OCR MEI Further Statistics B AS 2019 June — Question 5 14 marks

Exam BoardOCR MEI
ModuleFurther Statistics B AS (Further Statistics B AS)
Year2019
SessionJune
Marks14
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicT-tests (unknown variance)
TypeSingle sample t-test
DifficultyStandard +0.3 This is a straightforward one-sample t-test with standard steps: calculate sample statistics (mean and SD from 8 values), interpret a given normality check, perform a one-tailed t-test at 5% level, and name the z-test alternative. All steps are routine applications of standard procedures with no novel problem-solving required, making it slightly easier than average for A-level Further Maths statistics.
Spec5.05c Hypothesis test: normal distribution for population mean5.06b Fit prescribed distribution: chi-squared test
5 A technician is investigating whether a batch of nylon 66 (a particular type of nylon) is contaminated by another type of nylon.
The average melting point of nylon 66 is \(264 ^ { \circ } \mathrm { C }\). However, if the batch is contaminated by the other type of nylon the melting point will be lower. The melting points, in \({ } ^ { \circ } \mathrm { C }\), of a random sample of 8 pieces of nylon from the batch are as follows. \(\begin{array} { l l l l l l l l } 262.7 & 265.0 & 264.1 & 261.7 & 262.9 & 263.5 & 261.3 & 262.6 \end{array}\)
  1. Find
    The technician produces a Normal probability plot and carries out a Kolmogorov-Smirnov test for these data as shown in Fig. 5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1db026a2-dffc-4877-b927-247fbf0e7a78-5_560_1358_982_246} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure}
  2. Comment on what the Normal probability plot and the \(p\)-value of the test suggest about the data.
  3. In this question you must show detailed reasoning. Carry out a suitable test at the \(5 \%\) significance level to investigate whether the batch appears to be contaminated with another type of nylon.
  4. Name an alternative test that could have been carried out if the population standard deviation had been known.
Question 5:
AnswerMarks Guidance
5(a) Sample mean = 262.975
Sample standard deviation = 1.213 (1.2127...)B1
B1
AnswerMarks
[2]BC
BC
AnswerMarks
(b)Normal probability plot is roughly straight
Very high p-value
Both suggest that the data may be Normally
AnswerMarks
distributedE1
E1
E1
AnswerMarks
[3]Dep on at least one previous E
markDo not allow ‘strong
correlation’
AnswerMarks
(c)H : μ = 264 H : μ < 264
0 1
Where μ is the population mean melting
temperature
262.975264
Test statistic is
1.213/ 8
= 2.391
Refer to t
7
Critical value (1-tailed) at 5% level is 1.895
2.391 < 1.895 so significant (reject H )
0
Sufficient evidence to suggest that the batch might
AnswerMarks
be contaminated with another type of nylonB1
B1
M1
A1
M1
A1
M1
A1
AnswerMarks
[8]Hypotheses in words only must
include “population”.
For definition in context.
FT their mean and/or sd
Must be t
7
AnswerMarks
Or 2.391 > 1.895No marks for Wilcoxon test
unless in part (b) candidate
says ‘not Normal’
If two-tailed test used can get
all marks except first B1 with
critical value = 2.365
If critical value = 2.365 but
one-tailed test allow Max
M1A0M1E0 for last 4 marks
AnswerMarks Guidance
(d)A test based on the Normal distribution E1
[1]
Question 5:
5 | (a) | Sample mean = 262.975
Sample standard deviation = 1.213 (1.2127...) | B1
B1
[2] | BC
BC
(b) | Normal probability plot is roughly straight
Very high p-value
Both suggest that the data may be Normally
distributed | E1
E1
E1
[3] | Dep on at least one previous E
mark | Do not allow ‘strong
correlation’
(c) | H : μ = 264 H : μ < 264
0 1
Where μ is the population mean melting
temperature
262.975264
Test statistic is
1.213/ 8
= 2.391
Refer to t
7
Critical value (1-tailed) at 5% level is 1.895
2.391 < 1.895 so significant (reject H )
0
Sufficient evidence to suggest that the batch might
be contaminated with another type of nylon | B1
B1
M1
A1
M1
A1
M1
A1
[8] | Hypotheses in words only must
include “population”.
For definition in context.
FT their mean and/or sd
Must be t
7
Or 2.391 > 1.895 | No marks for Wilcoxon test
unless in part (b) candidate
says ‘not Normal’
If two-tailed test used can get
all marks except first B1 with
critical value = 2.365
If critical value = 2.365 but
one-tailed test allow Max
M1A0M1E0 for last 4 marks
(d) | A test based on the Normal distribution | E1
[1]
5 A technician is investigating whether a batch of nylon 66 (a particular type of nylon) is contaminated by another type of nylon.\\
The average melting point of nylon 66 is $264 ^ { \circ } \mathrm { C }$. However, if the batch is contaminated by the other type of nylon the melting point will be lower. The melting points, in ${ } ^ { \circ } \mathrm { C }$, of a random sample of 8 pieces of nylon from the batch are as follows.\\
$\begin{array} { l l l l l l l l } 262.7 & 265.0 & 264.1 & 261.7 & 262.9 & 263.5 & 261.3 & 262.6 \end{array}$
\begin{enumerate}[label=(\alph*)]
\item Find

\begin{itemize}
  \item the sample mean,
  \item the sample standard deviation.
\end{itemize}

The technician produces a Normal probability plot and carries out a Kolmogorov-Smirnov test for these data as shown in Fig. 5.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{1db026a2-dffc-4877-b927-247fbf0e7a78-5_560_1358_982_246}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{center}
\end{figure}
\item Comment on what the Normal probability plot and the $p$-value of the test suggest about the data.
\item In this question you must show detailed reasoning.

Carry out a suitable test at the $5 \%$ significance level to investigate whether the batch appears to be contaminated with another type of nylon.
\item Name an alternative test that could have been carried out if the population standard deviation had been known.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Statistics B AS 2019 Q5 [14]}}
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