Descriptive Statistics Practice Problems 1. The age in months at which a child first starts to walk is observed for a random group of children from a town in Brazil. The results are 14.3, 11.6, 12.2, 14., 2.4, 13.4, 12.9, 11.7, 13.1. (a) (i) Find the mean of the ages of these children. (ii) Find the standard deviation of the ages of these children. Find the median age. 2. The numbers of games played in each set of a tennis tournament were 9, 7, 8, 11, 9, 6, 1, 8, 12, 6, 8, 13, 7, 9, 1, 9, 1, 11, 12, 8, 7, 13, 1, 7, 7. The raw data has been organized in the frequency table below. games frequency 6 2 7 5 8 n 9 4 1 4 11 2 12 2 13 2 (Total 6 marks) (a) Write down the value of n. Calculate the mean number of games played per set. What percentage of the sets had more than 1 games? (d) What is the modal number of games? (Total 8 marks) 3. An atlas gives the following information about the approximate population of some cities in the year 2. The population of Nairobi has accidentally been left out. City Population in Millions Melbourne 3.2 Bangkok 7.2 Nairobi Paris 9.6 São Paulo 17.7 Tokyo 28. Seattle 2.1 The atlas tells us that the mean population for this group of cities is 1.1 million. (a) Calculate the population of Nairobi. Which city has the median population value? (Total 8 marks) 4. In the following ordered data, the mean is 6 and the median is 5. 2, b, 3, a, 6, 9, 1, 12 Find each of the following (a) the value of a; the value of b. (Total 8 marks) 5. The weight in kilograms of 12 students in a class are as follows. 63 76 99 65 63 51 52 95 63 71 65 83 (a) State the mode. C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 1 of 21
Calculate (i) the mean weight; (ii) the standard deviation of the weights. When one student leaves the class, the mean weight of the remaining 11 students becomes 7 kg. Find the weight of the student who left. (Total 5 marks) 6. Twenty students are asked how many detentions they received during the previous week at school. The results are summarised in the frequency distribution table below. Number of Number of fx detentions x students f 6 1 3 2 1 3 1 Total 2 (a) What is the modal number of detentions received? (i) Complete the table. (ii) Find the mean number of detentions received. 7. For the set of {8, 4, 2, 1, 2, 5, 9, 12, 2, 6} (a) calculate the mean; find the mode; find the median. 8. The mean of the ten numbers listed below is 5.5. 4, 3, a, 8, 7, 3, 9, 5, 8, 3 (a) Find the value of a. Find the median of these numbers. C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 2 of 21 (Total 4 marks) (Total 4 marks) (Total 4 marks) 9. The table shows the number of children in 5 families. Number of children Frequency Cumulative frequency 1 3 3 2 m 22 3 12 34 4 p q 5 5 48 6 2 5 T (a) Write down the value of T. Find the values of m, p and q. (Total 4 marks) 1. The temperatures in C, at midday in Geneva, were measured for eight days and the results are recorded below. 7, 4, 5, 4, 8, T, 14, 4 The mean temperature was found to be 7 C. (a) Find the value of T.
Write down the mode. Find the median. 11. The grades obtained by a group of 2 IB students are listed below: 6 2 5 3 5 5 6 2 6 1 7 6 2 4 2 4 3 4 5 6 (a) Complete the following table for the grades obtained by the students. (3) (Total 6 marks) Grade Frequency 1 2 3 2 4 5 4 6 7 1 Write down the modal grade obtained by the students. Calculate the median grade obtained by the students. One student is chosen at random from the group. (d) Find the probability that this student obtained either grade 4 or grade 5. 12. The figure below shows the lengths in centimetres of fish found in the net of a small trawler. 11 1 9 8 (a) Number of fish 7 6 5 4 3 2 1 1 1 Find the total number of fish in the net. 2 3 4 5 6 7 8 9 11 1213 Length (cm) Find (i) the modal length interval; (ii) the interval containing the median length; (iii) an estimate of the mean length. (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? C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 3 of 21 (Total 6 marks) (5) (3)
The fishing company must pay a fine if more than 1% of the catch have lengths less than 4cm. (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).8.6.4.2 (e) 2 4 6 8 1 C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 4 of 21 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.871. (ii) There is an exact linear relation between W and L. (iii) The line of regression of W on L has equation W =.12L +.8. (iv) There is negative correlation between the length and weight. (v) The value of r, the correlation coefficient, is approximately.998. (vi) The line of regression of W on L has equation W = 63.5L + 16.5. (Total 14 marks) 13. 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) t <15 15 t < 3 3 t < 45 45 t < 6 6 t < 75 75 t < 9 Number of people 21 32 35 41 27 11 (a) 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. (3) On graph paper, draw a fully labelled histogram to represent the data. (4) (Total 8 marks) 14. The histogram below shows the amount of money spent on food each week by 45 families. The amounts have been rounded to the nearest 1 dollars. (a) frequency 18 16 14 12 1 8 6 4 2 15 16 17 18 19 $ 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) $243 (ii) $2495 (iii) $25 (iv) $252 (v) $26 (Total 6 marks) 15. The heights in cm of the members of 4 volleyball teams A, B, C and D were taken and represented in the frequency histograms given below. frequency A frequency B 18 19 2 height (cm) 18 19 2 height (cm) frequency C frequency D 18 19 2 height (cm) 18 19 2 height (cm) The mean x and standard deviation σ of each team are shown in the following table. I II III IV x 194 189 188 195 σ 6.5 4.91 3.9 3.74 Match each pair of x and σ (I, II, III, or IV) to the correct team (A, B, C or D). x and σ Team I II III IV (Total 6 marks) 16. The following histogram shows the house prices in thousands of Australian dollars (AUD) of a random sample of houses in a certain town in Australia. (a) How many houses are there in the sample? C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 5 of 21
