New QMaths 11B CD-ROM
Additional exercise 2.5
Analysing data
1 An archaeologist discovered 25 coins that had been produced in the Middle Ages. The weights (in grams) of the coins were:
8.12 8.21 8.33 8.12 8.26 8.14 8.21 8.56 7.78 8.24 8.34 8.62 8.45 7.89 8.25 8.65 8.19 8.77 7.91 7.80 8.32 8.56 8.64 8.73 9.84 Draw a box-and-whisker diagram to show these results.
2 The numbers of typing mistakes in a page of typing were recorded for 24 business college students as follows:
4 1 0 12 9 4 5 7 2 0 1 9 5 9 10 21 4 7 6 3 5 4 8 0
a Calculate the mean number of mistakes made. b Draw a boxplot for this data.
c Arturo made 7 mistakes. How does he compare with the rest of the class?
3 In each case that follows, identify whether any of the data items given are outliers. a Median = 30, interquartile range = 10. Data items to check: 37, 35, 59, 11, 51, 16
b Median = 580, interquartile range = 110. Data items to check: 701, 736, 174, 503, 255, 672 c Mean = 100, standard deviation = 15. Data items to check: 138, 55, 123, 112, 107, 66 d Mean = 61.3, standard deviation = 9.4. Data items to check: 35.1, 42.7, 72.9, 72.7, 64.6, 85.4
4 a Find the mean, median and mode of the following data.
b Draw a histogram of the data and indicate on it the positions of the three measures of central tendency. c Explain the differences between the three measures in this case.
5 The minimum daily temperatures (°C) in Stanthorpe were recorded over a 50-day period.
3 4 7 8 2 5 6 3 1 7 1 5 2 3 7 3 5
1 4 2 4 3 6 4 0 3 6 3 1 6 4 3 1 7
8 3 4 9 6 2 3 5 8 3 9 1 7 2 3 5
a Draw up a cumulative frequency table of this data.
b What was the modal minimum daily temperature over this period? c What was the mean minimum daily temperature?
d Calculate the median minimum daily temperature. e What is the most likely minimum?
f What is the typical minimum?
6 The 8 employees at a small business are paid the following amounts per week:
$490 $560 $540 $540 $600 $520 $510 $1180
a For these wages find the:
i mean ii median iii mode.
b If the workers are granted a flat rise of $40 per week, what are the new:
i mean? ii median? iii mode?
c Do you think that the mean is an appropriate measure of central tendency in this case? d What should be used as the measure of a typical wage?
7 The 12 runners in a horse race are given the following weights by the handicapper:
No. 1 58 kg No. 2 56 kg No. 3 55 kg
No. 4 54.5 kg No. 5 54.5 kg No. 6 54 kg
No. 7 53 kg No. 8 52 kg No. 9 52 kg
No. 10 52 kg No. 11 50 kg No. 12 48 kg
x 1 2 3 4 5 6 7 8 9
f 6 8 7 5 4 2 1 2 2
New QMaths 11B CD-ROM
Additional exercise 2.5 Analysing data continued
a For the handicapper’s weights, find the:
i mean ii mode iii median.
b What is the typical weight?
c If the top weight (58 kg) is given a 2.5 kg allowance because it is ridden by an apprentice jockey, which (if any) of the three measures calculated will remain unchanged?
8 The mean height of the Brisbane Bullets basketball team is 2.01 m. If an American import who is 2.10 m tall joins the existing squad of 9 players, what will be the new mean height of the squad?
9 A vigoro player has scored this number of runs at each of her innings so far this season:
48 19 26 15 27 50 3
How many runs must she score in her next inning if she is to have an average of 35 for the 8 innings in the season?
10 The mean height of the 12 boys in a maths class is 179 cm. The mean height of the 16 girls is 172 cm. What is the mean height of all students in the class?
11 A teacher gave a quiz containing five questions, each worth 1 point. She found that 40% of the class scored 5 points, 25% scored 4 points, 15% scored 3 points, 10% scored 2 points, 5% scored 1 point and 5% scored 0.
a If there are 20 people in the class, calculate the mean score for the class. b What would be the mean score if there were 40 students?
c If you did not know the number of students in the group, would it still be possible to calculate the mean for the class?
12 The test scores of some students doing both Physics and Chemistry were as follows:
Physics: 43 57 59 64 78 56 43 49 34 28 42 55 67 69 62 54
Chemistry: 51 56 62 68 84 59 44 29 26 48 67 78 75 84 58 65 a Draw a back-to-back stem-and-leaf diagram to compare the results.
b Compare the results in Physics and Chemistry for these students.
