Mathematics
Maths Task- Univariate Data and Sequences

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PART 1: 1. What sort of data are each of these? Nominal, discrete or continuous? 2. (a) The suburb a person lives in _________________ (b) The speed of a car _________________ (c) The make and model of a car _________________ (d) The weights of pet animals _________________ (e) The number of goals scored _________________ The table shows the weekly rents in a survey of 50 one bedroom units around Thornbury. Rent ($) 180– Frequency 3 200– 6 220– 12 240– 10 260– 3 280– 5 300– 3 320- 6 340- 2 Relative frequency % relative frequency 1.00 100% 50 (a) Complete the relative frequency and % relative frequency columns (b) What percentage of flats have (i) rents less than $300 a week? (ii) rents of at least $240 a week? (iii) rents between $220 and $300 a week? (c) If another 30 flats were surveyed, based on this survey, how many would you expect to have rents less than $300 a week? 3. What type of shape does each frequency histogram have? Briefly explain what the shape suggests about the data. (a) Hours worked (b) (c ) Weight loss for dieters 4. The stem and leaf plot shows the grams of sugar in a serve of various breakfast cereals, rounded to the nearest gram. Kids cereals 9987 87764442 998877665542 6522 60 Key 15 = 15 0 1 2 3 4 5 Adults cereals 889 0568 00288 566889 00224446688 2 (a) How many cereals were surveyed? ____________ (b) Complete the table. Sugar Content Kids cereals Adults cereals Lowest Highest Mode (c) (i) Change the stem and leaf plots into frequency tables. KIDS ADULTS 0-9 0-9 10 - 19 10 - 19 20 - 29 20 - 29 30 - 39 30 - 39 40 - 49 40 - 49 50 - 59 50 - 59 Will the class interval of grams of sugar be open or closed? Give reasons. (ii) Show the sugar content as line graphs, both drawn on the same graph. PART 2 1. Find the sum of x, (that is x), the total frequency (n), the mean ( x ) and the median for the daily minimum temperatures in Canada, in °C. −2, 5, 10.5, −5, 6, 5, 5.5, −13, 14.5, −3, 0, −6.5, 8, 2 2. Find the mean ( x ) and the total frequency (n) for the following grouped data. (a) Pets in a family 1 2 3 5 8 Number of families 12 15 10 2 1 (b) Hours of TV 1 2 3 4 5 Number of teenagers 4 5 18 15 8 3. We use the mid-point when the data is in intervals. Fill in the mid-points in these tables and then find the mean and the total frequency, n. Temp. Mid-point 1 – 10 11 – 20 21 – 30 31 – 40 Weight (kg) Mid-point Number of days 3 14 15 8 Number of pets. 0– 5– 10– 15– 4. 5. 4 6 12 3 Over 14 games the number of goals we scored was 4, 9, 13, 11, 12, 3, 5, 10, 12, 8, 7, 12, 14, 15 (a) Write the goal scores in order from lowest to highest (b) What was our median goal score? (c) What was our mean goal score? (d) Here the median was a better measure of the average score than the mode or mean. Why? The table below shows the weights of 20 pets. Weight (kg) (x) 5 7 8 9 11 14 Total (a) Number of pets (f) 3 2 5 7 2 1 20 x f Weight up to Cumulative frequency Up to 5 kg Up to 7 kg Up to 8 kg Up to 9 kg Up to 11 kg Up to 14 kg Complete the cumulative frequency column. Use it to find the median weight. (b) Work out the mean weight. (c) What is the mode? (d) Out of the median, mean and mode which do you think is the best measure of average weight? Why? 6. The table shows the result of a survey of 90 people who were asked "How much did you spend the last time you went out?" Amount spent How many Spent less than Cumulative % cumulative (frequency) frequency frequency Less than $10 $0 − $19 12 Less than $20 $20 − $39 21 Less than $30 $40 − $59 16 Less than $40 15 $60 − $79 Less than $50 17 $80 − $99 Less than $60 9 $100 − $119 Total 90 (a) Complete the cumulative frequency and % cumulative frequency columns. (b) Plot the cumulative frequency graph and read off the median. (c) Complete the table below, and use it to work out the mean. Amount spent $0 − $19 $20 − $39 $40 − $59 $60 − $79 $80 − $99 $1000 − $119 Total Mid-point (x) Frequency (f) 12 21 16 15 17 9 90 Total spent for group x f (d) 7. For this data, what is your preferred measure of average? Why? The histogram below shows the numbers of tickets bought by 53 people in a lottery. (a) Based on the shape of the histogram is the mean or the median the better measure of the average number of tickets bought by each person? Why? 12 Frequency 10 8 6 Mean = 4.2 tickets Median = 3 tickets 4 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Tickets bought (b) 8. Use your preferred measure of average to answer this. If another 15 people buy tickets, how many tickets in total would we expect them to buy? Show approximately where