.611
(iii) | Degrees of freedom = 4 seen
Critical value = 7.779
Test statistic = 9.66
9.66 > 7.779 significant/reject H
0
The evidence suggests that there is some association
between smoking status and type of worker. | B1
B1
M1
A1
[4] | 3.3
1.1
2.2b
3.5a | No further marks from here if wrong.
M1 for sensible comparison with
7.779, leading to a conclusion
For non-assertive correct conclusion
in terms of H .
1 | See additional notes.
(iv) | For ‘Managerial’ (large) contributions (of 2.1358
and 4.8017) show that fewer than expected are
smokers and more are ex-smokers (and numbers of
never smoked are as expected).
For ‘Production line’ (small) contributions show the
numbers are as expected.
For ‘Administrative’ the (large) contribution (of
1.4964) suggests that fewer than expected are ex-
smokers. (The numbers of smokers and never
smoked are as expected). | E1
E1
E1
[3] | 3.5a
3.5a
3.5a | For correct comments relating to both
of the large contributions.
For correct comment relating to the
large contribution. | Must compare observed and
expected frequencies. No
marks for general comments
or for statements about
likelihood.
Max 1/3 if no mention of
contributions.
No ISW for additional
incorrect comments

Show LaTeX source

5 A random sample of workers for a large company were asked whether they are smokers, ex-smokers or have never smoked. The responses were classified by the type of worker: Managerial, Production line or Administrative.

Fig. 5 is a screenshot showing part of the spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted.

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
 & A & B & C & D & E & F \\
\hline
1 & \multicolumn{5}{|c|}{Observed frequencies} &  \\
\hline
2 &  & Smoker & Ex-smoker & Never smoked & Totals &  \\
\hline
3 & Managerial & 2 & 10 & 5 & 17 &  \\
\hline
4 & Production line & 18 & 15 & 21 & 54 &  \\
\hline
5 & Administrative & 13 & 6 & 14 & 33 &  \\
\hline
6 & Totals & 33 & 31 & 40 & 104 &  \\
\hline
7 & \multicolumn{6}{|c|}{} \\
\hline
8 &  & \multicolumn{3}{|c|}{Expected frequencies} &  &  \\
\hline
9 &  & 5.3942 & 5.0673 & 6.5385 &  &  \\
\hline
10 &  & 17.1346 &  & 20.7692 &  &  \\
\hline
11 &  & 10.4712 & 9.8365 & 12.6923 &  &  \\
\hline
12 &  & \multicolumn{3}{|c|}{} &  &  \\
\hline
13 &  & \multicolumn{3}{|c|}{Contributions to the test statistic} &  &  \\
\hline
14 &  & 2.1358 & 4.8017 & 0.3620 &  &  \\
\hline
15 &  & 0.0437 &  & 0.0026 &  &  \\
\hline
16 &  &  & 1.4964 & 0.1347 &  &  \\
\hline
17 &  &  &  & Test statistic & 9.66 &  \\
\hline
18 &  &  &  &  &  &  \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{center}
\end{table}
\begin{enumerate}[label=(\roman*)]
\item (A) State the sample size.\\
(B) State the null and alternative hypotheses for a test to investigate whether there is any association between type of worker and smoking status.
\item Showing your calculations, find the missing values in each of the following cells.

\begin{itemize}
  \item C 10
  \item C 15
  \item B 16
\item Complete the hypothesis test at the $10 \%$ level of significance.
\item Discuss briefly what the data suggest about smoking status for different types of workers. You should make a comment for each type of worker.
\end{itemize}
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Statistics A AS 2018 Q5 [13]}}

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