Standard +0.3 This is a straightforward chi-squared test question requiring basic calculations from a partially completed spreadsheet. Students need to recall the validity condition (expected frequencies ≥5), then calculate missing expected frequencies using row/column totals and compute chi-squared contributions using the standard formula (O-E)²/E. All required formulas are standard and the multi-step arithmetic is routine for Further Statistics students.
5 A researcher is investigating whether there is any relationship between the overall performance of a student at GCSE and their grade in A Level Mathematics. Their A Level Mathematics grade is classified as A* or A, B, C or lower, and their overall performance at GCSE is classified as Low, Middle, High.
Data are collected for a sample of 80 students in a particular area. The researcher carries out a chi-squared test. The screenshot below shows part of a spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted.
1
A
B
C
D
E
\multirow{2}{*}{
}
Observed frequency
A* or A
B
C or lower
Totals
3
Low
6
13
9
28
4
Middle
10
6
8
24
5
High
15
10
3
28
6
Totals
31
29
20
80
7
8
\multirow{2}{*}{}
9
A* or A
B
C or lower
10
Low
10.85
11
Middle
9.30
12
High
10.85
13
\multirow[b]{2}{*}{Contribution to the test statistic}
14
15
A* or A
B
C or lower
16
Low
2.1680
0.8002
0.5714
17
Middle
0.0527
0.8379
0.6667
18
High
1.5873
2.2857
2.2857
19
State what needs to be known about the sample for the test to be valid.
For the remainder of this question, you should assume that the test is valid.
Determine the missing values in each of the following cells.
Carry out a hypothesis test at the \(10 \%\) significance level to investigate whether there is any association between level of performance at GCSE and A Level Mathematics grade.
Discuss briefly what the data suggest about A Level Mathematics grade for different levels of performance at GCSE.
State one disadvantage of using a 10\% significance level rather than a 5\% significance level in a hypothesis test.
There is evidence (at the 10% level) to suggest that
there is association between A-Level Mathematics
Answer
Marks
grade and GCSE performance
B1
B1
B1
B1
M1
A1
Answer
Marks
[6]
3.4
3.3
1.1
1.1
2.2b
Answer
Marks
3.5a
Allow H : ‘A-Level Mathematics grade and GCSE
0
performance are independent’
H : ‘A-Level Mathematics grade and GCSE performance
1
are not independent’
Allow 8.972, 8.97 FT their C18
Or 24 ( 8 .9 7 ) 0 .9 3 8 =
For correct comparison of their test statistic with their
critical value leading to consistent conclusion
Or 0.938 > 0.90 so reject H
0
FT their test statistic but A0 if wrong critical value used or
if hypotheses reversed
Answer
Marks
Guidance
5
(d)
For Low performance the high contribution (of
2.1680) suggests fewer get A* or A than would be
expected (if there were no association).
For Middle performance the low contributions suggest
that things are as would be expected (if there were no
association).
For High performance the high contributions (of
1.5873 and 2.2857) suggest that more than expected
get A* or A and that fewer than expected get C or
lower (than would be expected if there were no
Answer
Marks
association).
E1
E1
E1
Answer
Marks
[3]
2.2b
3.5a
Answer
Marks
3.5a
If contributions not mentioned then award max SC B1 B1
for interpretations that are otherwise correct.
Answer
Marks
Guidance
5
(e)
The null hypothesis is more likely to be wrongly
rejected
E1
[1]
2.2a
The alternative hypothesis is more likely to be wrongly
accepted.
Or there is a greater chance of a false positive.
Question 5:
5 | (a) | The sample must be random | B1
[1] | 1.2
5 | (b) | C11: 8.70
(1 0 − 1 0 .1 5 ) 2
C18:
1 0 .1 5
= 0.0022 | B1
M1
A1
[3] | 1.1
1.1
1.1 | Allow 8.7
cao Answer correct to 4 d.p.
