frecuencia acumulada datos agrupados discretos y continuos estadistica estudios matematicos

1. 120 Mathematics students in a school sat an examination. Their scores (given as a percentage) were summarized on a cumulative frequency diagram. This diagram is given below.

(a) Complete the grouped frequency table for the students.

ExaminationScore x (%)

0 ≤ x ≤ 20 20 < x ≤ 40 40 < x ≤ 60 60 < x ≤ 80 80 < x ≤ 100

Frequency 14 26(3)

(b) Write down the mid-interval value of the 40 < x ≤ 60 interval.(1)

IB Questionbank Mathematical Studies 3rd edition 1

(c) Calculate an estimate of the mean examination score of the students.(2)

(Total 6 marks)

2. The cumulative frequency graph shows the amount of time in minutes, 200 students spend waiting for their train on a particular morning.

(a) Write down the median waiting time.(1)

(b) Find the interquartile range for the waiting time.(2)


The minimum waiting time is zero and the maximum waiting time is 45 minutes.

(c) Draw a box and whisker plot on the grid below to represent this information.

(3)(Total 6 marks)

3. The diagram shows the cumulative frequency graph for the time t taken to perform a certain task by 2000 men.

(a) Use the diagram to estimate


(i) the median time;

(ii) the upper quartile and the lower quartile;

(iii) the interquartile range.(4)

(b) Find the number of men who take more than 11 seconds to perform the task.(3)

(c) 55 % of the men took less than p seconds to perform the task. Find p.(2)

The times taken for the 2000 men were grouped as shown in the table below.

Time Frequency

5 ≤ t < 10 500

10 ≤ t < 15 850

15 ≤ t < 20 a

20 ≤ t < 25 b

(d) Write down the value of

(i) a;

(ii) b.(2)

(e) Use your graphic display calculator to find an estimate of

(i) the mean time;

(ii) the standard deviation of the time.(3)


Everyone who performs the task in less than one standard deviation below the mean will receive a bonus. Pedro takes 9.5 seconds to perform the task.

(f) Does Pedro receive the bonus? Justify your answer.(3)

(Total 17 marks)


4. A cumulative frequency graph is given below which shows the height of students in a school.

(a) Write down the median height of the students.(1)


(b) Write down the 25th percentile.(1)

(c) Write down the 75th percentile.(1)

The height of the tallest student is 195 cm and the height of the shortest student is 136 cm.

(d) Draw a box and whisker plot on the grid below to represent the heights of the students in the school.

(3)(Total 6 marks)


5. There are 120 teachers in a school. Their ages are represented by the cumulative frequency graph below.

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5A g e

1 3 0

1 2 0

11 0

1 0 0

9 0

8 0

7 0

6 0

5 0

4 0

3 0

2 0

1 0

0

Cum

ulat

ive

freq

uenc

y

(a) Write down the median age.(1)

(b) Find the interquartile range for the ages.(2)

(c) Given that the youngest teacher is 21 years old and the oldest is 72 years old, represent the information on a box and whisker plot using the scale below.

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5A g e

(3)(Total 6 marks)


6. A random sample of 200 females measured the length of their hair in cm. The results are displayed in the cumulative frequency curve below.

2 0 0

1 7 5

1 5 0

1 2 5

1 0 0

7 5

5 0

2 5

05 04 54 03 53 02 52 01 51 050

Cum

ulat

ive

freq

uenc

y

le n g th (cm )

(a) Write down the median length of hair in the sample.(1)

(b) Find the interquartile range for the length of hair in the sample.(2)

(c) Given that the shortest length was 6 cm and the longest 47 cm, draw and label a box and whisker plot for the data on the grid provided below.

5 04 54 03 53 02 52 01 51 050len g th (cm )

(3)(Total 6 marks)


7. The diagram below shows the cumulative frequency distribution of the heights in metres of 600 trees in a wood.

(a) Write down the median height of the trees.(1)

(b) Calculate the interquartile range of the heights of the trees.(2)


(c) Given that the smallest tree in the wood is 3 m high and the tallest tree is 28 m high, draw the box and whisker plot on the grid below that shows the distribution of trees in the wood.

