Represent this information on a Venn diagram.

Find the number of students who had none of the three choices for breakfast.

Write down the percentage of students who had fruit for breakfast.

Describe in words what the students in the set \(X \cap Y'\) had for breakfast.

Find the probability that a student had at least two of the three choices for breakfast.

Two students are chosen at random. Find the probability that both students had all three choices for breakfast.

Write down the null hypothesis, H₀, for this test.

Write down the number of degrees of freedom for this test.

Write down the critical value for this test.

Show that the expected number of females that have more than 5 meals per day is 13, correct to the nearest integer.

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

Decide whether H₀ must be accepted. Justify your answer.

(A1) for rectangle and three intersecting circles

(A1) for 10, (A1) for 8, 13 and 9, (A1) for 12, 15 and 17     (A4)

[4 marks]

100 – (9 +12 +13 +15 +10 +17 + 8) =16     (M1)(A1)(ft)(G2) 

Note: Follow through from their diagram.

[2 marks]

\(\frac{{51}}{{100}}(0.51)\)     (A1)(ft)

= 51%     (A1)(ft)(G2)

Note: Follow through from their diagram.

[2 marks]

Note: The following statements are correct. Please note that the connectives are important. It is not the same (had cereal) and (not bread) and (had cereal) or (not bread). The parentheses are not needed but are there to facilitate the understanding of the propositions.

(had cereal) and (did not have bread)

(had cereal only) or (had cereal and fruit only)

(had either cereal or (fruit and cereal)) and (did not have bread)     (A1)(A1)

Notes: If the statements are correct but the connectives are wrong then award at most (A1)(A0). For the statement (had only cereal) and (cereal and fruit) award (A1)(A0). For the statement had cereal and fruit award (A0)(A0).

[2 marks]

\(\frac{{54}}{{100}}(0.54,{\text{ 54 % }})\)     (A1)(ft)(A1)(ft)(G2)

Note: Award (A1)(ft) for numerator, follow through from their diagram, (A1)(ft) for denominator. Follow through from total or denominator used in part (c).

[2 marks]

\(\frac{{10}}{{100}} \times \frac{9}{{99}} = \frac{1}{{110}}(0.00909,{\text{ 0}}{\text{.909 % }})\)     (A1)(ft)(M1)(A1)(ft)(G2)

Notes: Award (A1)(ft) for their correct fractions, (M1) for multiplying two fractions, (A1)(ft) for their correct answer. Answer 0.009 with no working receives no marks. Follow through from denominator in parts (c) and (e) and from their diagram.

[3 marks]

H₀ : The (average) number of meals per day a student has and gender are independent     (A1)

Note: For “independent” accept “not associated” but do not accept “not related” or “not correlated”.

[1 mark]

2     (A1)

[1 mark]

5.99 (accept 5.991)     (A1)(ft)

Note: Follow through from their part (b).

[1 mark]

\(\frac{{28 \times 45}}{{100}} = 12.6 = 13\) or \(\frac{{28}}{{100}} \times \frac{{25}}{{100}} \times 100 = 12.6 = 13\)     (M1)(A1)(AG)

Notes: Award (M1) for correct formula and (A1) for correct substitution. Unrounded answer must be seen for the (A1) to be awarded.

[2 marks]

0.0321      (G2)

Note: For 0.032 award (G1)(G1)(AP). For 0.03 with no working award (G0).

[2 marks]

0.0321 < 5.99 or 0.984 > 0.05     (R1)

accept H₀     (A1)(ft)

Note: If reason is incorrect both marks are lost, do not award (R0)(A1).

[2 marks]

This question was in general well done. Candidates began the paper well by drawing the Venn diagram correctly. Some students omitted the rectangle (universal set) around the three circles. There were quite a few errors in (c) as some students forgot to convert their answers to percentages. Also describing in words what the students in \(X \cap Y'\) had for breakfast seemed to be difficult for the majority of the candidates. Some misread what Y was and even more missed the complement sign. However, the main problem in answering this question seemed to be the lack of knowledge in the relationship between set theory and logic (use of "and" and "or"). Combining probabilities caused problems to many. Common wrong answers were \(\frac{{10}}{{100}}\), \(\frac{{10}}{{100}} \times \frac{{10}}{{100}}\) or \(\frac{{10}}{{100}} + \frac{9}{{99}}\).

