State Ludmila’s null hypothesis.

Write down the number of degrees of freedom.

Find the expected frequency for the females studying Spanish.

Use your graphic display calculator to find the \({\chi ^2}\) test statistic for this data.

State whether Ludmila accepts the null hypothesis. Give a reason for your answer.

Draw a labelled scatter diagram for this data. Use a scale of \(2{\text{ cm}}\) to represent \(10{\text{ marks}}\) on the \(x\)-axis (\(S\)) and \(10{\text{ marks}}\) on the \(y\)-axis (\(F\)).

Use your graphic calculator to find

(i)     \({\bar S}\), the mean of \(S\) ;

(ii)    \({\bar F}\), the mean of \(F\) .

Plot the point \({\text{M}}(\bar S{\text{, }}\bar F)\) on your scatter diagram.

Use your graphic display calculator to find the equation of the regression line of \(F\) on \(S\) .

Draw the regression line on your scatter diagram.

Carletta’s mark on the Science examination was \(44\). She did not sit the French examination.

Estimate Carletta’s mark for the French examination.

Monique’s mark on the Science examination was 85. She did not sit the French examination. Her French teacher wants to use the regression line to estimate Monique’s mark.

State whether the mark obtained from the regression line for Monique’s French examination is reliable. Justify your answer.

\({{\text{H}}_0}:\) Choice of language is independent of gender.     (A1)

Notes: Do not accept “not related” or “not correlated”.

[1 mark]

\(2\)     (A1)

[1 mark]

\(\frac{{50 \times 69}}{{150}} = 23\)     (M1)(A1)(G2)

Notes: Award (M1) for correct substituted formula, (A1) for \(23\).

[2 marks]

\({\chi ^2} = 4.77\)     (G2)

Notes: If answer is incorrect, award (M1) for correct substitution in the correct formula (all terms).

[2 marks]

Accept \({{\text{H}}_0}\) since

\({\chi ^2}_{calc} < {\chi ^2}_{crit}(5.99)\) or \(p\)-value \((0.0923) > 0.05\)     (R1)(A1)(ft)

Notes: Do not award (R0)(A1). Follow through from their (d) and (b).

Award (A1) for correct scale and labels.

Award (A3) for all seven points plotted correctly, (A2) for 5 or 6 points plotted correctly, (A1) for 3 or 4 points plotted correctly.

(A4)

[4 marks]

(i)     \({\bar S}= 49.9\),     (G1)

(ii)    \({\bar F} = 47.3\)     (G1)

[2 marks]

\({\text{M}}(49.9{\text{, }}47.3)\) plotted on scatter diagram     (A1)(ft)

Notes: Follow through from (a) and (b).

[1 mark]

\(F = - 0.619S + 78.2\)     (G1)(G1)

Notes: Award (G1) for \( - 0.619S\), (G1) for \(78.2\). If the answer is not in the form of an equation, award (G1)(G0). Accept \(y = - 0.619x + 78.2\) .

OR

(F - 47.3 = - 0.619(S - 49.9))     (G1)(G1)

Note: Award (G1) for \( - 0.619\), (G1) for the coordinates of their midpoint used. Follow through from their values in (b).

[2 marks]

line drawn on scatter diagram     (A1)(ft)(A1)(ft)

Notes: The drawn line must be straight for any marks to be awarded. Award (A1)(ft) passing through their M plotted in (c). Award (A1)(ft) for correct \(y\)-intercept. Follow through from their \(y\)-intercept found in (d).

[2 marks]

\(F = - 0.619 \times 44 + 78.2\)     (M1)

\(= 51.0\) (allow \(51\) or \(50.9\))     (A1)(ft)(G2)(ft)

Note: Follow through from their equation.

OR

(M1) any indication of an acceptable graphical method.     (M1)

(A1)(ft) from their regression line.     (A1)(ft)(G2)(ft)

[2 marks]

not reliable     (A1)

Monique’s score in Science is outside the range of scores used to create the regression line.     (R1)

Note: Do not award (A1)(R0).

[2 marks]

Part A: Chi-square test

This question part was answered well by most candidates. The null hypothesis and degrees of freedom were mostly correct. Some candidates offered a conclusion supported by good justifications, but others still showed lack of the necessary knowledge to do that. Some responses to part d) incurred an accuracy penalty for not adhering to the required accuracy level.

