Calculate the mean test grade of the students;

Calculate the standard deviation.

Find the median test grade of the students.

Find the interquartile range.

Find the probability that this student scored a grade 5 or higher.

Given that the first student chosen at random scored a grade 5 or higher, find the probability that both students scored a grade 6.

Calculate the probability that a student chosen at random spent at least 90 minutes preparing for the test.

Calculate the expected number of students that spent at least 90 minutes preparing for the test.

\(\frac{{1(1) + 3(2) + 7(3) + 13(4) + 11(5) + 10(6) + 5(7)}}{{50}} = \frac{{230}}{{50}}\)     (M1)

Note:     Award (M1) for correct substitution into mean formula.

\( = 4.6\)     (A1)     (G2)

[2 marks]

\(1.46{\text{ }}(1.45602 \ldots )\)     (G1)

[1 mark]

5     (A1)

[1 mark]

\(6 - 4\)     (M1)

Note:     Award (M1) for 6 and 4 seen.

\( = 2\)     (A1)     (G2)

[2 marks]

\(\frac{{11 + 10 + 5}}{{50}}\)     (M1)

Note:     Award (M1) for \(11 + 10 + 5\) seen.

\( = \frac{{26}}{{50}}{\text{ }}\left( {\frac{{13}}{{25}},{\text{ }}0.52,{\text{ }}52\% } \right)\)     (A1)     (G2)

[2 marks]

\(\frac{{10}}{{{\text{their }}26}} \times \frac{9}{{49}}\)     (M1)(M1)

Note:     Award (M1) for \(\frac{{10}}{{{\text{their }}26}}\) seen, (M1) for multiplying their first probability by \(\frac{9}{{49}}\).

OR

\(\frac{{\frac{{10}}{{50}} \times \frac{9}{{49}}}}{{\frac{{26}}{{50}}}}\)

Note:     Award (M1) for \({\frac{{10}}{{50}} \times \frac{9}{{49}}}\) seen, (M1) for dividing their first probability by \(\frac{{{\text{their }}26}}{{50}}\).

\( = \frac{{45}}{{637}}{\text{ (}}0.0706,{\text{ }}0.0706436 \ldots ,{\text{ }}7.06436 \ldots \% )\)     (A1)(ft)     (G3)

Note:     Follow through from part (d).

[3 marks]

\({\text{P}}(X \geqslant 90)\)     (M1)

OR

     (M1)

Note:     Award (M1) for a diagram showing the correct shaded region \(( > 0.5)\).

\(0.773{\text{ }}(0.773372 \ldots ){\text{ }}0.773{\text{ }}(0.773372 \ldots ,{\text{ }}77.3372 \ldots \% )\)     (A1)     (G2)

[2 marks]

\(0.773372 \ldots  \times 50\)     (M1)

\( = 38.7{\text{ }}(38.6686 \ldots )\)     (A1)(ft)     (G2)

Note:     Follow through from part (f)(i).

[2 marks]