Write down the modal group for house prices. Find the probability of choosing a house at random that costs less than 6 AUD or more than 24 AUD. (d) Given that a house costs more than 12 AUD, find the probability that it costs between 18 and 24 AUD. (Total 6 marks) 17. The number of hours that a professional footballer trains each day in the month of June is represented in the following histogram. 1 9 8 7 number of days 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 number of hours (a) Write down the modal number of hours trained each day. Calculate the mean number of hours he trains each day. (Total 8 marks) 18. Fifty students at Layton High School recorded how much money each student in their class spent on video rentals this month (to the nearest dollar). The results are shown in the frequency table below: Class interval in $ Boundaries in $ Frequency 1 1.5 1.5 1 11 2 1.5 2.5 2 21 3 2.5 3.5 1 31 4 3.5 4.5 41 5 4.5 5.5 4 51 6 5.5 6.5 2 61 7 6.5 7.5 4 (a) On graph paper using a scale of 2 cm to represent each interval ($1.) on the horizontal axis and 1 cm to represent 5 people on the vertical axis, draw and clearly label a frequency histogram which displays the above information. (5) Answer the following questions: (i) Which class is the modal class? (ii) In which class is the median? Assuming these students spend the same amount on videos each month, find the probability that next month a student will spend an amount in the class interval: (i) From $21 to $3 inclusive on video rentals. (ii) $3 or less on video rentals. (iii) From $41 to $6 on video rentals, given that they spent more than $2 on video rentals. (iv) Not more than $6 on video rentals, given that they spent over $1 on video rentals. (6) (Total 13 marks) C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 6 of 21
19. The bar chart below shows the number of people in a selection of families. 1 8 Number of families 6 4 2 3 4 5 6 7 8 9 1 Number of people in a family (a) How many families are represented? Write down the mode of the distribution. Find, correct to the nearest whole number, the mean number of people in a family. (Total 4 marks) 2. The following table shows the age distribution of teachers who smoke at Laughlin High School. Ages Number of smokers 2 x < 3 5 3 x < 4 4 4 x < 5 3 5 x < 6 2 6 x < 7 3 (a) Calculate an estimate of the mean smoking age. On the following grid, construct a histogram to represent this data. (Total 4 marks) 21. A group of 25 females were asked how many children they each had. The results are shown in the histogram below. 1 9 8 7 6 Frequency 5 4 3 2 1 Number of Children per Female 1 2 3 4 Number of Children (a) Show that the mean number of children per female is 1.4. Show clearly that the standard deviation for this data is approximately 1.6. Another group of 25 females was surveyed and it was found that the mean number of children per C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 7 of 21 (3)
(d) (e) female was 2.4 and the standard deviation was 2. Use the results from parts (a) and to describe the differences between the number of children the two groups of females have. A female is selected at random from the first group. What is the probability that she has more than two children? Two females are selected at random from the first group. What is the probability that (i) both females have more than two children? (ii) (iii) only one of the females has more than two children? the second female selected has two children given that the first female selected had no children? (Total 15 marks) 22. David looked at a passage from a book. He recorded the number of words in each sentence as shown in the following frequency table. Class interval (number of words) Frequency f 1 5 16 6 1 28 11 15 26 16 2 14 21 25 1 26 3 3 31 35 1 36 4 41 45 2 (a) Find the class interval in which the median lies. Estimate, correct to the nearest whole number, the mean number of words in a sentence. (Total 4 marks) 23. There are 12 teachers in a school. Their ages are represented by the cumulative frequency graph below. Cumulative frequency 13 12 11 1 9 8 7 6 5 4 3 2 1 5 1 15 2 25 3 35 4 45 5 55 6 65 7 75 Age C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 8 of 21 (3)
(a) 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. 5 1 15 2 25 3 35 4 45 5 55 6 65 7 75 Age (3) (Total 6 marks) 24. A random sample of 2 females measured the length of their hair in cm. The results are displayed in the cumulative frequency curve below. 2 Cumulative frequency (a) 175 15 125 1 75 5 25 5 1 15 2 25 length (cm) Write down the median length of hair in the sample. 3 Find the interquartile range for the length of hair in the sample. 35 4 45 5 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 1 15 2 25 3 35 4 45 5 length (cm) (3) (Total 6 marks) 25. The diagram below shows the cumulative frequency distribution of the heights in metres of 6 trees in a wood. C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 9 of 21
(a) 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. 26. (a) 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. The boxplot below shows the statistics for a set of data. (3) (Total 6 marks) C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 1 of 21
