IBDP Maths AI: Topic: SL 4.1: Concepts of population, sample: IB style Questions SL Paper 1

Question

State which of the following sets of data are discrete.

(i) Speeds of cars travelling along a road.

(ii) Numbers of members in families.

(iii) Maximum daily temperatures.

(iv) Heights of people in a class measured to the nearest cm.

(v) Daily intake of protein by members of a sporting team.[2]

a.

The boxplot below shows the statistics for a set of data.

For this data set write down the value of

(i) the median

(ii) the upper quartile

(iii) the minimum value present[3]

b.

Write down three different integers whose mean is 10.[1]

c.
Answer/Explanation

Markscheme

(ii) and (iv) are discrete.     (A1)(A1)

Award (A1)(A0) for both correct and one incorrect.

Award (A1)(A0) for one correct and two incorrect.

Otherwise, (A0)(A0).     (C2)[2 marks]

a.

(i) Median = 10     (A1)

(ii) Q3 = 12     (A1)

(iii) Min value = 1 (±0.2)     (A1)     (C3)[3 marks]

b.

Any three different integers whose mean is 10 e.g. 9, 10, 11.     (A1)     (C1)[1 mark]

c.

Question

A survey was conducted of the number of bedrooms in \(208\) randomly chosen houses. The results are shown in the following table.

State whether the data is discrete or continuous.[1]

a.

Write down the mean number of bedrooms per house.[2]

b.

Write down the standard deviation of the number of bedrooms per house.[1]

c.

Find how many houses have a number of bedrooms greater than one standard deviation above the mean.[2]

d.
Answer/Explanation

Markscheme

Discrete     (A1)     (C1)[1 mark]

a.

For attempting to find \(\sum fx/\sum f\)     (M1)

\(2.73\)     (A1)     (C2)

Note: for (b) and (c), if both mean and standard deviation given to 2 significant figures.
Award (C1)(C0)(AP) for \(2.7\). Award (A1)(ft) for \(1.3\) ((AP) already deducted).
[2 marks]

b.

\(1.34\)     (A1)     (C1)

Note: for (b) and (c), if both mean and standard deviation given to 2 significant figures.
Award (C1)(C0)(AP) for \(2.7\). Award (A1)(ft) for \(1.3\) ((AP) already deducted).
[1 mark]

c.

Attempt to find their mean \( + \) their standard deviation (can be implied)     (M1)

\(23\), (ft) their mean and standard deviation.     (A1)(ft)     (C2)[2 marks]

d.

Question

The following table shows the number of errors per page in a 100 page document.

State whether the data is discrete, continuous or neither.[1]

a.

Find the mean number of errors per page.[2]

b.

Find the median number of errors per page.[2]

c.

Write down the mode.[1]

d.
Answer/Explanation

Markscheme

Discrete     (A1)     (C1)[1 mark]

a.

\(\frac{{0 + 24 + 40 + 51 + 44}}{{100}} = \frac{{159}}{{100}} = 1.59\)     (M1)(A1)     (C2)

Notes: Award (M1) for correctly substituted formula.

Award (M1)(A1) for 1 or 2 if 1.59 is seen.

Award (M0)(A0) for 1 or 2 seen with no working.[2 marks]

b.

1     (M1)(A1)     (C2)

Note: Award (M1) for attempt to order raw data (if frequency table not used) or (M1) for indicating halfway between 50th and 51st result or (M1) for 50th percentile seen.[2 marks]

c.

0     (A1)     (C1)[1 mark]

d.

Question

A survey was carried out on a road to determine the number of passengers in each car (excluding the driver). The table shows the results of the survey.

State whether the data is discrete or continuous.[1]

a.

Write down the mode.[1]

b.

Use your graphic display calculator to find

(i)     the mean number of passengers per car;

(ii)     the median number of passengers per car;

(iii)     the standard deviation.[4]

c.
Answer/Explanation

Markscheme

discrete     (A1)     (C1)[1 mark]

a.
\(0\)     (A1)     (C1)[1 mark]
b.

(i)     \(1.47\)   \((1.46666…)\)     (A2)

Note: Award (M1) for \(\frac{{176}}{{120}}\) seen.

     Accept \(1\) or \(2\) as a final answer if \(1.4666…\) or \(1.47\) seen.

(ii)     \(1.5\)     (A1)

(iii)     \(1.25\)   \((1.25122…)\)     (A1)     (C4)[4 marks]

c.

Question

In a particular week, the number of eggs laid by each hen on a farm was counted. The results are summarized in the following table.

State whether these data are discrete or continuous.[1]

a.

Write down

(i)     the number of hens on the farm;

(ii)     the modal number of eggs laid.[2]

b.

Calculate

(i)     the mean number of eggs laid;

(ii)     the standard deviation.[3]

c.
Answer/Explanation

Markscheme

discrete     (A1)     (C1)

a.

(i)     60     (A1)

(ii)     5     (A1)     (C2)

b.

(i)     \(\frac{{1 \times 4 + 2 \times 7 + 3 \times 12 \ldots }}{{60}}\)     (M1)

Notes: Award (M1) for an attempt to substitute into the “mean of a set of data” formula, with at least three correct terms in the numerator.

Denominator must be 60.

Follow through from part (b)(i), only if work is seen.

\( = 4.03{\text{ }}(4.03333 \ldots )\)     (A1)

Notes: Award at most (M1)(A0) for an answer of 4 but only if working seen.

(ii)     \(1.54{\text{ }}(1.53803 \ldots )\)     (A1)     (C3)

c.

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