Part A: Chi-square test

This question part was answered well by most candidates. The null hypothesis and degrees of freedom were mostly correct. Some candidates offered a conclusion supported by good justifications, but others still showed lack of the necessary knowledge to do that. Some responses to part d) incurred an accuracy penalty for not adhering to the required accuracy level.

Part A: Chi-square test

This question part was answered well by most candidates. The null hypothesis and degrees of freedom were mostly correct. Some candidates offered a conclusion supported by good justifications, but others still showed lack of the necessary knowledge to do that. Some responses to part d) incurred an accuracy penalty for not adhering to the required accuracy level.

Part A: Chi-square test

This question part was answered well by most candidates. The null hypothesis and degrees of freedom were mostly correct. Some candidates offered a conclusion supported by good justifications, but others still showed lack of the necessary knowledge to do that. Some responses to part d) incurred an accuracy penalty for not adhering to the required accuracy level.

Part A: Chi-square test

This question part was answered well by most candidates. The null hypothesis and degrees of freedom were mostly correct. Some candidates offered a conclusion supported by good justifications, but others still showed lack of the necessary knowledge to do that. Some responses to part d) incurred an accuracy penalty for not adhering to the required accuracy level.

Part B: Scatter plot and Regression line

Many candidates reversed the axes in a), but the points were mostly plotted well. The values of the coefficients of the equation of the regression line \(y = ax + b\) were often given not to the required 3 significant figure accuracy, and incurred a penalty. The regression line was often drawn not passing through point M and the y-intercept. The responses to the last part of the question were particularly weak, and many candidates were not able to offer a satisfactory reason to support their conclusion.

Part B: Scatter plot and Regression line

Many candidates reversed the axes in a), but the points were mostly plotted well. The values of the coefficients of the equation of the regression line \(y = ax + b\) were often given not to the required 3 significant figure accuracy, and incurred a penalty. The regression line was often drawn not passing through point M and the y-intercept. The responses to the last part of the question were particularly weak, and many candidates were not able to offer a satisfactory reason to support their conclusion.

Part B: Scatter plot and Regression line

Many candidates reversed the axes in a), but the points were mostly plotted well. The values of the coefficients of the equation of the regression line \(y = ax + b\) were often given not to the required 3 significant figure accuracy, and incurred a penalty. The regression line was often drawn not passing through point M and the y-intercept. The responses to the last part of the question were particularly weak, and many candidates were not able to offer a satisfactory reason to support their conclusion.

Part B: Scatter plot and Regression line

Many candidates reversed the axes in a), but the points were mostly plotted well. The values of the coefficients of the equation of the regression line \(y = ax + b\) were often given not to the required 3 significant figure accuracy, and incurred a penalty. The regression line was often drawn not passing through point M and the y-intercept. The responses to the last part of the question were particularly weak, and many candidates were not able to offer a satisfactory reason to support their conclusion.

Part B: Scatter plot and Regression line

Many candidates reversed the axes in a), but the points were mostly plotted well. The values of the coefficients of the equation of the regression line \(y = ax + b\) were often given not to the required 3 significant figure accuracy, and incurred a penalty. The regression line was often drawn not passing through point M and the y-intercept. The responses to the last part of the question were particularly weak, and many candidates were not able to offer a satisfactory reason to support their conclusion.

Part B: Scatter plot and Regression line

Many candidates reversed the axes in a), but the points were mostly plotted well. The values of the coefficients of the equation of the regression line \(y = ax + b\) were often given not to the required 3 significant figure accuracy, and incurred a penalty. The regression line was often drawn not passing through point M and the y-intercept. The responses to the last part of the question were particularly weak, and many candidates were not able to offer a satisfactory reason to support their conclusion.

Part B: Scatter plot and Regression line

Many candidates reversed the axes in a), but the points were mostly plotted well. The values of the coefficients of the equation of the regression line \(y = ax + b\) were often given not to the required 3 significant figure accuracy, and incurred a penalty. The regression line was often drawn not passing through point M and the y-intercept. The responses to the last part of the question were particularly weak, and many candidates were not able to offer a satisfactory reason to support their conclusion.