OCR MEI S2 2007 January — Question 4 18 marks

Exam BoardOCR MEI
ModuleS2 (Statistics 2)
Year2007
SessionJanuary
Marks18
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicChi-squared goodness of fit
TypeChi-squared test of independence
DifficultyStandard +0.3 This is a standard chi-squared test for independence question with routine calculations of expected frequencies, test statistics, and hypothesis testing. Part (i) requires a complete chi-squared test with a 2×2 table (straightforward), part (ii) extends to a 2×3 table with given test statistic (minimal calculation), and part (iii) asks for interpretation. All steps are textbook procedures with no novel problem-solving required, making it slightly easier than average.
Spec5.06a Chi-squared: contingency tables5.06b Fit prescribed distribution: chi-squared test
Two educational researchers are investigating the relationship between personal ambitions and home location of students. The researchers classify students into those whose main personal ambition is good academic results and those who have some other ambition. A random sample of 480 students is selected.
  1. One researcher summarises the data as follows.
    \multirow{2}{*}{Observed}Home location
    \cline{2-3}CityNon-city
    \multirow{2}{*}{Ambition}Good results102147
    \cline{2-3}Other75156
    Carry out a test at the 5\% significance level to examine whether there is any association between home location and ambition. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic. [9]
  2. The other researcher summarises the same data in a different way as follows.
    \multirow{2}{*}{Observed}Home location
    \cline{2-4}CityTownCountry
    \multirow{2}{*}{Ambition}Good results1028364
    \cline{2-4}Other756492
    1. Calculate the expected frequencies for both 'Country' cells. [2]
    2. The test statistic for these data is 10.94. Carry out a test at the 5\% level based on this table, using the same hypotheses as in part (i). [3]
    3. The table below gives the contribution of each cell to the test statistic. Discuss briefly how personal ambitions are related to home location. [2]
      \multirow{2}{*}{
      Contribution to the
      test statistic
      }      		Home location
      \cline{2-4}CityTownCountry
      \multirow{2}{*}{Ambition}Good results1.1290.5963.540
      \cline{2-4}Other1.2170.6433.816
  3. Comment briefly on whether the analysis in part (ii) means that the conclusion in part (i) is invalid. [2]
(i)
\(H_0:\) no association between ambition and home location;
\(H_1:\) some association between ambition and home location;
AnswerMarks Guidance
ObservedHome location
CityNon-city
AmbitionGood results 102
Other75 156
ExpectedHome location
CityNon-city
AmbitionGood results 91.82
Other85.18 145.82
Contribution to the test statisticHome location
CityNon-city
AmbitionGood results 1.129
Other1.217 0.711
\(X^2 = 3.716\), Refer to \(\chi_1^2\)
Critical value at 5% level = 3.841
Result is not significant
There is insufficient evidence to conclude that there is any association between home location and ambition.
AnswerMarks Guidance
NB if \(H_0, H_1\) reversed, or 'correlation' mentioned, do not award first B1or final B1 or final E1B1 in context; M1 A1 for attempt at expected values; M1 for valid attempt at (O-E)²/E; A1CAO for \(X^2\); B1 for 1 dof SOI; B1 CAO for cv; B1 dep on attempt at cv; E1 conclusion in context 8
(ii)(A)
Expected Country, Results = \(249 \times 156 / 480 = 80.93\)
AnswerMarks Guidance
Expected Country, Other = \(231 \times 156 / 480 = 75.08\)B1; B1 2
(ii)(B)
Refer to \(\chi_2^2\)
Critical value at 5% level = 5.991
Result is significant
AnswerMarks Guidance
There is evidence to conclude that there is association between home location and ambition.B1 for 2 dof SOI; B1 CAO for cv; E1 for conclusion in context 3
(ii)(C)
AnswerMarks Guidance
'Country' students are much less likely than city or town to have 'Results' as their main ambition. Low contributions show that city and town students do not appear to differ markedly in their ambitions.E1 for correct obs" for 'Country'; E1 for additional correct observation (must refer to contributions) 2
(iii)
AnswerMarks Guidance
Conclusion in (i) is valid if only categorizing home location into city and non-city. However if non-city is subdivided into town and country this additional subdivision gives the data more precision and allows the relationship in part (ii) (C) to be revealed.E1; E1 2
Total: 18 marks
## (i)
$H_0:$ no association between ambition and home location;
$H_1:$ some association between ambition and home location;

| Observed | Home location | | |
|----------|---|---|
| | City | Non-city |
| Ambition | Good results | 102 | 147 |
| | Other | 75 | 156 |

| Expected | Home location | | |
|----------|---|---|
| | City | Non-city |
| Ambition | Good results | 91.82 | 157.18 |
| | Other | 85.18 | 145.82 |

| Contribution to the test statistic | Home location | | |
|----------|---|---|
| | City | Non-city |
| Ambition | Good results | 1.129 | 0.659 |
| | Other | 1.217 | 0.711 |