2 4 6 8 1 12 14 16 18 2 data values For this data set write down the value of (i) the median; (ii) the upper quartile; (iii) the minimum value present. Write down three different integers whose mean is 1. 27. (a) The exam results for 1 boys are displayed in the following diagram: (Total 6 marks) (i) 1 2 3 4 5 6 7 8 9 1 Find the range of the results. (ii) Find the interquartile range. (iii) Write down the median. The exam results for 1 girls are displayed in the diagram below: 1 9 number of girls cumulative frequency 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 exam results (i) Write down the median. (ii) Find the inter quartile range. Write down the set of results that are the most spread out and give a reason for your answer. (Total 6 marks) 28. The local council has been monitoring the number of cars parked near a supermarket on an hourly basis. The results are displayed below. C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 11 of 21
Parked Cars/Hour Frequency Cumulative Frequency 19 3 3 2 39 15 18 4 59 25 w 6 79 35 78 8 99 17 95 (a) Write down the value of w. Draw and label the Cumulative Frequency graph for this data. Determine the median number of cars per hour parked near the supermarket. 29. The cumulative frequency table below shows the ages of 2 students at a college. Age Number of Students Cumulative Frequency 17 3 3 18 72 75 19 62 137 2 31 m 21 12 18 22 9 189 23 25 5 194 > 25 6 n (a) What are the values of m and n? How many students are younger than 2? Find the value in years of the lower quartile. (Total 8 marks) (Total 8 marks) 3. The table below shows the number and weight (w) of fish delivered to a local fish market one morning. weight (kg) frequency cumulative frequency.5 w <.7 16 16 C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 12 of 21
.7 w <.9 37 53.9 w < 1.1 44 c 1.1 w < 1.3 23 12 1.3 w < 1.5 1 13 (a) (i) Write down the value of c. (ii) On graph paper, draw the cumulative frequency curve for this data. Use a scale of 1 cm to represent.1 kg on the horizontal axis and 1 cm to represent 1 units on the vertical axis. Label the axes clearly. (4) (iii) Use the graph to show that the median weight of the fish is.95 kg. (i) The zoo buys all fish whose weights are above the 9 th percentile. How many fish does the zoo buy? (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) A restaurant buys all fish whose weights are within 1% of the median weight. (i) Calculate the minimum and maximum weights for the fish bought by the restaurant. (ii) Use your graph to determine how many fish will be bought by the restaurant. (3) (Total 16 marks) 31. 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. f 2 18 16 Number of students 14 12 1 8 6 4 2 5 1 15 2 25 3 35 4 45 5 55 6 Time in minutes (a) From the graph find (i) the median time; (ii) the interquartile range. (5) C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 13 of 21
The graph has been drawn from the data given in the table below. Time in minutes Number of students < x 5 2 5 < x 15 2 15 < x 2 p 2 < x 25 4 25 < x 35 6 35 < x 5 q 5 < x 6 1 Using the graph, find the values of p and q. Calculate an estimate of the mean time taken to finish the computer game. 32. The heights of 2 students are recorded in the following table. Height (h) in cm Frequency 14 h < 15 2 15 h < 16 28 16 h < 17 63 17 h < 18 74 18 h < 19 2 19 h < 2 11 2 h < 21 2 (a) Write down the modal group. Calculate an estimate of the mean and standard deviation of the heights. The cumulative frequency curve for this data is drawn below. (4) (Total 11 marks) (4) 2 18 16 number of students 14 12 1 8 6 4 2 14 15 16 17 18 19 2 21 height in cm Write down the median height. C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 14 of 21
(d) The upper quartile is 177.3 cm. Calculate the interquartile range. (e) Find the percentage of students with heights less than 165 cm. 33. The cumulative frequency graph below shows the examination scores of 8 students. (Total 1 marks) cumulative frequency 8 7 6 5 4 3 2 1 1 2 3 4 5 6 scores From the graph find (a) the median value; the interquartile range; the 35 th percentile; (d) the percentage of students who scored 5 or above on this examination. 34. The graph below shows the cumulative frequency for the yearly incomes of 2 people. 2 (Total 8 marks) 18 16 14 12 1 Cumulative frequency 8 6 4 2 5 1 15 2 25 3 35 Annual income in British pounds Use the graph to estimate (a) the number of people who earn less than 5 British pounds per year; the median salary of the group of 2 people; the lowest income of the richest 2% of this group. (Total 4 marks) C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 15 of 21
35. The table below shows the percentage, to the nearest whole number, scored by candidates in an examination. Marks (%) 9 1 19 2 29 3 39 4 49 5 59 6 69 7 79 8 89 9 1 Frequency 2 7 8 13 24 3 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 3 < 49.5 54 < 59.5 84 < 69.5 t < 79.5 95 < 89.5 98 < 1 1 (a) Calculate the values of s and of t. Using a scale of 1 cm to represent 1 marks on the horizontal axis, and 1 cm to represent 1 candidates on the vertical axis, draw a cumulative frequency graph. (3) Use your graph to estimate (i) the median mark; (ii) the lower quartile; (iii) the pass mark, if 4% of the candidates passed. (4) (Total 9 marks) 36. The following table shows the times, to the nearest minute, taken by 1 students to complete a mathematics task. Time (t) minutes 11 15 16 2 21 25 26 3 31 35 36 4 Number of students 7 13 25 28 2 7 (a) Construct a cumulative frequency table. (Use upper class boundaries 15.5, 2.5 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 1 students on the vertical axis. (3) 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) 37. A marine biologist records as a frequency distribution the lengths (L), measured to the nearest centimetre, of 1 mackerel. The results are given in the table below. Length of mackerel Number of (L cm) mackerel 27 < L 29 2 29 < L 31 4 31 < L 33 8 33 < L 35 21 C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 16 of 21
