11 Descriptive Statistics Test [73 marks]

1a.

The time, in minutes, that students in a school spend on their homework per day is presented inthe following box-and-whisker diagram.

Time, in minutes, students spend on their homework per day

Find

(i) the longest amount of time spent on homework per day;

(ii) the interquartile range.

Markscheme(i) 300 (minutes) OR 5 hours (A1)

Note: If answer given in hours, the unit must be seen.

(ii) 220 – 100 (M1)

Notes: Award (M1) for the two quartiles seen.

= 120 (minutes) OR 2 hours (A1) (C3)

Note: If answer given in hours, the unit must be seen.

1b. State the statistical term corresponding to the value of 140 minutes.

Markschememedian (time spent on homework per day) (A1) (C1)

Note: Do not accept middle or medium etc.

1c. Find the percentage of students who spend

(i) between 100 and 140 minutes per day on their homework;

(ii) more than 100 minutes per day on their homework.

[3 marks]

[1 mark]

[2 marks]

Markscheme(i) 25 (A1)

(ii) 75 (A1) (C2)

2a.

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

State whether these data are discrete or continuous.

Markschemediscrete (A1) (C1)

2b. Write down

(i) the number of hens on the farm;

(ii) the modal number of eggs laid.

Markscheme(i) 60 (A1)

(ii) 5 (A1) (C2)

2c. Calculate

(i) the mean number of eggs laid;

(ii) the standard deviation.

[1 mark]

[2 marks]

[3 marks]

Markscheme(i) (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.

(A1)

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

(ii) (A1) (C3)

1×4+2×7+3×12…60

= 4.03 (4.03333…)

1.54 (1.53803…)

3a.

The distribution of rainfall in a town over 80 days is displayed on the following box-and-whisker diagram.

Write down the median rainfall.

Markscheme (A1) (C1)

[1 mark]

43 (mm)

3b. Write down the minimum rainfall.

Markscheme (A1) (C1)

[1 mark]

10 (mm)

3c. Find the interquartile range.

[1 mark]

[1 mark]

[2 marks]

Markscheme (A1)

(A1) (C2)

Note: Award (A1) for identifying correct quartiles, (A1) for correct subtraction of thequartiles.

[2 marks]

48 − 20

= 28

3d. Write down the number of days the rainfall will be

(i) between 43 mm and 48 mm;

(ii) between 20 mm and 59 mm.

Markscheme(i) 20 (days) (A1)

(ii) 60 (days) (A1) (C2)

[2 marks]

4a.

The IB grades attained by a group of students are listed as follows.

Find the median grade.

Markscheme (M1)

Note: Award (M1) for correct ordered set.

(A1) (C2)

6 4 5 3 7 3 5 4 2 5

2 3 3 4 4 5 5 5 6 7

(Median =) 4.5

4b. Calculate the interquartile range.

[2 marks]

[2 marks]

[2 marks]

Markscheme (M1)

Note: Award (M1) for correct quartiles seen.

(A1) (C2)

5 − 3

= 2

4c. Find the probability that a student chosen at random from the group scored at least agrade .

Markscheme (A2) (C2)

4

(0.7, 70%)710

5a.

Two groups of 40 students were asked how many books they have read in the last two months.The results for the first group are shown in the following table.

The quartiles for these results are 3 and 5.

Write down the value of the median for these results.

Markscheme (A1)(C1)4

5b. Draw a box-and-whisker diagram for these results on the following grid.

[2 marks]

[1 mark]

[3 marks]

Markscheme

(A1)(ft)(A1)(A1)

(C3)

Notes: Award (A1)(ft) for correct median, (A1) for correct quartiles and box, (A1) forendpoints 2 and 8 joined by a straight line that does not cross the box. Follow through fromtheir median from part (a).

5c. The results for the second group of 40 students are shown in the following box-and-whisker diagram.

Estimate the number of students in the second group who have read at least 6 books.

Markscheme (M1)

Notes: Award (M1) for OR .

(A1) (C2)

40 × 0.25

40 × 25% 40 − 40 × 75%

10

6a.

The number of passengers in the first ten carriages of a train is listed below.

6 , 8 , 6 , 3 , 8 , 4 , 8 , 5 , p , p

The mean number of passengers per carriage is 5.6.

Calculate the value of p.

[2 marks]

[2 marks]

Markscheme (M1)

Notes: Accept equivalent forms. Award (M1) for correct substitutions in mean formula.

4 (A1) (C2)

= 5.648+2p

10

6b. Find the median number of passengers per carriage.

MarkschemeCorrectly rearranging the list with their p (M1)

5.5 (A1)(ft) (C2)

Note: Follow through from their value of p in part (a).

6c. If the passengers in the eleventh carriage are also included, the mean number ofpassengers per carriage increases to 6.0.

Determine the number of passengers in the eleventh carriage of the train.

Markscheme (M1)

Notes: Accept equivalent forms. Award (M1) for correct substitutions in mean formula.

OR

(M1)

10 (A1)(ft) (C2)

Note: Follow through from their answer to part (a).

= 6.056+x

11

48+2× their part (a)+x

11

The table shows the number of bicycles owned by 50 households.

[2 marks]

[2 marks]

7a.

The table shows the number of bicycles owned by 50 households.

