WJEC Further Unit 2 Specimen — Question 7 12 marks

Exam BoardWJEC
ModuleFurther Unit 2 (Further Unit 2)
SessionSpecimen
Marks12
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Mark schemeDownload PDF ↗
TopicChi-squared test of independence
TypeStandard 2×3 contingency table
DifficultyModerate -0.5 This is a standard chi-squared test of independence question requiring routine calculations: stating hypotheses (textbook recall), computing one expected value using the standard formula (row total × column total / grand total), and calculating two chi-squared contributions using (O-E)²/E. All steps are mechanical applications of well-practiced formulas with no problem-solving or conceptual insight required. Slightly easier than average due to its purely computational nature, though the multi-part structure and context prevent it from being trivial.
Spec5.06a Chi-squared: contingency tables
The Pew Research Center's Internet Project offers scholars access to raw data sets from their research. One of the Pew Research Center's projects was on teenagers and technology. A random sample of American families was selected to complete a questionnaire. For each of their children, between and including the ages of 13 and 15, parents of these families were asked: Do you know your child's password for any of [his/her] social media accounts? Responses to this question were received from 493 families. The table below provides a summary of their responses.
Age (years)\multirow{2}{*}{Total}
\cline{2-4} Parent know password131415
Yes767567218
No66103106275
Total142178173493
  1. A test for significance is to be undertaken to see whether there is an association between whether a parent knows any of their child's social media passwords and the age of the child.
    1. Clearly state the null and alternative hypotheses.
    2. Obtain the expected value that is missing from the table below, indicating clearly how it is calculated from the data values given in the table above.
    Expected values:
    Age (years)
    \cline{2-4} Parent knows password131415
    Yes62.7978.7176.50
    No99.2996.50
    1. Obtain the two chi-squared contributions that are missing from the table below.
    Chi-squared contributions:
    Age (years)
    \cline{2-4} Parent knows password131415
    Yes0.1751.180
    No2.2030.935
    The following output was obtained from the statistical package that was used to undertake the analysis: Pearson chi-squared (2) = 7.409 \quad \(p\)-value = 0.0305
    1. Indicate how the degrees of freedom have been calculated for the chi-squared statistic.
    2. Interpret the output obtained from the statistical test in terms of the initial hypotheses. [10]
  2. Comment on the nature of the association observed, based on the contributions to the test statistic calculated in (a). [2]
AnswerMarks Guidance
(a)(i) \(H_0\): There is no association between parents knowing their child's social media passwords and age of child. \(H_1\): There is an association between parents knowing their child's social media passwords and age of child.B1 AO3
(ii) Expected values:
AnswerMarks Guidance
Parent knows passwordAge (years)
1314 15
Yes62.79 78.71
No79.21 99.29
\(142 \times \frac{275}{493} = 79.21\) OR \(275 \times \frac{142}{493} = 79.21\)M1, A1 AO2, AO1
(iii) Chi-squared contributions:
AnswerMarks Guidance
Parent knows passwordAge (years)
1314 15
Yes2.779 0.175
No2.203 0.139
M1, A1AO2, AO1 M1A0 for one correct \(\chi^2\) contribution; FT 'their observed values'
(iv) 2 degrees of freedom from \((3 - 1) \times (2 - 1)\)B1 AO1
(v) Since p-value < 0.05, Reject \(H_0\); Strong evidence to suggest there is an association between parents knowing their child's social media passwords and age.B1, B1, B1, B1 AO1, AO2, AO2, AO3
(b) Largest contribution for 13-year-olds especially for yes. It seems more parents than expected know passwords for their 13-year-old children.E1, E1 AO2, AO2 
**(a)(i)** $H_0$: There is no association between parents knowing their child's social media passwords and age of child. $H_1$: There is an association between parents knowing their child's social media passwords and age of child. | B1 | AO3 | Or $H_0$: Parents knowing their child's social media passwords is independent of age; $H_1$: Parents knowing their child's social media passwords is not independent of age

**(ii)** Expected values:

| Parent knows password | Age (years) | | |
|---|---|---|---|
| | 13 | 14 | 15 |
| Yes | 62.79 | 78.71 | 76.50 |
| No | 79.21 | 99.29 | 96.50 |