35 < L 37 3 37 < L 39 18 39 < L 41 12 41 < L 43 5 1 (a) 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. (3) Use the cumulative frequency curve to find an estimate, to the nearest cm for (i) the median length of mackerel; (ii) the interquartile range of mackerel length. (Total 9 marks) 38. 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 5 45 4 35 number of chickens 3 25 2 15 1 5 1 2 3 weight (kg) 4 5 (a) 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. (Total 6 marks) 39. A survey was conducted of the number of bedrooms in 28 randomly chosen houses. The results are shown in the following table. Number of bedrooms 1 2 3 4 5 6 Number of houses 41 6 52 32 15 8 (a) State whether the data is discrete or continuous. Write down the mean number of bedrooms per house. C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 17 of 21 6
Write down the standard deviation of the number of bedrooms per house. (d) Find how many houses have a number of bedrooms greater than one standard deviation above the mean. (Total 6 marks) 4. The number of bottles of water sold at a railway station on each day is given in the following table. Day 1 2 3 4 5 6 7 8 9 1 11 12 Temperature (T ) 21 2.7 2 19 18 17.3 17 17.3 18 19 2 2.7 21 Number of bottles sold (n) 15 141 126 125 98 11 93 99 116 121 119 134 141 (a) Write down (i) the mean temperature; (ii) the standard deviation of the temperatures. Write down the correlation coefficient, r, for the variables n and T. Comment on your value for r. (d) The equation of the line of regression for n on T is n = dt 1. (i) Write down the value of d. (ii) Estimate how many bottles of water will be sold when the temperature is 19.6. (e) On a day when the temperature was 36 Peter calculates that 314 bottles would be sold. Give one reason why his answer might be unreliable. (Total 8 marks) 41. 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) 4 49 5 59 6 69 7 79 Frequency 6 18 14 4 (a) On the grid below draw a histogram to show the above weight distribution. Write down the mid-interval value for the 4 49 interval. Find an estimate of the mean weight of the members of the club. C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 18 of 21
(d) Write down an estimate of the standard deviation of their weights. (Total 6 marks) 42. Five pipes labelled, 6 metres in length, were delivered to a building site. The contractor measured each pipe to check its length (in metres) and recorded the following; 5.96, 5.95, 6.2, 5.95, 5.99. (a) (i) Find the mean of the contractor s measurements. (ii) Calculate the percentage error between the mean and the stated, approximate length of 6 metres. Calculate 3.89 5 8.73.5, giving your answer (i) correct to the nearest integer; (ii) in the form a 1 k, where 1 a < 1, k. (Total 6 marks) 43. (a) Write down the following numbers in increasing order. 3.5, 1.6 1 19, 673, 6.73 1 5,.673 1 6, π, 9.8 1 18. Write down the median of the numbers in part (a). State which of the numbers in part (a) is irrational. (Total 6 marks) 44. In an experiment a vertical spring was fixed at its upper end. It was stretched by hanging different weights on its lower end. The length of the spring was then measured. The following readings were obtained. Load (kg) x 1 2 3 4 5 6 7 8 Length (cm) y 23.5 25 26.5 27 28.5 31.5 34.5 36 37.5 (a) Plot these pairs of values on a scatter diagram taking 1 cm to represent 1 kg on the horizontal axis and 1 cm to represent 2 cm on the vertical axis. (4) (i) Write down the mean value of the load ( x ). (ii) Write down the standard deviation of the load. (iii) Write down the mean value of the length ( y ). (iv) Write down the standard deviation of the length. Plot the mean point ( x, y ) on the scatter diagram. Name it L. It is given that the covariance S xy is 12.17. (d) (i) Write down the correlation coefficient, r, for these readings. (ii) Comment on this result. (e) Find the equation of the regression line of y on x. (f) Draw the line of regression on the scatter diagram. (g) (i) Using your diagram or otherwise, estimate the length of the spring when a load of 5.4 kg is applied. (ii) Malcolm uses the equation to claim that a weight of 3 kg would result in a length of 62.8 cm. Comment on his claim. C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 19 of 21
(Total 18 marks) 45. Peter has marked 8 exam scripts. He has calculated the mean mark for the scripts to be 62.1. Maria has marked 6 scripts with a mean mark of 56.8. (a) Peter discovers an error in his marking. He gives two extra marks each to eleven of the scripts. Calculate the new value of the mean for Peter s scripts. After the corrections have been made and the marks changed, Peter and Maria put all their scripts together. Calculate the value of the mean for all the scripts. (Total 8 marks) 46. A group of students has measured the heights of 9 trees. The class calculate the mean height to be x = 12.4 m with standard deviation s = 5.35 m. One student notices that two of the measurements, 44.5 m and 43.2 m, are much too big and must be wrong. (a) How many standard deviations away from the mean of 12.4 is the value 44.5? The incorrect measurements of 44.5 m and 43.2 m must be removed from the data. Calculate the new value of x after removing the two unwanted values. (Total 8 marks) 47. A shopkeeper wanted to investigate whether or not there was a correlation between the prices of food 1 years ago in 1992, with their