IB Questionbank Mathematical Studies 3rd edition Grouped discrete 184 min 183 marks 1. The weights in kg, of 80 adult males, were collected and are summarized in the box and whisker plot shown below. Write down the median weight of the males. Calculate the interquartile range. Estimate the number of males who weigh between 61 kg and 66 kg.
(d) Estimate the mean weight of the lightest 40 males. 2. 31 pupils in a class were asked to estimate the number of sweets in a jar. The following stem and leaf diagram gives their estimates. Stem 4 5 6 7 8 Leaf 2, 4, 7, 8, 9 1, 1, 2, 3, 8, 9 0, 2, 2, 4, 6, 6, 7, 8, 8 0, 0, 1, 3, 4, 5, 5, 7 1, 2, 2 Key: 4 7 represents 47 sweets For the pupils estimates, write down (i) (ii) (iii) the median; the lower quartile; the upper quartile.
Draw a box and whisker plot of the pupils estimates using the grid below. 3. The stem and leaf diagram below shows the lengths of 22 metal components in cm. Stem Leaf 1 2, 2, 3, 7 2 4, 4, 4, 8, 9, 9 3 6, 7, 7 4 1, 1, 1, 1, 3, 5, 6 5 0, 1 Key: 1 2 means 1.2 cm Write down the modal length of the metal components. Find the median length of the metal components.
Calculate the interquartile range of the lengths of the metal components. 4. 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. Complete the grouped frequency table for the students. Examination Score x (%) 0 x 20 20 < x 40 40 < x 60 60 < x 80 80 < x 100 Frequency 14 26
Write down the mid-interval value of the 40 < x 60 interval. Calculate an estimate of the mean examination score of the students. 5. The cumulative frequency graph shows the amount of time in minutes, 200 students spend waiting for their train on a particular morning. Write down the median waiting time.
Find the interquartile range for the waiting time. The minimum waiting time is zero and the maximum waiting time is 45 minutes. Draw a box and whisker plot on the grid below to represent this information. 6. 56 students were given a test out of 40 marks. The teacher used the following box and whisker plot to represent the marks of the students.
Write down (i) (ii) (iii) the median mark; the 75 th percentile mark; the range of marks. (4) Estimate the number of students who achieved a mark greater than 32. 7. The weights of 90 students in a school were recorded. The information is displayed in the following table. Weight (kg) Number of students 40 w < 50 7 50 w < 60 28 60 w < 70 35 70 w < 80 20 Write down the mid interval value for the interval 50 w < 60. Use your graphic display calculator to find an estimate for (i) (ii) the mean weight; the standard deviation. Find the weight that is 3 standard deviations below the mean.
8. The diagram shows the cumulative frequency graph for the time t taken to perform a certain task by 2000 men. Use the diagram to estimate (i) (ii) (iii) the median time; the upper quartile and the lower quartile; the interquartile range. (4) Find the number of men who take more than 11 seconds to perform the task. 55 % of the men took less than p seconds to perform the task. Find p.
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. (e) Use your graphic display calculator to find an estimate of (i) (ii) the mean time; the standard deviation of the time. 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. (Total 17 marks)
9. A cumulative frequency graph is given below which shows the height of students in a school. Write down the median height of the students.
Write down the 25 th percentile. Write down the 75 th percentile. 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.
10. There are 120 teachers in a school. Their ages are represented by the cumulative frequency graph below. 130 120 110 100 90 Cumulative frequency 80 70 60 50 40 30 20 10 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 Age Write down the median age. Find the interquartile range for the ages. 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 10 15 20 25 30 35 40 45 50 55 60 65 70 75 Age
11. Complete the following table of values for the height and weight of seven students. Values Mode Median Mean Standard deviation Height (cm) 151, 158, 171, 163, 184, 148, 171 164 11.7 Weight (kg) 53, 61, 58, 82, 45, 72, 82 82 61 (4) The ages (in months) of seven students are 194, 205, 208, 210, 200, 226, 223. Represent these values in an ordered stem and leaf diagram.
12. The table below shows the number of words in the extended essays of an IB class. Draw a histogram on the grid below for the data in this table. 20 15 Frequency 10 5 0 3000 3200 3400 3600 3800 4000 4200 4400 Number of words Write down the modal group. The maximum word count is 4000 words. Write down the probability that a student chosen at random is on or over the word count.
13. The following histogram shows the weights of a number of frozen chickens in a supermarket. The weights are grouped such that 1 weight < 2, 2 weight < 3 and so on. 55 50 45 40 35 number of chickens 30 25 20 15 10 5 0 0 1 2 3 4 5 6 weight (kg) On the graph above, draw in the frequency polygon. Find the total number of chickens. Write down the modal group. Gabriel chooses a chicken at random. (d) Find the probability that this chicken weighs less than 4 kg.
14. A random sample of 200 females measured the length of their hair in cm. The results are displayed in the cumulative frequency curve below. 200 175 Cumulative frequency 150 125 100 75 50 25 0 0 5 10 15 20 25 30 35 40 45 50 length (cm) Write down the median length of hair in the sample. Find the interquartile range for the length of hair in the sample. 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. 0 5 10 15 20 25 30 35 40 45 50 length (cm)
15. The diagram below shows the cumulative frequency distribution of the heights in metres of 600 trees in a wood. Write down the median height of the trees. Calculate the interquartile range of the heights of the trees.
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. 16. The following table shows the age distribution of teachers who smoke at Laughlin High School. Ages Number of smokers 20 x < 30 5 30 x < 40 4 40 x < 50 3 50 x < 60 2 60 x < 70 3 Calculate an estimate of the mean smoking age.