This question was in general well done. Candidates began the paper well by drawing the Venn diagram correctly. Some students omitted the rectangle (universal set) around the three circles. There were quite a few errors in (c) as some students forgot to convert their answers to percentages. Also describing in words what the students in \(X \cap Y'\) had for breakfast seemed to be difficult for the majority of the candidates. Some misread what Y was and even more missed the complement sign. However, the main problem in answering this question seemed to be the lack of knowledge in the relationship between set theory and logic (use of "and" and "or"). Combining probabilities caused problems to many. Common wrong answers were \(\frac{{10}}{{100}}\), \(\frac{{10}}{{100}} \times \frac{{10}}{{100}}\) or \(\frac{{10}}{{100}} + \frac{9}{{99}}\).

This question was in general well done. Candidates began the paper well by drawing the Venn diagram correctly. Some students omitted the rectangle (universal set) around the three circles. There were quite a few errors in (c) as some students forgot to convert their answers to percentages. Also describing in words what the students in \(X \cap Y'\) had for breakfast seemed to be difficult for the majority of the candidates. Some misread what Y was and even more missed the complement sign. However, the main problem in answering this question seemed to be the lack of knowledge in the relationship between set theory and logic (use of "and" and "or"). Combining probabilities caused problems to many. Common wrong answers were \(\frac{{10}}{{100}}\), \(\frac{{10}}{{100}} \times \frac{{10}}{{100}}\) or \(\frac{{10}}{{100}} + \frac{9}{{99}}\).

This question was in general well done. Candidates began the paper well by drawing the Venn diagram correctly. Some students omitted the rectangle (universal set) around the three circles. There were quite a few errors in (c) as some students forgot to convert their answers to percentages. Also describing in words what the students in \(X \cap Y'\) had for breakfast seemed to be difficult for the majority of the candidates. Some misread what Y was and even more missed the complement sign. However, the main problem in answering this question seemed to be the lack of knowledge in the relationship between set theory and logic (use of "and" and "or"). Combining probabilities caused problems to many. Common wrong answers were \(\frac{{10}}{{100}}\), \(\frac{{10}}{{100}} \times \frac{{10}}{{100}}\) or \(\frac{{10}}{{100}} + \frac{9}{{99}}\).

This question was in general well done. Candidates began the paper well by drawing the Venn diagram correctly. Some students omitted the rectangle (universal set) around the three circles. There were quite a few errors in (c) as some students forgot to convert their answers to percentages. Also describing in words what the students in \(X \cap Y'\) had for breakfast seemed to be difficult for the majority of the candidates. Some misread what Y was and even more missed the complement sign. However, the main problem in answering this question seemed to be the lack of knowledge in the relationship between set theory and logic (use of "and" and "or"). Combining probabilities caused problems to many. Common wrong answers were \(\frac{{10}}{{100}}\), \(\frac{{10}}{{100}} \times \frac{{10}}{{100}}\) or \(\frac{{10}}{{100}} + \frac{9}{{99}}\).

This question was in general well done. Candidates began the paper well by drawing the Venn diagram correctly. Some students omitted the rectangle (universal set) around the three circles. There were quite a few errors in (c) as some students forgot to convert their answers to percentages. Also describing in words what the students in \(X \cap Y'\) had for breakfast seemed to be difficult for the majority of the candidates. Some misread what Y was and even more missed the complement sign. However, the main problem in answering this question seemed to be the lack of knowledge in the relationship between set theory and logic (use of "and" and "or"). Combining probabilities caused problems to many. Common wrong answers were \(\frac{{10}}{{100}}\), \(\frac{{10}}{{100}} \times \frac{{10}}{{100}}\) or \(\frac{{10}}{{100}} + \frac{9}{{99}}\).