prices today. He chose 8 everyday items and the prices are given in the table below. sugar milk eggs rolls tea bags coffee potatoes flour 1992 price $1.44 $.8 $2.16 $1.8 $.92 $3.16 $1.32 $1.12 22 price $2.2 $1.4 $2.64 $3. $1.32 $2.28 $1.92 $1.44 (a) Calculate the mean and the standard deviation of the prices (i) in 1992; (ii) in 22. (i) Given that s xy =.314, calculate the correlation coefficient. (ii) Comment on the relationship between the prices. Find the equation of the line of the best fit in the form y = mx + c. (3) (d) What would you expect to pay now for an item costing $2.6 in 1992? (e) Which item would you omit to increase the correlation coefficient? (Total 14 marks) 48. The heights and weights of 1 students selected at random are shown in the table below. Student 1 2 3 4 5 6 7 8 9 1 Height 155 161 173 15 182 165 17 185 175 145 x cm Weight 5 75 8 46 81 79 64 92 74 18 y kg (a) Plot this information on a scatter graph. Use a scale of 1 cm to represent 2 cm on the x-axis and 1 cm to represent 1 kg on the y-axis. (4) Calculate the mean height. Calculate the mean weight. (d) It is given that S xy = 44.31. (i) By first calculating the standard deviation of the heights, correct to two decimal places, C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 2 of 21 (4) (4)
(e) (f) (ii) (iii) show that the gradient of the line of regression of y on x is.276. Calculate the equation of the line of best fit. Draw the line of best fit on your graph. Use your line to estimate (i) the weight of a student of height 19 cm; (ii) the height of a student of weight 72 kg. It is decided to remove the data for student number 1 from all calculations. Explain briefly what effect this will have on the line of best fit. 49. The following table gives the heights and weights of five sixteen-year-old boys. Name Height Weight Blake 182 cm 73 kg Jorge 173 cm 68 kg Chin 162 cm 6 kg Ravi 178 cm 66 kg Derek 19 cm 75 kg (a) Find (i) the mean height; (ii) the mean weight. Plot the above data on the grid below and draw the line of best fit. (6) (Total 15 marks) height (cm) 19 185 18 175 17 165 16 6 65 7 75 weight (kg) (Total 4 marks) C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 21 of 21
IB Math Studies 1: Statistics - Scratch 213-14 Descriptive Statistics Practice Problems: MarkScheme 1. (a) (i) mean = 13.7 (M1)(A1) (G2) (ii) sd = 2.52 (M1)(A1) (G2) For attempting to put their numbers in order (M1) 13.1 (A1) (G2) 2. (a) n = 4 (A2) (C2) Mean number of games is 9.8 (accept 9). (M1)(A1) (C2) Note: Award (M1) for indicating a sum of games times frequency (possibly curtailed by dots) or for 227 seen. 6 1 = 24% 25 1 (M1)(A1) (C2) Note: Award (M1)(A) if 6 is replaced by 1. No other alternative. (d) Modal number of games is 7. (A2) (C2) 3. (a) 3.2 + 7.2 + x + 9.6 + 17.7 + 28. + 2.1 µ = = 1.1. 7 (M1) Hence 67.8 + x = 1.1 7 = 7.7. (A1)(M1) x = 7.7 67.8 = 2.27 (accept 2.27 or 2.3 million). (M1)(A1) (C5) Median is the middle value which is 7.2. (M1)(A1) Bangkok. (A1) (C3) Note: Award (M)(A)(A) for Paris. 4. (a) a + 6 = 5 2 (M1)(A1) a + 6 = 1 (A1) a = 4 (A1) (C4) 42 + a + b = 6 (M1) 8 42 + a + b = 48 (A1) a + b = 6 4 + b = 6 (A1) b = 2 (A1) (C4) 5. (a) 63 kg (A1) 1 (i) 7.5 kg (G1) (ii) 14.6 kg (also accept 15.2 kg) (G1) 2 Total weight of 12 students = 846 kg Total weight of 11 students = 11 7 = 77 kg (M1) Weight of student who left = 846 77 = 76 kg (A1) 2 6. (a) Mode = 2 (A1) (i) x f fx 1 2 3 6 3 1 1 3 2 3 Total 2 26 (A1) Note: Award (A1) for three or more correct bold entries. 26 (ii) Mean = 2 (M1) [6] [8] [8] [8] [5] OR Note: Award (M1) for dividing fx total by 2. = 1.3 (A1) C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 1 of 17
IB Math Studies 1: Statistics - Scratch 213-14 Mean = 1.3 7. (a) 6 Mean = 1 = 6 (A1) (C1) Mode = 2 (A1) (C1) 2, 2, 2, 4, 5, 6, 8, 9, 1, 12 Median = 5 + 6 (M1) 2 = 5.5 (A1) (C2) 8. (a) 4 + 3 + a + 8 + 7 + 3 + 9 + 5 + 8 + 3 5.5 = 1 (M1) 55 = 5 + a 5 = a (A1) (C2) 3, 3, 3, 4, 5, 5, 7, 8, 8, 9 (M1) Median = 5 (A1) (C2) Note: Award (M1) for arranging scores in ascending or descending order. Follow through with candidate s a 9. (a) T = 5 (A1) m = 19 (A1) p = 9 (A1) (d) q = 43 (A1) 7 + 4 + 5 + 4 + 8 + T + 14 + 4 1. (a) = 7 (A1)(A1) 8 Note: Award (A1) for sum +T, (A1) for 56 or 7 8 or 8 in the denominator and 7 seen. 11. (a) T = 1 (A1) (C3) 4 (A1) (C1) 4, 4, 4, 5, 7, 8, 1, 14 (M1) Note: Award (M1) for arranging their numbers in order. Median = 6 (A1)(ft) (C2) Grade Frequency 1 1 2 4 3 4 3 5 (4) 6 5 7 Note: Award (A1) for three correct. Award (A) for two or fewer correct. Mode = 6 (A1)(ft) (C1) Median = 4.5 (M1) Note: (M1) for attempt to order raw data (if frequency table not used) (A1)(ft) or (M1) halfway between 1 th and 11 th result. (C2) (A2) (C2) (C2) [4] [4] [4] [4] [6] C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 2 of 17
IB Math Studies 1: Statistics - Scratch 213-14 (d) 7 (.35, 35 %) (A1)(ft) (C1) 2 12. Unit penalty (UP) is applicable where indicated. (a) Total = 2 + 3 + 5 + 7 + 11 + 5 + 6 + 9 + 2 + 1 (M1) Note: (M1) is for a sum of frequencies. = 51 (A1)(G2) 2 (i) modal interval is 6 7 (A1) Note: Award (A) for 65 (ii) median is length of fish no. 26, (M1) also 6 7 (A1)(G2) Note: Can award (A1)(ft) or (G2)(ft) for 65 if (A) was awarded for 65 in part (i). 