you would expect the mode, median and mean to be located in the following histograms. (b) frequency frequency (a) goal scores waiting time 9. Activity (a) What happens to the shape of the frequency distribution pattern when you use larger number of measurements? (b) What would happen to the pattern if you didn’t shake the coins well each time? (c) If you use large numbers of measurements will you often see big differences between the mean and median, or unusual results? Explain. PART 3 1. For each of the following data sets. (a) Find the median, QL , QU and IQR. (b) Identify any outliers by showing the 1.5 x IQR calculations. (c) Draw a box plot. Be sure to include a scaled line with your boxplot. Data set 1 Temperatures over 14 days at a ski resort 3, −2, 5, 1, –2, 0, 10, 2, 3, 2, 0, −1, 3, 0 Data set 2 Weights of 27 students. Key 7|3 = 73 kg Stem 5 6 7 8 Leaf 269 1235566777789 11233678 027 Data set 3 Number of phone lines operated by 50 companies. Number of phone lines 1 2 4 5 8 10 12 15 Number of companies (frequency) 1 2 3 3 8 20 8 5 50 Cumulative frequency Fill in the cumulative frequency column before finding the median and the quartiles. 2. There are four box plots shown below. Two of them relate to the histograms shown. Match the histograms with their correct box plots. Give reasons. Histogram 1 Box plot 1 Box plot 2 Goal score Histogram 2 Box plot 3 Box plot 4 Light bulb hours of life 3. (a) Draw the shapes of histograms that match these box plots. (b) Goal score 0 1 2 3 4 5 6 7 8 9 10 11 12 Sally’s team Brett’s team (a) Sally’s team Fill in this table Brett’s team Range QL QU IQR Median (b) When drawing the box plot we forgot to check for outliers. Find any outliers. (c) Which team scored better? Why? (d) (i) In how many games did Sally’s team score more than 6 goals? (ii) In how many games did Brett’s team score more than 5 goals? PART 4 1. Fill in the following tables, then work out the mean, variance and standard deviation of each sample. Show your workings. Sample A: Weights of 12 pets Weight (kg), x x2 1.4 7 9 11. 2 9.3 20. 5 7 10 12. 6 5 4.1 6  Sample B: Number of cars seen each hour over a 24 hour period Cars seen (x) 0 Total Number of hours (f) 2 3 4 7 4 13 6 25 5 30 3 xf x2  f Sample C: The daily temperatures over our 60 day camping holiday Set up your own table for this one and show how you work out the mean and standard deviation. Temperature, °C 0− 10 − 20 − 30 − 40 − Total x Number of days 4 13 21 19 3 60 xf x2  f 2. A survey of 100 teams found the mean score to be 14 goals per game with a standard deviation of 4.3 goals. (a) (b) The histogram of the survey is almost symmetrical. We can therefore conclude that (i) about 68% of teams score between __________ and ____________ goals. (ii) about 95% of teams score between _________ and ___________ goals. (iii) about 99.7% of teams score between ___________ and ___________ goals. (i) Is a score of 3 goals an unusual score? Why? (ii) Is a score of 18 goals an unusual score? Why? 3. Number of seedlings The histogram shows the number of seedlings which grow using a growth solution. Show on the histogram (i) the 68% confidence interval (ii) the 95% confidence interval (iii) the 99.7% confidence interval 10 8 Mean dose of growth solution = 8.9 units 6 4 Std dev. = 3.7 units 2 0 0 2 4 6 8 10 12 14 Units of growth 16 solution PART 5 1. For each of the following sequences (a) Find a formula for the nth term, tn (b) Evaluate t5 and t11 (i) 17, 13, 9, … (ii) –5, 2, 9, 16, … 2. In an arithmetic sequence the third term is 1 and the eighth term is 9 (a) Find the term, a, and the common difference d (b) 3. Find which term is 25 (Find n where tn = 25) Use your graphics calculator or use the formulae to find: (a) the first 10 terms of the sequence with the rule tn = 5n + 1 (b) the 35th term of the same sequence. (c) the sum of the first 12 terms of the sequence. 4. An employee receives $12000 in her 1st year and is to be given a yearly increase of $1050. (a) How much will she receive in her 8th year? (b) 5. How much will she have earned altogether A builder wishes to tile a section of a roof in the shape of an isosceles triangle (like the ladder rungs in Lesson 3). In order to fix the tiles, it is necessary to lay timber battens across the roof parallel to one another. If the shortest batten is 1 metre long, the longest batten is 5 metres long and 16 battens are required, find: 1m (a) the amount by which adjacent battens differ in length. 5m (b) the total length of the batten required. ...
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Alex Z (997)
UC Berkeley

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