(3)(Total 6 marks)

8. The local council has been monitoring the number of cars parked near a supermarket on an hourly basis. The results are displayed below.

Parked Cars/Hour Frequency Cumulative Frequency

0–19 3 3

20–39 15 18

40–59 25 w

60–79 35 78

80–99 17 95

(a) Write down the value of w.


(b) Draw and label the Cumulative Frequency graph for this data.

(c) Determine the median number of cars per hour parked near the supermarket.

(Total 8 marks)


9. The cumulative frequency table below shows the ages of 200 students at a college.

Age Number of Students Cumulative Frequency

17 3 3

18 72 75

19 62 137

20 31 m

21 12 180

22 9 189

23–25 5 194

> 25 6 n

(a) What are the values of m and n?

(b) How many students are younger than 20?

(c) Find the value in years of the lower quartile.

(Total 8 marks)

10. The table below shows the number and weight (w) of fish delivered to a local fish market one morning.

weight (kg) frequency cumulative frequency

0.50 ≤ w < 0.70 16 16

0.70 ≤ w < 0.90 37 53

0.90 ≤ w < 1.10 44 c

1.10 ≤ w < 1.30 23 120

1.30 ≤ w < 1.50 10 130

(a) (i) Write down the value of c.(1)


(ii) On graph paper, draw the cumulative frequency curve for this data. Use a scale of 1 cm to represent 0.1 kg on the horizontal axis and 1 cm to represent 10 units on the vertical axis. Label the axes clearly.

(4)

(iii) Use the graph to show that the median weight of the fish is 0.95 kg.(1)

(b) (i) The zoo buys all fish whose weights are above the 90th percentile.How many fish does the zoo buy?

(2)

(ii) A pet food company buys all the fish in the lowest quartile. What is the maximum weight of a fish bought by the company?

(3)

(c) A restaurant buys all fish whose weights are within 10% of the median weight.

(i) Calculate the minimum and maximum weights for the fish bought by the restaurant.

(2)

(ii) Use your graph to determine how many fish will be bought by the restaurant.(3)

(Total 16 marks)


11. The cumulative frequency graph has been drawn from a frequency table showing the time it takes a number of students to complete a computer game.

2 0 0

1 8 0

1 6 0

1 4 0

1 2 0

1 0 0

8 0

6 0

4 0

2 0

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0T im e in m in u te s

Num

ber o

f stu

dent

s

f

(a) From the graph find

(i) the median time;

(ii) the interquartile range.(5)


The graph has been drawn from the data given in the table below.

Time in minutes Number of students

0 < x ≤ 5 20

5 < x ≤ 15 20

15 < x ≤ 20 p

20 < x ≤ 25 40

25 < x ≤ 35 60

35 < x ≤ 50 q

50 < x ≤ 60 10

(b) Using the graph, find the values of p and q.(2)

(c) Calculate an estimate of the mean time taken to finish the computer game.(4)

(Total 11 marks)

12. The heights of 200 students are recorded in the following table.

Height (h) in cm Frequency

140 ≤ h < 150 2

150 ≤ h < 160 28

160 ≤ h < 170 63

170 ≤ h < 180 74

180 ≤ h < 190 20

190 ≤ h < 200 11

200 ≤ h < 210 2

(a) Write down the modal group.(1)

(b) Calculate an estimate of the mean and standard deviation of the heights.(4)


The cumulative frequency curve for this data is drawn below.