In general this part question was well answered. The major concerns of the examining team were the following:

• In (f) many students wrote down the expected values table (from the GDC) and highlighted the correct expected value, 12.6. As this is a "show that" question the use of the GDC is not expected and therefore no marks are awarded for this working. Instead it is expected the use of the formula for the expected value with the correct substitutions.
• In (e) surprisingly many candidates found the \(\chi _{calc}^2\) through the use of the formula. Unfortunately this led to some incorrect answers and also to a bad use of time. The question clearly says "use your graphic display calculator" and it is worth 2 marks therefore a student should not spend more than 2 minutes to answer this part question. Time management is essential in this type of examinations and the IB rule is one minute – one mark.

In general this part question was well answered. The major concerns of the examining team were the following:

• In (f) many students wrote down the expected values table (from the GDC) and highlighted the correct expected value, 12.6. As this is a "show that" question the use of the GDC is not expected and therefore no marks are awarded for this working. Instead it is expected the use of the formula for the expected value with the correct substitutions.
• In (e) surprisingly many candidates found the \(\chi _{calc}^2\) through the use of the formula. Unfortunately this led to some incorrect answers and also to a bad use of time. The question clearly says "use your graphic display calculator" and it is worth 2 marks therefore a student should not spend more than 2 minutes to answer this part question. Time management is essential in this type of examinations and the IB rule is one minute – one mark.

In general this part question was well answered. The major concerns of the examining team were the following:

• In (f) many students wrote down the expected values table (from the GDC) and highlighted the correct expected value, 12.6. As this is a "show that" question the use of the GDC is not expected and therefore no marks are awarded for this working. Instead it is expected the use of the formula for the expected value with the correct substitutions.
• In (e) surprisingly many candidates found the \(\chi _{calc}^2\) through the use of the formula. Unfortunately this led to some incorrect answers and also to a bad use of time. The question clearly says "use your graphic display calculator" and it is worth 2 marks therefore a student should not spend more than 2 minutes to answer this part question. Time management is essential in this type of examinations and the IB rule is one minute – one mark.

In general this part question was well answered. The major concerns of the examining team were the following:

• In (f) many students wrote down the expected values table (from the GDC) and highlighted the correct expected value, 12.6. As this is a "show that" question the use of the GDC is not expected and therefore no marks are awarded for this working. Instead it is expected the use of the formula for the expected value with the correct substitutions.
• In (e) surprisingly many candidates found the \(\chi _{calc}^2\) through the use of the formula. Unfortunately this led to some incorrect answers and also to a bad use of time. The question clearly says "use your graphic display calculator" and it is worth 2 marks therefore a student should not spend more than 2 minutes to answer this part question. Time management is essential in this type of examinations and the IB rule is one minute – one mark.

In general this part question was well answered. The major concerns of the examining team were the following:

• In (f) many students wrote down the expected values table (from the GDC) and highlighted the correct expected value, 12.6. As this is a "show that" question the use of the GDC is not expected and therefore no marks are awarded for this working. Instead it is expected the use of the formula for the expected value with the correct substitutions.
• In (e) surprisingly many candidates found the \(\chi _{calc}^2\) through the use of the formula. Unfortunately this led to some incorrect answers and also to a bad use of time. The question clearly says "use your graphic display calculator" and it is worth 2 marks therefore a student should not spend more than 2 minutes to answer this part question. Time management is essential in this type of examinations and the IB rule is one minute – one mark.

In general this part question was well answered. The major concerns of the examining team were the following:

• In (f) many students wrote down the expected values table (from the GDC) and highlighted the correct expected value, 12.6. As this is a "show that" question the use of the GDC is not expected and therefore no marks are awarded for this working. Instead it is expected the use of the formula for the expected value with the correct substitutions.
• In (e) surprisingly many candidates found the \(\chi _{calc}^2\) through the use of the formula. Unfortunately this led to some incorrect answers and also to a bad use of time. The question clearly says "use your graphic display calculator" and it is worth 2 marks therefore a student should not spend more than 2 minutes to answer this part question. Time management is essential in this type of examinations and the IB rule is one minute – one mark.