Write down the value of

(i) t ;

(ii) w .

Markscheme(i) 8 (A1)

(ii) 48 (A1)(ft) (C2)

Note: Follow through from their t, even if no workings seen as long as w < 50.

7b. Indicate with a tick ( ) whether the following statements are True or False.

Markscheme

(A1)(A1)(A1)(A1) (C4)

The cumulative frequency graph shows the heights, in cm, of 80 young trees.

[2 marks]

[4 marks]

8a.

The cumulative frequency graph shows the heights, in cm, of 80 young trees.

Write down the median height of the trees.

Markscheme54 (cm) (A1) (C1)

8b. Write down the 75 percentile.

Markscheme58 (A1) (C1)

th

8c. Find the interquartile range.

[1 mark]

[1 mark]

[2 marks]

Markscheme58, 45 (A1)

Note: Award (A1) for 45 seen.

58 − 45

= 13 (A1)(ft) (C2)

Note: Follow through from part (b).

8d. Estimate the number of trees that are more than 40 cm in height.

Markscheme80 − 14 (M1)

Note: Award (M1) for 14 seen.

= 66 (A1) (C2)

9a.

A survey was conducted to determine the length of time,

, in minutes, people took to drink their coffee in a café. The information is shown in the following grouped frequency

table.

Write down the total number of people who were surveyed.

Markscheme (A1)

[1 mark]

t

60

9b. Write down the mid-interval value for the group.10 < t ⩽ 15

[2 marks]

[1 mark]

[1 mark]

Markscheme (A1)

[1 mark]

12.5

9c. Find an estimate of the mean time people took to drink their coffee.

Markscheme (M1)

Note: Award (M1) for an attempt to substitute their mid-interval values (consistent with theiranswer to part (b)) into the formula for the mean.

Award (M1) where a table is constructed with their (consistent) mid-interval values listedalong with the frequencies.

(A1)(ft)(G2)

Note: Follow through from their answer to part (b).

[2 marks]

3×2.5+5×7.5+…+10×27.560

= ( , 17.9, 17.9166 …)107560

21512

9d. The information above has been rewritten as a cumulative frequency table.

Write down the value of and the value of .

Markscheme (A1)(A1)

[2 marks]

a b

a = 34, b = 60

This information is shown in the following cumulative frequency graph.

[2 marks]

[2 marks]

9e. This information is shown in the following cumulative frequency graph.

For the people who were surveyed, use the graph to estimate

(i) the time taken for the first people to drink their coffee;

(ii) the number of people who take less than minutes to drink their coffee;

(iii) the number of people who take more than minutes to drink their coffee.

40

8

23

[4 marks]

Markscheme(i) (A1)

Note: Accept .

Accept any answer between and .

(Accept 21.5, but do not accept 21.)

(ii) (A1)

Note: Accept . Do not accept .

Answer must be an integer.

(iii) (M1)

(A1)(G2)

Notes: Award (M1) for subtraction from . Accept .

Answer must be an integer.

[4 marks]

⩽ 21.25 minutes

21.25

21 21.5

5

< 6 6

60 − 45

= 15

60 15 ± 1

10a.

The table below shows the distribution of test grades for 50 IB students at Greendale School.

Calculate the mean test grade of the students;

Markscheme (M1)

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

(A1) (G2)

[2 marks]

=1(1)+3(2)+7(3)+13(4)+11(5)+10(6)+5(7)50

23050

= 4.6

10b. Calculate the standard deviation.

[2 marks]

[1 mark]

Markscheme (G1)

[1 mark]

1.46 (1.45602…)

10c. Find the median test grade of the students.

Markscheme5 (A1)

[1 mark]

10d. Find the interquartile range.

Markscheme (M1)

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

(A1) (G2)

[2 marks]

6 − 4

= 2

10e.

A student is chosen at random from these 50 students.

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

Markscheme (M1)

Note: Award (M1) for seen.

(A1) (G2)

[2 marks]

11+10+550

11 + 10 + 5

= ( , 0.52, 52%)2650

1325

A second student is chosen at random from these 50 students.

[1 mark]

[2 marks]

[2 marks]

10f.

A second student is chosen at random from these 50 students.

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

Markscheme (M1)(M1)

Note: Award (M1) for seen, (M1) for multiplying their first probability by .

OR

Note: Award (M1) for seen, (M1) for dividing their first probability by .

(A1)(ft) (G3)

Note: Follow through from part (d).

[3 marks]

×10their 26

949

10their 26

949

×1050

949

2650

×1050

949

their 2650

= (0.0706, 0.0706436… , 7.06436… %)45637

10g.

The number of minutes that the 50 students spent preparing for the test was normally distributedwith a mean of 105 minutes and a standard deviation of 20 minutes.

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

[3 marks]

[2 marks]

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Markscheme (M1)

OR

(M1)

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

(A1) (G2)

[2 marks]

P(X ⩾ 90)

(> 0.5)

0.773 (0.773372 …) 0.773 (0.773372 … , 77.3372… %)

10h. Calculate the expected number of students that spent at least 90 minutes preparingfor the test.

Markscheme (M1)

(A1)(ft) (G2)

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

[2 marks]

0.773372 … × 50

= 38.7 (38.6686…)

[2 marks]