2 25+ 3 35+ 5 45+ 7 55+... UP (iii) mean is (M1) 51 UP = 69.5 cm (3s.f.) (A1)(ft)(G1) 5 Note: (M1) is for a sum of (frequencies multiplied by mid-point values) divided by candidate s answer from part (a). Accept mid-points 25.5, 35.5 etc or 24.5, 34.5 etc, leading to answers 7. or 69. (3s.f.) respectively. Answers of 69., 69.5 or 7. (3s.f.) with no working can be awarded (G1). UP (i) standard deviation is 21.8 cm (G1) Note: For any other answer without working, award (G). If working is present then (G)(AP) is possible. (ii) 69.5 + 3 21.8 = 134.9 > 12 (M1) no fish (A1)(ft)(G1) 3 Note: For no fish without working, award (G1) regardless of answer to (i). Follow through from (i) only if method is shown. (d) 5 fish are less than 4 cm in length, (M1) 5 46 Award (M1) for any of,,.98 or 9.8%,.92, 9.2% or 5.1 seen. 51 51 hence no fine. (A1)(ft) 2 Note: There is no G mark here and (M)(A1) is never allowed. The followthrough is from answer in part (a). (e) (i) and (iii) are correct. (A1)(A1) 2 13. Unit penalty (UP) is applicable in question part only. (a) 45 t < 6 (A1) 1 42.4 minutes (G2) UP 21.6 minutes (G1) 3 [6] [14] (A4) 4 (8 15 + 16 16 + 11 17 + 7 18 + 3 19) 14. (a) mean = (M1)(M1) 45 Note: Award (M1) for five correct products shown or implied in the numerator, (M1) for denominator 45. [8] C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 3 of 17
IB Math Studies 1: Statistics - Scratch 213-14 15. = $165.78 per week (allow $166) (A1) (C3) Notes: For 165. 7 ɺ 7 or 165 award (C3) for exact answer. 9 For 165.77 award (C2) and no (AP). For 165.77 with no working award (C2)(A)(AP). $164.99 ($165) (A1) (C1) 16 $155 = $248 and 16 164.99 = $2639.84 ($264) (M1) Note: The (M1) is for a sensible attempt to calculate both bounds or for showing division by 16 of any of the values (i) to (v). $243 is not possible (A1)(ft) (C2) Note: Follow through if wrong modal group is used in. Team x and σ I B II C III D IV A (A6) (C6) Note: Award (A6) for all correct, (A4) for 2 correct or for 3 correct and 1 blank, (A2) for 1 correct but (A) if the same letter appears 4 times. 16. (a) 19 (A1) (C1) 6 12 thousand dollars (A1) (C1) 32 For correct numerator (A1) 19 For correct denominator (A1) (C2) 1 (d) For correct numerator (A1) 39 For correct denominator (A1) (C2) 17. (a) 6 hours (accept (5.5 6.5)) (A2) (C2) ( 4 4 + 5 5 + 6 9 + 7 8 + 8 4) 3 (M1)(A2)(A1) 183 = 3 = 6.1 (A2) (C6) Note: Award (M1) for method, (A2) for all 5 terms in numerator correct. ((A1) for 3 or 4 terms in the numerator correct), (A1) for denominator. 18. (a) Number of people (frequency) 25 2 15 1 5 Amount of Money spent on Video Rentals per Month.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 Dollars (class boundaries) [6] [6] [6] [8] C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 4 of 17
IB Math Studies 1: Statistics - Scratch 213-14 (A5) 5 Notes: Award (A1) for correct scales, (A1) for bars connected (where possible), (A1) for axes labelled correctly, (A2) for bars all correctly drawn. If one bar is incorrect, award (A1), if 2 or more bars are incorrect, award (A). 3.5 4.5 counts as a bar, height zero. A1so, allow horizontal scale to be 1, 2, 3, etc, with bars moved.5 to the right. (i) 11 2 (accept 1.5 2.5) (A1) (ii) 11 2 (accept 1.5 2.5) (A1) 2 (i) 1 1 = or 2% 5 5 (A1) (ii) 4 4 = or 8% 5 5 (A1) (iii) 6 3 = or 3% 2 1 (A2) Note: Award (A1) for correct numerator and (A1) for correct denominator. (iv) 36 9 = or 9% (A2) 6 4 1 Note: Award (A1) for correct numerator and (A1) for correct denominator. 19. (a) 3 (A1) 6 (A1) [13] 1 ((3 2) + (4 4) +... + (1 1)) = 5.9 3 (M1) = 6 (nearest whole number) (A1) 2. (a) ( 25)5 + (35)4 + (45)3 + (55)2 + (65)3 mean = 17 (M1) Note: Award (M1) for using mid-interval values. mean = 41.5 (A1) [4] frequency (# of smokers) 21. (a) Mean = 5 4 3 2 1 2 3 4 5 6 7 ages Note: Award (A1) for correct intervals, (A1) for correct bar lengths (A2) 5 + 1 1 + 6 2 + 3 3 + 1 4 = 1.4 (M2)(AG) 2 25 Note: Award (M1) for the numerator and (M1) for the denominator. [4] _ ) f ( x x 2 2 2 2 = 5( 1.4) + 1(1 1.4) + 6(2 1.4) + 3(3 1.4) 2 + 1(4 1.4) 2 = 28 (M2) _ ) 2 Note: Award (M1) for ( x x appropriate frequencies. values, and (M1) for multiplying by the C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 5 of 17
IB Math Studies 1: Statistics - Scratch 213-14 S.D. = 28 25 (M1) = 1.6 (AG) 3 Award (R1) for each acceptable reason, e.g. Group 2 has more children in total. Group 2 has a larger number of children per female. Group 2 generally have larger families. (R2) 2 (d) 3 +1 P(> 2 children) = (M1) 4 = 25 25 (A1) 2 (e) (i) P(both females have > 2 children) = 4 3 (M1) 25 24 12 1 = or or.2 6 5 (A1) (ii) 4 21 P(only 1 female has > 2 children) = 2 25 24 (M2) 4 21 Note: Award (M1) for, (M1) for multiplying by 2. 25 24 168 21 = or or.28 (A1) 6 75 6 1 (iii) P(second has 2 children first has ) = or or.25 (A1) 6 24 4 22. (a) Interval 11 15 (A1) Mid-intervals 3, 8, 13, 18... (M1) Note: Award (M1) for all correct numbers. Σxf = 48 + 224 + 338 +... Note: Award (M1) for attempt to obtain sum. Mean = 13 23. (a) Median = 45 (A1) Note: Accept 45.5 (C1) 53 37 for identifying correct quartiles (A1) = 16 for correct answer to subtraction (A1)(ft) Note: (ft) on their quartiles (C2) (M1) (A1) [15] [4] Median marked correctly. Box with ends at candidate s quartiles. End points at 21 and 72 joined to box with straight lines. Note: Award (A) if lines go right through the box. (A1)(ft) (A1)(ft) (A1) (C3) 24. Unit penalty (UP) is applicable where indicated. UP (a) 26cm (A1) (C1) 33 19 for identifying correct quartiles. (A1) [6] C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 6 of 17
