Resources: For this week the theory work is in the PDF file: Week 1/2 Notes & Exercises Knowledge Checklist

• Numerical and categorical

• Discrete and continuous

• Nominal and ordinal

• Stem and leaf plot

• Histogram

• Bar chart

• Frequency (and cumulative frequency) table

• Tally

• Dot plot

• Outlier

• Symmetric

• Skewness

• Reading and interpreting graphs Order

1. Read the notes and view the on-line resources

2. Complete the questions in the Notes and Exercises

3. Complete the Mathspace and then the Investigation

4. Submit your completed the brief for marking

There are questions to be answered in Week 1/2 Notes & Exercises Introduction to stem and leaf plots http://www.youtube.com/watch?v=OaJXJduRiIE How to construct and use a divided bar graph http://www.youtube.com/watch?v=oTYJx_mvcug Interpreting graphs https://www.youtube.com/watch?v=Sbzw653gW10 Mathspace: Statistics Review https://mathspace.co/student/tasks/SubtopicCustomTask-740318/

See the end of the brief ☺

Quizzes On Mathspace https://mathspace.co/student/tasks/SubtopicCustomTask-740318/

Investigation

Theoretical Components

Practical Components

Goals:

• Review the statistical investigation process (MAT01)

• Classify a categorical variable as ordinal or nominal and use tables and bar charts to organise and display the data (MAT02)

• Classify a numerical variable as discrete or continuous (MAT03)

• Chose from dot plot, stem plot, bar chart or histogram to display and describe the distribution of a numerical dataset, it’s shape (symmetric versus positively or negatively skewed), location and spread, and outliers, and interpret this information in the context of the data (MAT04)

Goals

Learning Brief MA2: Univariate data, Trigonometry, Linear equations

Week 1/2 Term 3 2020

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2020 MATHEMATICAL APPLICATIONS 2

WEEK 1/2 NOTES & EXERCISES Univariate Data

Types of data This era has been called the information age. The success of any venture, large or small, relies upon the accurate sourcing, processing and analysing of information. These tasks are assigned to statisticians. In this unit we learn statistical methods which can be used in the processing and analysis of data. There are several different types of data that may result from a survey or experiment. The type of data will influence how they can be analysed and how useful they might be to a researcher. A distinction is made between categorical data and numerical data. Let us consider these questions and their responses: ‘Do you own a pet?’ The answer is ‘yes’ or ‘no’. A categorical answer ‘How many pets do you own?’ The answer is a number. A numerical answer Consider the following examples: Example 1: A survey asks for the religious beliefs of all the students at a large multicultural school. The researcher divides the responses into the categories: Christianity, Islam, Buddhism, Hindu, Other and No belief. Example 2: A researcher asks people in a telephone survey if they agree with the statement: ‘The Prime Minister is doing a good job’. Their responses must be: (a) strongly agree, (b) agree, (c) no opinion, (d) disagree or (e) strongly disagree. Example 3: A researcher counts the number of people travelling in each car on a busy section of highway. Example 4: A quality control manager checks the weights of 50 sacks of flour. Examples 1 and 2 illustrate categorical data. Examples 3 and 4, numerical data. Categorical data Categorical data are observations or records (or ‘pieces of data’) that fit some qualitative category. Data of this type do not involve numbers or measurements. Categorical data may be either nominal or ordinal. If there is no order associated with the categories formed by the data then they are termed nominal data but if the data categories may be aligned to some qualitative scale then they are termed ordinal data. Example 1 (student’s religion) is an example of nominal data because there is no inherent order that the religions can fit into. Example 2, however, could be termed ordinal data. The categories formed have a definite order from strong agreement through to strong disagreement. Numerical data The categories into which numerical data are sorted will be quantitative amounts. Numerical data may be classified as being either discrete or continuous. Discrete data are observations or records that can take only certain set values. Example 3 is an illustration of discrete data. The researcher in this example would expect to obtain numbers like 1, 2, 3, 4 and only these values. It is not possible to have, say, 3.62 people in any car! The data records obtained from this survey are limited strictly to whole numbers. Other examples of discrete data might be the number of matches in each of a sample of boxes of matches, the shoe size of

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every person in the classroom and the number of pets in each household in the street. Discrete data often arise as the result of counting objects. Example 4 is an illustration of continuous data. The quality control manager in this example might find that the data she collected consisted of numbers like 20.005 kg, 19.998 kg and 20.0054 kg. In fact the weights could give rise to any number at all! The results would be limited only by the accuracy of the measuring device used. Other examples of continuous data are the heights of everyone in the classroom, the length of each person’s index finger, and the time taken to complete a task. Continuous data often arise as the result of taking measurements. Exercise 1 Q1. State whether the data formed in each of the following situations would be categorical or numerical.

a) The number of matches in each box is counted for a large sample of boxes.

b) The sex of respondents to a questionnaire is recorded as either M or F.

c) The lengths of 40 cod are recorded by a fisheries inspector.

d) The occurrence of hot, warm, mild and cool weather for each day in January is recorded.

e) Cinema critics are asked to judge a film by awarding it a rating from one star to five stars.