5 | (c) | H : No association between A-Level Mathematics
0
grade and GCSE performance
H : Some association between A-Level Mathematics
1
grade and GCSE performance
Degrees of freedom = 4
Test statistic = 2.1680 + 0.8002 + ... + 2.2857 = 8.9721
Critical value = 7.779
8.972 > 7.779 so reject H
0
There is evidence (at the 10% level) to suggest that
there is association between A-Level Mathematics
grade and GCSE performance | B1
B1
B1
B1
M1
A1
[6] | 3.4
3.3
1.1
1.1
2.2b
3.5a | Allow H : ‘A-Level Mathematics grade and GCSE
0
performance are independent’
H : ‘A-Level Mathematics grade and GCSE performance
1
are not independent’
Allow 8.972, 8.97 FT their C18
Or 24 ( 8 .9 7 ) 0 .9 3 8 =
For correct comparison of their test statistic with their
critical value leading to consistent conclusion
Or 0.938 > 0.90 so reject H
0
FT their test statistic but A0 if wrong critical value used or
if hypotheses reversed
5 | (d) | For Low performance the high contribution (of
2.1680) suggests fewer get A* or A than would be
expected (if there were no association).
For Middle performance the low contributions suggest
that things are as would be expected (if there were no
association).
For High performance the high contributions (of
1.5873 and 2.2857) suggest that more than expected
get A* or A and that fewer than expected get C or
lower (than would be expected if there were no
association). | E1
E1
E1
[3] | 2.2b
3.5a
3.5a | If contributions not mentioned then award max SC B1 B1
for interpretations that are otherwise correct.
5 | (e) | The null hypothesis is more likely to be wrongly
rejected | E1
[1] | 2.2a | The alternative hypothesis is more likely to be wrongly
accepted.
Or there is a greater chance of a false positive.
5 A researcher is investigating whether there is any relationship between the overall performance of a student at GCSE and their grade in A Level Mathematics. Their A Level Mathematics grade is classified as A* or A, B, C or lower, and their overall performance at GCSE is classified as Low, Middle, High.
Data are collected for a sample of 80 students in a particular area. The researcher carries out a chi-squared test. The screenshot below shows part of a spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
1 & A & B & C & D & E \\
\hline
\multicolumn{2}{|l|}{\multirow{2}{*}{\begin{tabular}{l}
1 \\
2 \\
\end{tabular}}} & \multicolumn{3}{|c|}{Observed frequency} & \\
\hline
& & A* or A & B & C or lower & Totals \\
\hline
3 & Low & 6 & 13 & 9 & 28 \\
\hline
4 & Middle & 10 & 6 & 8 & 24 \\
\hline
5 & High & 15 & 10 & 3 & 28 \\
\hline
6 & Totals & 31 & 29 & 20 & 80 \\
\hline
7 & \multicolumn{5}{|c|}{} \\
\hline
8 & \multirow{2}{*}{} & \multicolumn{3}{|c|}{} & \\
\hline
9 & & A* or A & B & C or lower & \\
\hline
10 & Low & 10.85 & & & \\
\hline
11 & Middle & 9.30 & & & \\
\hline
12 & High & 10.85 & \multicolumn{2}{|c|}{} & \\
\hline
13 & \multicolumn{4}{|r|}{\multirow[b]{2}{*}{Contribution to the test statistic}} & \\
\hline
14 & & & & & \\
\hline
15 & & A* or A & B & C or lower & \\
\hline
16 & Low & 2.1680 & 0.8002 & 0.5714 & \\
\hline
17 & Middle & 0.0527 & 0.8379 & 0.6667 & \\
\hline
18 & High & 1.5873 & \multicolumn{2}{|c|}{\begin{tabular}{l}
2.2857 \\
2.2857 \\
\end{tabular}} & \\
\hline
19 & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item State what needs to be known about the sample for the test to be valid.
For the remainder of this question, you should assume that the test is valid.
\item Determine the missing values in each of the following cells.
\begin{itemize}
\item C11
\item C18
\item In this question you must show detailed reasoning.
\end{itemize}
Carry out a hypothesis test at the $10 \%$ significance level to investigate whether there is any association between level of performance at GCSE and A Level Mathematics grade.
\item Discuss briefly what the data suggest about A Level Mathematics grade for different levels of performance at GCSE.
\item State one disadvantage of using a 10\% significance level rather than a 5\% significance level in a hypothesis test.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics A AS 2022 Q5 [14]}}