13 The relative humidities at 3 pm in two towns over a period of 2 weeks were:
Town A: 33 35 67 45 48 67 84 56 58 57 45 48 68 56
Town B: 45 48 67 78 79 84 65 58 43 59 69 89 78 69 a Draw a back-to-back stemplot of these humidities.
b Compare the results and suggest which town got more storms.
14 The advertised prices of houses in two different areas of Brisbane are shown below. Construct a back-to-back stemplot and comment on the results.
CARINDALE OPEN TODAY
PRICE SLASHEDfor quick sale
Entertainer’s 2-storey, 4 brms, study, pool $389K 22Smith St Carindale
CARINDALE Close to parks, immac. presentation, 3 brms, 2 baths, large rumpus, 2 car LU $319K 9 Ryan St Carindale URGENT SALE Choice of 2 x 2
brm t’houses, lounge, family, courtyard, LU car accom, internal access, pool, security, on site mgr $200K 2/2A Bank St Carindale CARINDALE HTS New 2 brm
t’house, no GST $254K
19 May St Carindale NEW LISTING Immac, lowset,
4 brms, close to park $309K
52 Cardin St Carindale
VOGUE LIVING 1 yr old, d/storey 36sq. 4 brms + study, 3 baths, 2 fam rooms, pool $423K82 Peel Rd Carindale CARINDALE Huge 4 brms,
ensuite, study, separate rumpus, large kitchen + family room 681sqm Great entertainer $488K
33 Oak St Carindale CARINDALE L’set 2 rumpus
rooms, 4 brms, ens, fam. rm, quality fitout. Must see $379K
16 Berry St Carindale CARINDALE 40sq, 5 brms,
3 bath, rumpus, dbl. gar, $389K8 Mary St Carindale CARINDALE Huge 47sq. with 6 brms, 3 with ens, study, family room, pool, 18 mths old $440K
52 Chestnut St Carindale STYLISHLY PRIVATE
4 dbl brms, huge ensuite, parents retreat, lg. pool + pergola $498K
53 Mard St Carindale DEAN REALTY
PH: 4822 3111 ALL HRS
INSPECT TODAY
CLAYFIELDGreat New Listing.
5 brm fam. home, 2 bthrms, rumpus, study, granite kitch, pool, 809sqm $695K 25Bent St Clayfield
CLAYFIELD Renov. 4 brm + study, 2 storey Q’lander, mod. kitch, North v’dah over pool 809sqm $640K 9 Ryan St Clayfield CLAYFIELDLarge 2 brm, ens,
sec. leafy out-look, loc’n demand high rental $230K 14 Avondale St Clayfield
GRAND COLONIAL 6 brm, 4 bthrms, views, pool, 2216sqm $884K 82 Woodville Rd Clayfield CLAYFIELD 4 brm, ens, rumpus,
pool, $320K 2 Bright Ave Clayfield
CLAYFIELD New 2 storey 4 brm brick, large open plan, IG pool $620K 18 Scott St Clayfield CLAYFIELD 4 brm, Colonial, hilltop pos’n, 784sqm $625K
23 Parker Ave Clayfield CLASSIC 3 brm colonial, superb
rear deck/pool, $398K
5 Ireland St Clayfield CLAYFIELD 4 brm h’set
Q’lander, good pos’n, needs some TLC $449K 153 Union St Clayfield
CLAYFIELD 5 year old, 2 storey brick, 2 way duct a/con, 3-5 brm, 2 bthrms, spacious living, multi car accom. $990K
24 Tennis Grove Clayfield OPPORTUNITY 3 brm + ens,
hard work done, new roof and gutters, restump, insulation, sec. windows/doors, corner pos’n. 449sqm. $388K 19 Aroona St Clayfield
SMITH & GRAY
Phone: 3288 2233
New QMaths 11B CD-ROM
Additional exercise 2.5 Analysing data continued
15 The maximum temperatures (°C) in two Queensland towns over a fortnight were:
Town A: 22 24 28 32 35 27 24 29 28 29 27 31 23 30
Town B: 25 26 26 24 26 28 28 26 29 30 27 24 29 31 a Draw boxplots to compare the temperatures.
b Which town is most likely to be near the sea?
16 The number of words in each sentence of some prose was counted as part of a readability test. The data was recorded as follows by a researcher:
20 36 14 11 20 19 27 21 14 25 18 17 45 22 17
14 15 20 21 18 17 16 13 23 27 10 18 23 31 40
a Calculate the mean sentence length and standard deviation of the length. b Draw a boxplot of the data.
