Standard +0.3 This is a straightforward chi-squared test question requiring calculation of expected frequencies using the standard formula (row total × column total / grand total) and computing one contribution to the test statistic using (O-E)²/E. These are routine procedures taught in any statistics course with no conceptual difficulty or novel insight required.
10 A scientist is researching dietary fat intake and cholesterol level. A random sample of 60 people is selected and their dietary fat intakes and cholesterol levels are measured. Dietary fat intakes are classified as low, medium and high, and cholesterol levels are classified as normal and high.
The scientist decides to carry out a chi-squared test to investigate whether there is any association between dietary fat intake and cholesterol level. Tables \(\mathbf { 1 0 . 1 }\) and \(\mathbf { 1 0 . 2 }\) show the data and some of the expected frequencies for the test.
\begin{table}[h]
Complete the table of expected frequencies in the Printed Answer Booklet.
Determine the contribution to the chi-squared test statistic for people with normal cholesterol level and high dietary fat intake, giving your answer to \(\mathbf { 4 }\) decimal places.
The contributions to the chi-squared test statistic for the remaining categories are shown in Table 10.3.
\begin{table}[h]
H : some association between dietary fat intake and
1
cholesterol level
Test statistic = 6.4508
Degrees of freedom = 2
Critical value = 5.991
6.4508 > 5.991 so reject H
0
There is sufficient evidence to suggest that there is some
association between dietary fat intake and cholesterol
Answer
Marks
level.
B1
B1
B1
B1
M1
A1
Answer
Marks
[6]
2.5
1.1
3.3
1.1
2.2b
Answer
Marks
3.5a
For both Allow independent/not independent. Do NOT allow
relationship in place of association
BC Accept awrt 6.45
Can be implied by correct critical value
Allow χ2 (6.4508)=0.9603
2
With the above allow comparison with 95% so reject for H M1
0
Must have correct test statistic and critical value
If hypotheses wrong way around allow MAX B0B1B1B1M0A0
Answer
Marks
Guidance
10
(d)
For low dietary fat, the contribution of 1.0563 shows
that more people than expected have normal
cholesterol level whereas the contribution of 1.2071
shows that fewer than expected have high cholesterol
level.
For medium dietary fat, the numbers are as expected.
For high dietary fat, the contribution of 1.8240 shows
that fewer people than expected have normal
cholesterol level whereas the contribution of 2.0846
shows that more than expected have high cholesterol
Answer
Marks
level
B3,2,1
,0
Answer
Marks
[3]
2.2b
3.5a
Answer
Marks
3.5a
B3 for 5 correct contribution comments
B2 for 4 correct contribution comments
B1 for two correct comments
Allow slightly fewer with higher and slightly more with normal
Question 10:
10 | (a) | 32×12 28×12
Low-normal= =6.4, Low-high= =5.6
60 60
32×31
Medium-normal= =16.5333
60
28×31
Medium-high= =14.4667
60 | B1
B1
[2] | 3.4
1.1 | B1 for any one correct
B1 for the other 3 (by subtraction)
10 | (b) | (5−9.0667)2
9.0667
= 1.8240 | M1
A1
[2] | 1.1
1.1 | (O−E)2
For
E
10 | (c) | DR
H : no association between dietary fat intake and
0
cholesterol level
H : some association between dietary fat intake and
1
cholesterol level
Test statistic = 6.4508
Degrees of freedom = 2
Critical value = 5.991
6.4508 > 5.991 so reject H
0
There is sufficient evidence to suggest that there is some
association between dietary fat intake and cholesterol
level. | B1
B1
B1
B1
M1
A1
[6] | 2.5
1.1
3.3
1.1
2.2b
3.5a | For both Allow independent/not independent. Do NOT allow
relationship in place of association
BC Accept awrt 6.45
Can be implied by correct critical value
Allow χ2 (6.4508)=0.9603
2
With the above allow comparison with 95% so reject for H M1
0
Must have correct test statistic and critical value
If hypotheses wrong way around allow MAX B0B1B1B1M0A0
10 | (d) | For low dietary fat, the contribution of 1.0563 shows
that more people than expected have normal
cholesterol level whereas the contribution of 1.2071
shows that fewer than expected have high cholesterol
level.
For medium dietary fat, the numbers are as expected.
For high dietary fat, the contribution of 1.8240 shows
that fewer people than expected have normal
cholesterol level whereas the contribution of 2.0846
shows that more than expected have high cholesterol
level | B3,2,1
,0
[3] | 2.2b
3.5a
3.5a | B3 for 5 correct contribution comments
B2 for 4 correct contribution comments
B1 for two correct comments
Allow slightly fewer with higher and slightly more with normal
10 A scientist is researching dietary fat intake and cholesterol level. A random sample of 60 people is selected and their dietary fat intakes and cholesterol levels are measured. Dietary fat intakes are classified as low, medium and high, and cholesterol levels are classified as normal and high.
The scientist decides to carry out a chi-squared test to investigate whether there is any association between dietary fat intake and cholesterol level. Tables $\mathbf { 1 0 . 1 }$ and $\mathbf { 1 0 . 2 }$ show the data and some of the expected frequencies for the test.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{3}{|c|}{Dietary fat intake} & \\
\hline
& & Low & Medium & High & Total \\
\hline
\multirow{2}{*}{Cholesterol level} & Normal & 9 & 18 & 5 & 32 \\
\hline
& High & 3 & 13 & 12 & 28 \\
\hline
& Total & 12 & 31 & 17 & 60 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 10.1}
\end{center}
\end{table}
\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | l | c | c | c | }
\hline
\multicolumn{2}{|c|}{Expected frequency} & \multicolumn{3}{c|}{Dietary fat intake} \\
\cline { 3 - 5 }
& Low & Medium & High & \\
\hline
\multirow{2}{*}{\begin{tabular}{ c }
Cholesterol \\
level \\
\end{tabular}} & Normal & & & 9.0667 \\
\cline { 2 - 5 }
& High & & & 7.9333 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 10.2}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Complete the table of expected frequencies in the Printed Answer Booklet.
\item Determine the contribution to the chi-squared test statistic for people with normal cholesterol level and high dietary fat intake, giving your answer to $\mathbf { 4 }$ decimal places.
The contributions to the chi-squared test statistic for the remaining categories are shown in Table 10.3.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | l | c | c | c | }
\hline
\multicolumn{2}{|c|}{\begin{tabular}{ c }
Contribution to the \\
test statistic \\
\end{tabular}} & \multicolumn{3}{|c|}{Dietary fat intake} \\
\cline { 2 - 5 }
& Low & Medium & High & \\
\hline
\multirow{2}{*}{\begin{tabular}{ c }
Cholesterol \\
level \\
\end{tabular}} & Normal & 1.0563 & 0.1301 & \\
\cline { 2 - 5 }
& High & 1.2071 & 0.1487 & 2.0846 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 10.3}
\end{center}
\end{table}
\item In this question you must show detailed reasoning.
Carry out the test at the 5\% significance level.
\item For each level of dietary fat intake, give a brief interpretation of what the data suggest about the level of cholesterol.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2022 Q10 [13]}}