2 0 0

1 8 0

1 6 0

1 4 0

1 2 0

1 0 0

8 0

6 0

4 0

2 0

01 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 2 1 0

h e ig h t in cm

num

ber o

f stu

dent

s

(c) Write down the median height.(1)

(d) The upper quartile is 177.3 cm. Calculate the interquartile range.(2)

(e) Find the percentage of students with heights less than 165 cm.(2)

(Total 10 marks)


13. The cumulative frequency graph below shows the examination scores of 80 students.

8 0

7 0

6 0

5 0

4 0

3 0

2 0

1 0

1 0 2 0 3 0 4 0 5 0 6 0sco res

cu m u la tiv efre q u e n c y

0

From the graph find

(a) the median value;

(b) the interquartile range;

(c) the 35th percentile;

(d) the percentage of students who scored 50 or above on this examination.

(Total 8 marks)


14. The graph below shows the cumulative frequency for the yearly incomes of 200 people.

2 0 0

1 8 0

1 6 0

1 4 0

1 2 0

1 0 0

8 0

6 0

4 0

2 0

00 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0

A n n u a l in c o m e in B ritish p o u n d s

C u m u la tiv efre q u e n c y

Use the graph to estimate

(a) the number of people who earn less than 5000 British pounds per year;

(b) the median salary of the group of 200 people;

(c) the lowest income of the richest 20% of this group.

(Total 4 marks)


15. The table below shows the percentage, to the nearest whole number, scored by candidates in an examination.

Marks (%) 0–9 10–19 20–29 30–39 40–49 50–59 60–69 70–79 80–89 90–100

Frequency 2 7 8 13 24 30 6 5 3 2

The following is the cumulative frequency table for the marks.

Marks (%) Cumulative frequency

< 9.5 2

< 19.5 9

< 29.5 s

< 39.5 30

< 49.5 54

< 59.5 84

< 69.5 t

< 79.5 95

< 89.5 98

< 100 100

(a) Calculate the values of s and of t.(2)


(b) Using a scale of 1 cm to represent 10 marks on the horizontal axis, and 1 cm to represent 10 candidates on the vertical axis, draw a cumulative frequency graph.

(3)

(c) Use your graph to estimate

(i) the median mark;

(ii) the lower quartile;

(iii) the pass mark, if 40% of the candidates passed.(4)

(Total 9 marks)

16. The following table shows the times, to the nearest minute, taken by 100 students to complete a mathematics task.

Time (t) minutes 11–15 16–20 21–25 26–30 31–35 36–40

Number of students 7 13 25 28 20 7

(a) Construct a cumulative frequency table. (Use upper class boundaries 15.5, 20.5 and so on.)

(2)


(b) On graph paper, draw a cumulative frequency graph, using a scale of 2 cm to represent 5 minutes on the horizontal axis and 1 cm to represent 10 students on the vertical axis.

(3)

(c) Use your graph to estimate

(i) the number of students that completed the task in less than 17.5 minutes;

(ii) the time it will take for 43

of the students to complete the task.

(2)(Total 7 marks)

17. The table shows the number of children in 50 families.

Number ofchildren

Frequency Cumulativefrequency

1 3 3

2 m 22

3 12 34

4 p q

5 5 48

6 2 50

T

(a) Write down the value of T.

(b) Find the values of m, p and q.

(Total 4 marks)


18. A marine biologist records as a frequency distribution the lengths (L), measured to the nearest centimetre, of 100 mackerel. The results are given in the table below.

Length of mackerel(L cm)

Number ofmackerel

27 < L ≤ 29 2

29 < L ≤ 31 4

31 < L ≤ 33 8

33 < L ≤ 35 21

35 < L ≤ 37 30

37 < L ≤ 39 18

39 < L ≤ 41 12

41 < L ≤ 43 5

100

(a) Construct a cumulative frequency table for the data in the table.(2)

(b) Draw a cumulative frequency curve.

Hint: Plot your cumulative frequencies at the top of each interval.(3)

(c) Use the cumulative frequency curve to find an estimate, to the nearest cm for

(i) the median length of mackerel;(2)

(ii) the interquartile range of mackerel length.(2)

(Total 9 marks)


frecuencia acumulada datos agrupados discretos y continuos estadistica estudios matematicos

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