On the following grid, construct a histogram to represent this data. (Total 4 marks) 17. The distribution of the weights, correct to the nearest kilogram, of the members of a football club is shown in the following table. Weight (kg) 40 49 50 59 60 69 70 79 Frequency 6 18 14 4
On the grid below draw a histogram to show the above weight distribution. Write down the mid-interval value for the 40 49 interval. Find an estimate of the mean weight of the members of the club. (d) Write down an estimate of the standard deviation of their weights. 18. State which of the following sets of data are discrete. (i) (ii) (iii) (iv) (v) Speeds of cars travelling along a road. Numbers of members in families. Maximum daily temperatures. Heights of people in a class measured to the nearest cm. Daily intake of protein by members of a sporting team.
The boxplot below shows the statistics for a set of data. 2 4 6 8 10 12 14 16 18 20 data values For this data set write down the value of (i) (ii) (iii) the median; the upper quartile; the minimum value present. Write down three different integers whose mean is 10.
19. The figure below shows the lengths in centimetres of fish found in the net of a small trawler. Number of fish 11 10 9 8 7 6 5 4 3 2 1 1 0 10 20 30 40 50 60 70 80 90 110 120 130 Length (cm) Find the total number of fish in the net. Find (i) the modal length interval; (ii) (iii) the interval containing the median length; an estimate of the mean length. (5) (i) Write down an estimate for the standard deviation of the lengths. (ii) How many fish (if any) have length greater than three standard deviations above the mean? The fishing company must pay a fine if more than 10% of the catch have lengths less than 40cm. (d) Do a calculation to decide whether the company is fined.
A sample of 15 of the fish was weighed. The weight, W was plotted against length, L as shown below. 1.2 1 W (kg) 0.8 0.6 0.4 0.2 (e) 0 20 40 60 80 100 L (cm) Exactly two of the following statements about the plot could be correct. Identify the two correct statements. Note: You do not need to enter data in a GDC or to calculate r exactly. (i) The value of r, the correlation coefficient, is approximately 0.871. (ii) There is an exact linear relation between W and L. (iii) The line of regression of W on L has equation W = 0.012L + 0.008. (iv) There is negative correlation between the length and weight. (v) The value of r, the correlation coefficient, is approximately 0.998. (vi) The line of regression of W on L has equation W = 63.5L + 16.5. (Total 14 marks)
20. The birth weights, in kilograms, of 27 babies are given in the diagram below. 1 7, 8, 9 key 1 7 = 1.7 kg 2 1, 2, 2, 3, 5, 5, 7, 8, 9 3 0, 1, 3, 4, 5, 5, 6, 6, 7, 9 4 1, 1, 2, 3, 7 Calculate the mean birth weight. Write down: (i) (ii) the median weight; the upper quartile. The lower quartile is 2.3 kg. On the scale below draw a box and whisker diagram to represent the birth weights. 1 2 3 4 5 Weight (kg)
21. A random sample of 167 people who own mobile phones was used to collect data on the amount of time they spent per day using their phones. The results are displayed in the table below. Time spent per day (t minutes) 0 t <15 15 t < 30 30 t < 45 45 t < 60 60 t < 75 75 t < 90 Number of people 21 32 35 41 27 11 State the modal group. Use your graphic display calculator to calculate approximate values of the mean and standard deviation of the time spent per day on these mobile phones. On graph paper, draw a fully labelled histogram to represent the data. (4) (Total 8 marks)
22. The heights (cm) of seedlings in a sample are shown below. 6 3, 7 key 6 3 represents 63 cm 7 2, 5, 8 8 3, 6, 6, 8, 8 9 2, 5, 7, 8 10 3, 6, 6 11 2, 2 State how many seedlings are in the sample. Write down the values of (i) (ii) the median; the first and third quartile. (d) Calculate the range. Using the scale below, draw a box and whisker plot for this data. 60 70 80 90 100 110 120 23. 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 Construct a cumulative frequency table. (Use upper class boundaries 15.5, 20.5 and so on.)
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. Use your graph to estimate (i) the number of students that completed the task in less than 17.5 minutes; 3 (ii) the time it will take for of the students to complete the task. 4 (Total 7 marks) 24. The table shows the number of children in 50 families. Number of children Write down the value of T. Find the values of m, p and q. Frequency Cumulative frequency 1 3 3 2 m 22 3 12 34 4 p q 5 5 48 6 2 50 T (Total 4 marks)
25. 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 of mackerel 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 Construct a cumulative frequency table for the data in the table. Draw a cumulative frequency curve. Hint: Plot your cumulative frequencies at the top of each interval. Use the cumulative frequency curve to find an estimate, to the nearest cm for (i) (ii) the median length of mackerel; the interquartile range of mackerel length. (Total 9 marks)
26. The histogram below shows the amount of money spent on food each week by 45 families. The amounts have been rounded to the nearest 10 dollars. frequency 18 16 14 12 10 8 6 4 2 0 150 160 170 180 190 $ Calculate the mean amount spent on food by the 45 families. Find the largest possible amount spent on food by a single family in the modal group. State which of the following amounts could not be the total spent by all families in the modal group: (i) $2430 (ii) $2495 (iii) $2500 (iv) $2520 (v) $2600
27. The following stem and leaf diagram gives the weights in kg of 34 eight year-old children. The median weight is 30.3 kg. Find the value of t. Key: 26 1 reads 26.1kg Write down the lower quartile weight. The value of the upper quartile is 31.6 kg and there are no outliers. Draw a box and whisker plot of the data using the axis below. 26 27 28 29 30 31 32 33 34 Weight (kg)