$X^2 = 3.716$, Refer to $\chi_1^2$

Critical value at 5% level = 3.841

Result is not significant

There is insufficient evidence to conclude that there is any association between home location and ambition.

NB if $H_0, H_1$ reversed, or 'correlation' mentioned, do not award first B1or final B1 or final E1 | B1 in context; M1 A1 for attempt at expected values; M1 for valid attempt at (O-E)²/E; A1CAO for $X^2$; B1 for 1 dof SOI; B1 CAO for cv; B1 dep on attempt at cv; E1 conclusion in context | 8

## (ii)(A)
Expected Country, Results = $249 \times 156 / 480 = 80.93$

Expected Country, Other = $231 \times 156 / 480 = 75.08$ | B1; B1 | 2

## (ii)(B)
Refer to $\chi_2^2$

Critical value at 5% level = 5.991

Result is significant

There is evidence to conclude that there is association between home location and ambition. | B1 for 2 dof SOI; B1 CAO for cv; E1 for conclusion in context | 3

## (ii)(C)
'Country' students are much less likely than city or town to have 'Results' as their main ambition. Low contributions show that city and town students do not appear to differ markedly in their ambitions. | E1 for correct obs" for 'Country'; E1 for additional correct observation (must refer to contributions) | 2

## (iii)
Conclusion in (i) is valid if only categorizing home location into city and non-city. However if non-city is subdivided into town and country this additional subdivision gives the data more precision and allows the relationship in part (ii) (C) to be revealed. | E1; E1 | 2

---

**Total: 18 marks**
Two educational researchers are investigating the relationship between personal ambitions and home location of students. The researchers classify students into those whose main personal ambition is good academic results and those who have some other ambition. A random sample of 480 students is selected.

\begin{enumerate}[label=(\roman*)]
\item One researcher summarises the data as follows.

\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\multirow{2}{*}{\textbf{Observed}} & \multicolumn{2}{|c|}{\textbf{Home location}} \\
\cline{2-3}
& City & Non-city \\
\hline
\multirow{2}{*}{Ambition} & Good results & 102 & 147 \\
\cline{2-3}
& Other & 75 & 156 \\
\hline
\end{tabular}
\end{center}

Carry out a test at the 5\% significance level to examine whether there is any association between home location and ambition. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic. [9]

\item The other researcher summarises the same data in a different way as follows.

\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{\textbf{Observed}} & \multicolumn{3}{|c|}{\textbf{Home location}} \\
\cline{2-4}
& City & Town & Country \\
\hline
\multirow{2}{*}{Ambition} & Good results & 102 & 83 & 64 \\
\cline{2-4}
& Other & 75 & 64 & 92 \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}[label=(\Alph*)]
\item Calculate the expected frequencies for both 'Country' cells. [2]

\item The test statistic for these data is 10.94. Carry out a test at the 5\% level based on this table, using the same hypotheses as in part (i). [3]

\item The table below gives the contribution of each cell to the test statistic. Discuss briefly how personal ambitions are related to home location. [2]

\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{\begin{tabular}{c}\textbf{Contribution to the}\\\textbf{test statistic}\end{tabular}} & \multicolumn{3}{|c|}{\textbf{Home location}} \\
\cline{2-4}
& City & Town & Country \\
\hline
\multirow{2}{*}{Ambition} & Good results & 1.129 & 0.596 & 3.540 \\
\cline{2-4}
& Other & 1.217 & 0.643 & 3.816 \\
\hline
\end{tabular}
\end{center}
\end{enumerate}

\item Comment briefly on whether the analysis in part (ii) means that the conclusion in part (i) is invalid. [2]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI S2 2007 Q4 [18]}}
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