$142 \times \frac{275}{493} = 79.21$ OR $275 \times \frac{142}{493} = 79.21$ | M1, A1 | AO2, AO1 | Or any equivalent correct method

**(iii)** Chi-squared contributions:

| Parent knows password | Age (years) | | |
|---|---|---|---|
| | 13 | 14 | 15 |
| Yes | 2.779 | 0.175 | 1.180 |
| No | 2.203 | 0.139 | 0.935 |

| M1, A1 | AO2, AO1 | M1A0 for one correct $\chi^2$ contribution; FT 'their observed values'

**(iv)** 2 degrees of freedom from $(3 - 1) \times (2 - 1)$ | B1 | AO1

**(v)** Since p-value < 0.05, Reject $H_0$; Strong evidence to suggest there is an association between parents knowing their child's social media passwords and age. | B1, B1, B1, B1 | AO1, AO2, AO2, AO3 | B1 for < 0.05; B1 for Reject $H_0$; B1 for strong evidence; B1 for relating back to hypothesis

**(b)** Largest contribution for 13-year-olds especially for yes. It seems more parents than expected know passwords for their 13-year-old children. | E1, E1 | AO2, AO2 | [12]
The Pew Research Center's Internet Project offers scholars access to raw data sets from their research.

One of the Pew Research Center's projects was on teenagers and technology. A random sample of American families was selected to complete a questionnaire. For each of their children, between and including the ages of 13 and 15, parents of these families were asked:

\textbf{Do you know your child's password for any of [his/her] social media accounts?}

Responses to this question were received from 493 families. The table below provides a summary of their responses.

\begin{tabular}{|l|c|c|c|c|}
\hline
& \multicolumn{3}{|c|}{Age (years)} & \multirow{2}{*}{Total} \\
\cline{2-4}
Parent know password & 13 & 14 & 15 & \\
\hline
Yes & 76 & 75 & 67 & 218 \\
\hline
No & 66 & 103 & 106 & 275 \\
\hline
Total & 142 & 178 & 173 & 493 \\
\hline
\end{tabular}

\begin{enumerate}[label=(\alph*)]
\item A test for significance is to be undertaken to see whether there is an association between whether a parent knows any of their child's social media passwords and the age of the child.
\begin{enumerate}[label=(\roman*)]
\item Clearly state the null and alternative hypotheses.
\item Obtain the expected value that is missing from the table below, indicating clearly how it is calculated from the data values given in the table above.
\end{enumerate}

Expected values:
\begin{tabular}{|l|c|c|c|}
\hline
& \multicolumn{3}{|c|}{Age (years)} \\
\cline{2-4}
Parent knows password & 13 & 14 & 15 \\
\hline
Yes & 62.79 & 78.71 & 76.50 \\
\hline
No & & 99.29 & 96.50 \\
\hline
\end{tabular}

\begin{enumerate}[label=(\roman*)]
\setcounter{enumii}{2}
\item Obtain the two chi-squared contributions that are missing from the table below.
\end{enumerate}

Chi-squared contributions:
\begin{tabular}{|l|c|c|c|}
\hline
& \multicolumn{3}{|c|}{Age (years)} \\
\cline{2-4}
Parent knows password & 13 & 14 & 15 \\
\hline
Yes & & 0.175 & 1.180 \\
\hline
No & 2.203 & & 0.935 \\
\hline
\end{tabular}

The following output was obtained from the statistical package that was used to undertake the analysis:

Pearson chi-squared (2) = 7.409 \quad $p$-value = 0.0305

\begin{enumerate}[label=(\roman*)]
\setcounter{enumii}{3}
\item Indicate how the degrees of freedom have been calculated for the chi-squared statistic.
\item Interpret the output obtained from the statistical test in terms of the initial hypotheses. [10]
\end{enumerate}

\item Comment on the nature of the association observed, based on the contributions to the test statistic calculated in (a). [2]
\end{enumerate}

\hfill \mbox{\textit{WJEC Further Unit 2  Q7 [12]}}
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