A page taken from another source had the following sentence lengths:
12 16 18 25 34 19 13 15 18 20 23 25 24 14 16
c Draw another boxplot to compare this data with the first set. d Find the mean and standard deviation of the second set of data. e Compare the difficulty of the passages in terms of sentence length.
17 Theo and Martina both sell insurance polices for the same company. The stemplot shows the number of policies each sold per month over an 11-month period.
a Find the mean, range, interquartile range and standard deviation for each person.
b Comment on their relative sales records.
18 Carrie submitted her last 15 golf score cards to the club handicapper so that her handicap could be reviewed. The scores were:
85 90 86 80 95 92 81 86 87 88 89 90 91 92 82
a Find the summary statistics of her results. b Are there any outliers?
c Draw a boxplot of her results.
Carrie’s friend Kathy had the following results:
86 102 81 90 105 78 89 86 80 91 84 82 79 81 88
d Find the summary statistics for Kathy’s results. e Are there any outliers in Kathy’s scores? f Draw another boxplot to compare their results.
g Whom would you expect to win a match between these friends?
19 Two golf ball manufacturers produce balls with a nominal diameter of 3.70 cm. A sample of 20 balls produced by each company was measured. The results (in centimetres) were:
Beaut Ball Company:
3.78 3.76 3.69 3.70 3.75 3.68 3.68 3.71 3.76 3.67 3.73 3.71 3.71 3.67 3.74 3.65 3.72 3.70 3.68 3.66
E-Zee Ball Company:
3.70 3.68 3.61 3.76 3.77 3.70 3.81 3.71 3.62 3.69 3.66 3.79 3.64 3.73 3.68 3.73 3.77 3.71 3.68 3.66
Statistically compare the outputs of the companies, and comment on the effectiveness of the manufacturing processes employed by each.
Insurance policies sold in a month
Theo Stem Martina
1 9 7 4 3 7 6 5 1 1
4 0 1 2 3 4
7 8 8 9 1 1 1 2 35 0
New QMaths 11B CD-ROM
Additional exercise 2.5 Analysing data continued
20 Travelling to work, I have two alternative routes. Both pass through sets of traffic lights. Over 8 journeys by each route, I record the time (minutes) spent waiting at lights. The results are shown below. Use statistical methods to recommend a route, and give reasons.
Route A: 15 15 16 10 17 20 14 13
Route B: 13 16 15 15 13.5 14 16.5 17
21 The percentages of some candidates in an examination are given below:
62 48 72 50 21 49 23 38 51 26 41 37 36 40
32 64 98 31 39 32 31 65 45 38 3 53 61 33
55 62 17 48 61 14 49 18 75 57 25 53 31 48
a Determine whether there are any outliers by treating the data as discrete. b Determine whether there are any outliers by treating the data as continuous.
22 In order to make comparisons easier, histograms are sometimes set out in a back-to-back way as is shown here. The histograms show the proportion of Aboriginal and Torres Strait Islander population and total population by age group in Queensland in 1999. Comment on any relevant comparisons that can be made concerning the age distributions shown.
23 Petrona got 15/40 for Film and TV and 32/50 for Speech and Drama. She said the Film and TV test was harder because only half the class got better than 10/40 but most of the Speech and Drama class passed. In fact, for the Film and TV test the mean was 9.8 and the standard deviation was 4.6, while for Speech and Drama the mean was 27.2 and the standard deviation was 7.4. Which subject did she do better in, compared to the rest of the class?
24 David is 198 cm tall and weighs 104 kg. Use the information in the table to determine whether David is thin or fat compared with the population.
Population mean Standard deviation
Height 174 cm 18 cm
Weight 84 kg 13 kg
75 and over 70–74 65–69 60–64 55–59 50–54 45–49 40–44 35–39 30–34 25–29 20–24 15–19 10–14 5–9 0–4
0 5
10 0 5 10
Percentage
15 15
Aboriginals and Total of Queensland
Proportion of ATSI population and total population
Torres Strait Islanders Age group (years)
Percentage
by age group: Queensland, 1999
New QMaths 11B CD-ROM
Additional exercise 2.5
Answers
1
2 a 5.67 mistakes b
c He is worse than a little more than half the class.
3 a Outliers are 51, 59. b Outliers are 174, 255.
c Outliers are 55, 138. d Outliers are 35.1, 85.4.
4 a Mean = 3.7, median = 3, mode = 2 b c The distribution is asymmetrical.
