Standard +0.3 This is a straightforward paired sample confidence interval question requiring interpretation of computer output and basic calculations. Parts (a)-(c) test understanding of paired design and CI interpretation (standard bookwork), while part (d) involves reverse-engineering mean and SD from a given CI using standard formulas. All techniques are routine for Further Statistics students with no novel problem-solving required.
7 A physiotherapist is investigating hand grip strength in adult women under 30 years old. She thinks that the grip strength of the dominant hand will be on average 2 kg higher than the grip strength of the non-dominant hand.
The physiotherapist selects a random sample of 12 adult women under 30 years old and measures the grip strength of each of their hands. She then uses software to produce a \(95 \%\) confidence interval for the mean difference in grip strength between the two hands (dominant minus nondominant), as shown in Fig. 7.
\begin{table}[h]
Explain why the physiotherapist used the same people for testing their dominant and nondominant grip strengths.
State any assumptions necessary in order to construct the confidence interval shown in Fig. 7.
Explain whether the confidence interval supports the physiotherapist's belief.
The physiotherapist then finds some data which have previously been collected on grip strength using a sample of 100 adult women. A 95\% confidence interval, based on this sample and calculated using a Normal distribution, for the mean difference in grip strength between the two hands (dominant minus non-dominant) is (1.94, 2.84).
The pairing will eliminate any differences in grip
strengths between different people and so will only
compare the grip strengths of the dominant and non-
Answer
Marks
dominant hands
E1
E1
Answer
Marks
[2]
2.2b
2.2b
Give 1 mark for any valid comment
For 2 marks must include pairing
Answer
Marks
Guidance
7
(b)
The parent population of differences must be Normally
distributed
E1
E1
Answer
Marks
[2]
1.1
1.2
For Normally distributed
For full answer including ‘differences’
Answer
Marks
Guidance
7
(c)
It does because the confidence interval contains 2
[1]
3.5a
7
(d)
(i)
SD
0.45=1.96×
100
Answer
Marks
Sample SD = 2.30 (2.2959…)
B1
M1
A1
Answer
Marks
[3]
1.1
3.1b
1.1
Answer
Marks
Guidance
7
(d)
(ii)
since only a random sample enables proper inference
Answer
Marks
about the population to be undertaken
B1
B1
Answer
Marks
[2]
3.2b
2.4
Do not allow eg a random sample is
less likely to be biased
Question 7:
7 | (a) | The pairing will eliminate any differences in grip
strengths between different people and so will only
compare the grip strengths of the dominant and non-
dominant hands | E1
E1
[2] | 2.2b
2.2b | Give 1 mark for any valid comment
For 2 marks must include pairing
7 | (b) | The parent population of differences must be Normally
distributed | E1
E1
[2] | 1.1
1.2 | For Normally distributed
For full answer including ‘differences’
7 | (c) | It does because the confidence interval contains 2 | E1
[1] | 3.5a
7 | (d) | (i) | Sample mean difference = 2.39
SD
0.45=1.96×
100
Sample SD = 2.30 (2.2959…) | B1
M1
A1
[3] | 1.1
3.1b
1.1
7 | (d) | (ii) | The sample must be random
since only a random sample enables proper inference
about the population to be undertaken | B1
B1
[2] | 3.2b
2.4 | Do not allow eg a random sample is
less likely to be biased
7 A physiotherapist is investigating hand grip strength in adult women under 30 years old. She thinks that the grip strength of the dominant hand will be on average 2 kg higher than the grip strength of the non-dominant hand.
The physiotherapist selects a random sample of 12 adult women under 30 years old and measures the grip strength of each of their hands. She then uses software to produce a $95 \%$ confidence interval for the mean difference in grip strength between the two hands (dominant minus nondominant), as shown in Fig. 7.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|}
\hline
\multicolumn{2}{|c|}{T Estimate of a Mean} \\
\hline
Confidence Level & 0.95 \\
\hline
\multicolumn{2}{|l|}{Sample} \\
\hline
\multicolumn{2}{|c|}{\multirow{3}{*}{\begin{tabular}{l}
Mean 2.79 \\
s □ 3.92 \\
N \\
\end{tabular}}} \\
\hline
& \\
\hline
& \\
\hline
\multicolumn{2}{|l|}{Result} \\
\hline
\multicolumn{2}{|l|}{T Estimate of a Mean} \\
\hline
Mean & 2.79 \\
\hline
s & 3.92 \\
\hline
SE & 1.13161 \\
\hline
N & 12 \\
\hline
df & 11 \\
\hline
Lower Limit & 0.29935 \\
\hline
Upper Limit & 5.28065 \\
\hline
Interval & $2.79 \pm 2.49065$ \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Explain why the physiotherapist used the same people for testing their dominant and nondominant grip strengths.
\item State any assumptions necessary in order to construct the confidence interval shown in Fig. 7.
\item Explain whether the confidence interval supports the physiotherapist's belief.
\item The physiotherapist then finds some data which have previously been collected on grip strength using a sample of 100 adult women. A 95\% confidence interval, based on this sample and calculated using a Normal distribution, for the mean difference in grip strength between the two hands (dominant minus non-dominant) is (1.94, 2.84).
\begin{enumerate}[label=(\roman*)]
\item For this sample, find
\begin{itemize}
\item the mean difference
\item the standard deviation of the differences.
\item Explain what you would need to know about the nature of this sample if you wanted to draw conclusions about the mean difference in grip strength in the population of adult women.
\end{itemize}
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2021 Q7 [10]}}