IB Math Studies 1: Statistics - Scratch 213-14 UP =14cm. (A1)(ft) (ft) on their quartiles. (C2) 5 1 15 2 25 length (cm) correct median correct quartiles and box endpoints at 6 and 47, joined to box by straight lines. 3 35 4 45 5 (A1)(ft) (A1)(ft) (A1) (C3) 25. Unit penalty (UP) applies in part (a) in this question (a) Median = 11m (A1) (C1) Interquartile range = 14 1 (A1) = 4 (A1)(ft) (C2) Note: (M1) for taking a sensible difference or for both correct quartile values seen. [6] correct median (A1)(ft) correct quartiles and box (A1)(ft) endpoints at 3 and 28, joined to box by straight lines (A1) (C3) Note: Award (A) if the lines go right through the box. Award final (A1) if the whisker goes to 2 with an outlier at 28 26. (a) (ii) and (iv) are discrete. (A1)(A1) (C2) Notes: Award (A1)(A) for both correct and one incorrect. Award (A1)(A) for one correct and two incorrect. Otherwise, (A)(A). (i) Median = 1 (A1) (ii) Q3 = 12 (A1) (iii) Min value = 1(±.2) (A1) (C3) Any three different integers whose mean is 1 eg 9, 1, 11. (A1) (C1) 27. (a) (i) 95 6 = 89 (A1) (ii) 73 5 = 23 (A1) (iii) 6 (A1) (C3) (i) 62 (A1) (ii) 73 43 = 3 (A1) (C2) The girls as the IQR is larger (R1) (C1) 28. (a) w = 43 (C1) [6] [6] [6] C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 7 of 17
IB Math Studies 1: Statistics - Scratch 213-14 1 cars parked per hour 8 cumulative frequency 6 4 2 2 4 6 8 1 number of cars (A5) Notes: Award (A1) for labels ( cumulative is not essential). Award (A1) for scales. Award (A2) for all points correct, (A1)(A) for 4 points correct. Award (A1) for neat curve or straight line segments. The curve must extend to zero for the last (A1). The points must be at 19, 39, etc (ft from candidate s scale). If the points are displaced consistently, (eg to 2,4 etc or to mid-points) then award (A)(A1) ft for points. If the scale is marked as 19,39, etc, this is allowed as long as is not included, is shown displaced right by one unit. Scale marked as an interval (eg 19) is acceptable, if written within the interval. Bar graph or histogram receive (A) for curve but if bars are at correct height and terminate at 19, 39, etc or have correct points marked on them, then award (A2) for points or (A)(A1) ft if points are consistently incorrect. median = 63 ± 2 (M1)(A1) (C2) Note: Award (M1) for a horizontal line drawn at height 48. Answer must be an integer, follow through from candidate s graph. 29. (a) m = 137 + 31 = 168, (M1)(A1) n = 194 + 6 = 2 or just n = 2 (M1)(A1) (C4) 137 students are aged below 2. (A2) (C2) 25% of 2 is 5. (M1) 5th student is 18 years old. (A1) (C2) 3. (a) (i) c = 97 (A1) 1 (ii) (not drawn to scale) Weights of fish cumulative frequency 14 12 1 8 6 4 2 (C5) [8] [8].5.7.9 1.1 1.3 1.5 weight (kg) (A4) 4 Note: Award (A1) for correct scales and axes labelled, (A2) for 5 correct points, C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 8 of 17
IB Math Studies 1: Statistics - Scratch 213-14 (A1) for 3 to 4 correct, (A1) for the curve. (iii) median is.95 kg line drawn correctly on diagram (M1)(AG) 1 (i) 9 th percentile... 13.9 = 117 (M1) The zoo buys 13 fish (±2). (A1) or (G2) 2 (ii) First quartile = 32/33 fish. (M1) Maximum weight =.79 kg (±.3). (M1)(A1) 3 (i) maximum :.95 1.1 = 1.45 kg (1.5 to 3 s.f.) (A1) minimum :.95.9 =.855 kg (A1) 2 (ii) number of fish bought by restaurant = 88 46 = 42(±4) (M1)(M1)(A1) 3 31. (a) (i) 25 minutes (±2 minutes) (A2) (ii) Lower quartile = 18 (±1 minute) (A1) Upper quartile = 32 (±1 minute) (A1) Interquartile range = 32 18 = 14 minutes (±2 minutes) (A1) OR Accept [18 to 32] as interval for the interquartile range. (A3) 5 p = 2 (A1) q = 3 (A1) 2 Midpoint Frequency M f 2.5 2 5 1 2 2 17.5 2 35 22.5 4 9 3 6 18 42.5 3 1275 55 1 55 (A1) Total = 2 Total = 5125 (A1) [16] Mean = 5125 = 25.625 (exact) or 25.6 (3 s.f.) (M1)(A1) 2 Note: Not every step needs to be seen to get the marks. OR Mean = 25.625 or 25.6 (using GDC) (G4) 4 32. (a) Modal group =17 h < 18 (A1) 1 Mean = 171 (G2) Standard deviation = 11.1 (G2) 4 Median = 171 (±1) (A1) 1 (d) Lower quartile = 164.5 (±1) (A1) Inter-quartile range = 177.3 164.5 = 12.8 (A1) 2 (e) number = 52 (±2) (A1) [11] 52 percentage = 1 = 26% (A1) 2 2 33. [1] C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 9 of 17
IB Math Studies 1: Statistics - Scratch 213-14 cumulative frequency 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 scores (a) 3 (A2) (C2) P75 = 46, P25 = 2 (M1) 46 2 = 26 (A1) (C2).35(8) = 28, P35 = 23 (A2) (C2) (d) 1 = 12.5% (A2) (C2) 8 Note: Allow ±2 for each measurement. Note: Allow ft for, and (d) if percentile scores were figured on the basis of 1 instead of 8. 34. (a) 19 or 2 people (A1) Median salary = 15 GBP (A1) 8% of 2 = 16 23 ± 5 (M1) (A1) 35. (a) s = 17, t = 9 (A1)(A1) 2 (M1)(M2) 3 [8] [4] 1 9 8 Cumulative frequency 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 Marks Note: Award (M1) for both axes with correct scales and correctly labelled. C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 1 of 17
IB Math Studies 1: Statistics - Scratch 213-14 36. (a) Award (M2) for 8, 9, 1 points plotted correctly, (M1) for 5, 6, 7 points plotted correctly, (M) for 4 or less. Accept a polygon or a curve. (i) Median mark = 48 (±1) (A1) (ii) Lower quartile = 36 (±1) (A1) (iii) Pass mark if 4% pass = 51 (±1) (M1)(A1) 4 Note: Follow through with candidate s own graph. Award (M)(A1) ft if candidate correctly finds the grade (44) where 4% fail. number of students Time less than (mins) Cumulative frequency 1.5 15.5 2.5 25.5 3.5 35.5 4.5 7 2 45 73 93 1 (A2) 2 Note: Award (A1) for each correct column 1 9 8 7 6 5 4 3 2 1 c(i) c(ii) [9] 37. (a) 5 1 15 2 25 3 35 4 time (A3) 3 Note: Award (A1) for the correct scale and labelling. Award (A2) for plotting 6 or 7 points correctly, (A1) for plotting 4 or 5 points correctly. (i) 12 ± 1 students (allow ft) (A1) (ii) 31 ±.5 minutes (allow ft) (A1) 2 L (cm) f Σf 29 2 2 31 4 6 33 8 14 35 21 35 37 3 65 39 18 83 41 12 95 43 5 1 (A2) 2 Notes: Award (Al) for four correct entries in the column headed Σf. Award (A2) for all 8 correct. [7] C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 11 of 17