5 a
b 3°C c 4.16°C d 3.7°C (continuous) e 3°C f 3.7°C
6 a i $617.50 ii $540 iii $540
b i $657.50 ii $580 iii $580
c No—it does not reflect the central measure very well. d Median
7 a i 53.25 kg ii 52 kg iii 53.5 kg
b 53.5 kg (median) c Both mode and median will be unchanged.
8 2.019 m 9 92 runs 10 175 cm
11 a 3.7 point b 3.7 points c Yes
12 a
Minimum temperature (°C) f c.f.
0 1 2 3 4 5 6 7 8 9
1 6 5 12 6 5 5 5 3 2
1 7 12 24 30 35 40 45 48 50
Students’ test scores
Physics Stem Chemistry
8 4 9 3 3 2 9 7 6 5 4 9 7 4 2 8
2 3 4 5 6 7 8
6 9
4 8 1 6 8 9 2 5 7 8 5 8 4 4
7.5 8 8.5 9 9.5 10
Weights of coins (g)
0 4 8 12 16 20
Number of mistakes in one page
Students’ typing mistakes
Frequenc
y
8
6
4
2
0
1 2 3 4 5 6 7 8 9 x
Mode Median Mean
New QMaths 11B CD-ROM
Additional exercise 2.5 Answers continued
b The results in Chemistry are a little higher, but more spread out. Most of these students find Chemistry easier than Physics, except for 1 or 2 students.
13 a
b The humidity is generally higher in town B, so this town probably gets more storms.
14
The prices of houses at Clayfield are generally more expensive than those at Carindale. A couple of houses are very much more expensive.
15 a
b Town B has more even temperatures, so it is probably nearer the sea.
16 a Mean = 21.1 words, σ= 8.03 words
b c
d Mean = 19.5 words, σ= 5.67 words
e The first passage is slightly harder (more words/sentence), but has greater variation in sentence length.
17 a For Theo: mean = 20.7 policies, range = 43 policies, IQR = 12 policies, σ= 10.17 policies For Martina: mean = 20.5 policies, range = 23 policies, IQR = 5 policies, σ= 5.37 policies b Their sales records are about the same, but Martina is more consistent.
18 a Mean = 87.6, σ= 4.19, modes = 86, 90 and 92, median = 88, range = 15, IQR = 6
b No c
d Mean = 86.8, σ= 7.66, modes = 81 and 86, median = 86, range = 27, IQR = 9
Humidity at 3 pm (%)
Town A Stem Town B
5 3 8 8 5 5 8 7 6 6 8 7 7
4 3 4 5 6 7 8
3 5 8 8 9 5 7 9 9 8 8 9 4 9
Advertised house prices ($’000)
Carindale Stem Clayfield
54, 00 89, 89, 79, 19, 09 98, 88, 40, 23
2 3 4 5 6 7 8 9
30 20, 88, 98 49
20, 25, 40, 95
84 90
22 24 26 28 30 32
Maximum temperature (°C) 34 Town B
Town A
Temperatures in two Qld towns
10 15 20 35 40 45
Words per sentence
25 30
Passage 1
Sentence length
Passage 2
10 15 20 35 40 45
Words per sentence
25 30
Sentence length
75 80 85 90 95 100
Carrie
Golf scores
Key: 5⏐4 = 45 = 4⏐5
New QMaths 11B CD-ROM
Additional exercise 2.5 Answers continued
e Yes, 105 f
g Kathy, as she mostly scores lower than Carrie
19 Beaut Ball: mean = 3.708 cm, σ= 0.036 cm E-Zee Ball: mean = 3.705 cm, σ= 0.054 cm
Beaut Ball has a more efficient operation than E-Zee Ball because the standard deviation is smaller, although the mean is a little further from 3.70 cm.
20 For route A: mean = 15 minutes, σ= 2.74 minutes, range = 10 minutes For route B: mean = 15 minutes, σ= 1.35 minutes, range = 4 minutes
Route B is better, because it has the same mean but is more consistent, so there is less chance of being very late.
21 a Using median = 43 and IQR = 24, 98% is an outlier. b Using mean = 43.6 and σ= 18.35, 98% is an outlier.
22 There are relatively more Aborigines and Torres Strait Islanders in the 0–29 age brackets and relatively fewer in the 40+ group. Life expectancy for the ATSI group is lower than for the total Queensland population.
23 She did better in Film and TV as she was 1.13 standard deviations above the mean, whereas in Speech and Drama she was only 0.65 standard deviations above the mean.
24 David is fatter than the average population as his weight is 1.54 standard deviations above the mean, whereas his height is only 1.33 standard deviations above the mean.
75 80 85 90 95 100 105
Kathy
Golf scores