IB Math Studies 1: Statistics - Scratch 213-14 cumulative frequency 1 9 8 7 6 5 4 3 2 1 27 29 31 33 35 37 39 41 43 length L (cm) Notes: Award (Al) for both axes and correct scale. Award [½ mark] for each correctly plotted point and round up to a maximum of [2 marks]. (A3) 3 (i) Median length of mackerel = 36 cm ±.2 cm (M1) = 36 cm (A1) (ii) Interquartile range of mackerel length = 3.8 ±.2 cm (M1) = 4 cm (A1) 4* *(read from candidate s curve) 38. Unit penalty (UP) is applicable where indicated. 55 5 45 4 35 number of chickens 3 25 2 15 1 5 1 2 3 4 weight (kg) (a) Dashed or solid line (using a ruler) passing through mid-points of the 4 bars. (A1) 1 1 Starting and finishing at, and 5, respectively. 2 2 (A1) (C2) 96 (A1) (C1) UP 3 weight < 4 kg. (Accept 3 4 kg) (A1) (C1) (d) For adding three heights or subtracting 14 from 96 (M1) 82 41 (.854 or, 85.4%) (ft) from. 96 48 (A1)(ft) (C2) 39. (a) Discrete (A1) (C1) For attempting to find fx / f (M1) 5 6 [9] [6] C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 12 of 17
IB Math Studies 1: Statistics - Scratch 213-14 2.73 (A1) (C2) 1.34 (A1) (C1) Notes: for and, if both mean and standard deviation given to 2 significant figures Award (C1)(C)(AP) for 2.7. Award (A1)(ft) for 1.3 ((AP) already deducted) (d) Attempt to find their mean + their standard deviation (can be implied) (M1) 23, (ft) their mean and standard deviation. (A1)(ft) (C2) 4. (a) (i) 19.2 (G1) (ii) 1.45 (G1) r =.942 (G1) Strong, positive correlation. (A1)(ft)(A1)(ft) (d) (i) d = 11.5 (G1) (ii) n = 11.5 19.6 1 (A1)(ft) = 125 (accept 126) Note: Answer must be a whole number (e) It is unreliable to extrapolate outside the values given (outlier). (R1) 41. Unit penalty (UP) applies in part in this question. (a) [6] [8] (A1)(A1) (C2) Note: (A1) for all correct heights, (A1) for all correct end points (39.5, 49.5 etc.). Histogram must be drawn with a ruler (straight edge) and endpoints must be clear. Award (A1) only if both correct histogram and correct frequency polygon drawn. 44.5 (A1) (C1) 44.5 6 + 54.5 18 +... Mean = 42 (M1) Note: (M1) for a sum of frequencies multiplied by midpoint values divided by 42. UP = 58.3 kg (A1)(ft) (C2) Note: Award (A1)(A)(AP) for 58. (d) Standard deviation = 8.44 (A1) (C1) Note: If is given as 45 then award 45 (A) 58.8 kg (M1)(A1)(ft) or (C2)(ft) if no working seen. (d) 8.44 (C1) 42. (a) (i) Mean = (5.96 + 5.95 + 6.2 + 5.95 + 5.99) / 5 = 5.974 (5.97) (A1) (ii) error % error = 1% actual value 6 5.974 = 1% =.435% (M1)(A1)(ft) (C3) 5.974 [6] C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 13 of 17
IB Math Studies 1: Statistics - Scratch 213-14 Note: (M1) for correctly substituted formula. Allow.53% as follow through from 5.97 Note: An answer of.433% is incorrect. number is 29.45728613 (i) Nearest integer = 29 (A1) (ii) Standard form = 2.95 1 1 (accept 2.9 1 1 ) (A1)(ft)(A1) (C3) Notes: Award (A1) for each correct term Award (A1)(A) for 2.95 1 43. (a) 1.6 1 19, 9.8 1 18, π, 3.5,.673 1 6, 673, 6.73 1 5 (A4) (C4) Notes: Award (A1) for π before 3.5 Award (A1) for 1.6 1 19 before 9.8 1 18 44. (a) Award (A1) for the three numbers containing 673 in the correct order. Award (A1) for the pair with negative indices placed before 3.5 and π and the remaining three numbers placed after (independently of the other three marks). Award (A3) for numbers given in correct decreasing order. Award (A2) for decreasing order with at most 1 error The median is 3.5. (A1)(ft) (C1) Note: Follow through from candidate s list. π is irrational. (A1) (C1) [6] [6] (A4) Note: Award (A1) for correct scales and labels, (A3) for correct points, (A2) for 7 or 8 correct, (A1) for 5 or 6 correct. (i) 4 (G1) (ii) 2.58 (G1) (iii) 3 (G1) (iv) 4.78 (G1) Note: If wrong version of s.d. used in (ii), can (ft) in (iv) (5.7). L correctly plotted on graph and named (A1)(ft) (d) (i) r =.986 (.987) (G1) (ii) (very) strong positive correlation (R1)(ft)(R1)(ft) (e) y = 1.83x + 22.7 (y = 1.825x + 22.7) (G1)(G1) Award (G1) for y= 1.83x (1.825x), (G1) for 22.7 (f) Line drawn on graph. (A1)(A1)(ft) Note: Award (A1) for passing through the mean point, (A1) for y intercept between 22 and 23. (g) (i) 32.6 cm (A1)(ft) C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 14 of 17
IB Math Studies 1: Statistics - Scratch 213-14 (ii) Note: Allow margin of error of.2 from value on candidate s diagram. Not possible to find an answer as the value lies too far outside the given set of data. (R1) 45. (a) 8 62.1 + 2 11 = 499 (M1)(A1) Note: Award (M)(A) if 2 11 is subtracted and ft the remainder of the question to answers of 61.825 (or 61.8) and 59.7 respectively. 499 = 62.375 (or 62.4 to 3 s.f.) (M1)(A1) (C4) 8 499 + 56.8 6 = 8398 (M1)(A1) [18] 8398 = 6. (3 s.f.) (M1)(A1) (C4) 14 Note: An answer of 6 (2 s.f.) with no working receives (G2) or with working using 499 receives (M1)(A1)(M1)(A) AP, however, if 8 62.4 is used then 6 is an exact answer and can receive all the marks. 46. (a) 44.5 12.4 5.35 = 6 (M1)(A1) (C2) 9 12.4 = 1116 (M1)(A1) 1116 44.5 43.2 = 128.3 (M1)(A1) 128.3 = 11.7 88 (M1)(A1) (C6) Note: Award (M)(A) then ft for 88 12.4. 128.3 Award (M)(A) for. 9 47. (a) (i) 1992 mean = $1.59, Sd = $.727 (or.73) (A1) (A1) (accept.777 or.78) (ii) 22 mean = $1.98, Sd = $.635 (or.64) (A1) (A1) 4 (accept.679 or.68) (i) r =.314 =.664 or.314 =.672 or.314.73.64.727.635.777.679 =.588 (M1) (A1) OR R =.672 (G2) (ii) There is a weak positive correlation (R1) (R1) 4 y 1.98 =.314 ( x 1.59) 2 (.73) (M1) y =.582x + 1.5 (A1) (A1) OR y = 1.98 =.314 2 (.727) ( x 1.59) (M1) y =.587x + 1.5 (A1) (A1) OR y =.588x + 1.5 (G3) 3 (d) y =.582 2.6 + 1.5 = $2.56 (A1) OR y =.587 2.6 + 1.5 = $2.58 (A1) OR y =.588 2.6 + 1.5 [8] [8] C:\Users\Bob\Documents\Dropbox\Desert\Studies\2Stats\TestsQuizzesPractice\MSIBStatisticsPractice.doc on 12/31/13 at 1:56 PM Page 15 of 17