Q2. State whether the categorical data formed by each of the following situations are nominal or ordinal.

a) Individual members of a group of people are asked to indicate their level of agreement with the statement: ‘I support open euthanasia’. Each respondent must select one of the following responses: strongly agree, agree, unsure, disagree, or strongly disagree.

b) Students are asked to write down the name of their favourite rock band.

c) The mode of transport that each student uses to get to school is recorded.

d) Members of a book club rate a new book as either ‘a great read’, or ‘quite readable’, or ‘not recommended’.

e) The achievement level of each student in a class is recorded as either ‘excellent’, ‘good’, ‘fair’ or ‘poor’.

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Q3. State whether the numerical data formed by each of the following situations are discrete or continuous.

a) The heights of 60 tomato plants at a plant nursery

b) The number of jellybeans in each of 50 packets

c) The time taken for each student in a class of 6-year-olds to tie their own shoes

d) The petrol consumption rate of a large sample of cars

e) The IQ (intelligence quotient) of each student in a class Q4. Sort the following CensusAtSchool question topics according to whether they will yield categorical or numerical data.

Amount of money earned last week Arm span Concentration exercise (seconds) Dominant hand reaction time Favourite sport Height (cm) Hours slept per night Birthdate

Language mostly spoken at home Foot length School post code State/Territory live in Travel method to school Travel time to school Year level Opinions on environmental conservation

Categorical

Numerical

(label D for discrete or C for continuous)

Why is data, that contains numbers, such as post codes and birthdates, considered categorical?

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Displaying Data

Data represented graphically is visually appealing, and we are more likely to notice patterns if data is presented in a statistical graph. A statistical display should be simple and interesting and make an impact, but should not mislead the reader.

Stem Plot A stem-and-leaf plot, or stem plot for short, is a way of displaying a set of data. This site provides a good introduction to stem and leaf plots. http://www.youtube.com/watch?v=OaJXJduRiIE

It is best suited to data which contain up to about 50 observations (or records). The following stem plot shows the ages of people attending an advanced computer class

Stem Leaf 1 6 2 2 2 3 3 0 2 4 6 4 2 3 6 7 5 3 7 6 1

The ages of the members of the class are 16, 22, 22, 23, 30, 32, 34, 36, 42, 43, 46, 47, 53, 57 and 61. A stem plot is constructed by breaking the numerals of a record into two parts — the stem, which in this case is the first digit, and the leaf, which is always the last digit. Example The number of cars sold in a week at a large car dealership over a 20-week period

is given below.

16 12 8 7 26 32 15 51 29 45 19 11 6 15 32 18 43 31 23 23

Construct a stem plot to display the number of cars sold in a week at the

dealership.

Solution: In this example the observations are one- or two digit numbers and so the

stems will be the digits referring to the ‘tens’, and the leaf part will be the digits

referring to the units. Work out the lowest and highest numbers in the data in order

to determine what the stems will be.

Lowest number = 6 Highest number = 51 Use stems from 0–5.

Before we construct an ordered stem plot, construct an unordered stem plot by listing the leaf digits in the order they appear in the data.

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Now rearrange the leaf digits in numerical order to create an ordered stem plot. Include a key so that the data can be understood by anyone viewing the stem plot.

Example A set of golf scores for a group of professional golfers trialling a new 18-hole golf course is shown on the following stem plot.

Produce another stem plot for this data by splitting the stems into halves. By splitting the stem 6 into halves, any leaf digits in the range 0–4 appear next to the first 6, and any leaf digits in the range 5–9 appear next to the second 6. Likewise for the stem 7.

Exercise 2 Q1. In the following, write down all the pieces of data shown on the stem plot.

a) b)

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Q2. The stem and leaf plot represents the points scored per match by the Sharks in a baseball season.

a) Redraw as an ordered stem and leaf plot. b) How many matches were played in the season? c) What was the highest score for a match? d) What was the lowest score for a match? Q3. The money (to the nearest dollar) earned each week by a busker over an 18-week period is shown below. Construct a stem plot for the busker’s weekly earnings.

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Q4. A random sample of 20 screws is taken and the length of each is recorded to the nearest millimetre.

Construct a stem plot for screw length using the stems 1 and 2 split into halves. Frequency Histograms and Polygons A histogram is used to represent quantitative data and is a column graph with no spaces between the columns. The height of each column represents the frequency of the scores. A frequency polygon is a line graph that plots score against frequency and can also be drawn by joining the midpoints of the tops of histogram columns. Example A pair of dice was rolled 50 times and the sum recorded. Construct a frequency distribution table, then draw a frequency histogram and polygon for the data.

With a large set of numbers it is necessary to construct a Frequency Distribution Table. A Tally column, in which the scores are marked of in order, is useful as it helps reduce the possibility of missing a number. This table makes it easy to construct a histogram. The scores are placed on the horizontal line and the frequencies on the vertical line.

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Note that the polygon is drawn through the centre of the columns. Exercise 3 Q1. Forty primary school students went on an excursion. Their ages were:

a) Draw up a frequency distribution table using scores 5, 6, 7, …, 12. b) Use your table to construct a frequency histogram and polygon next to the

table (as in the example).

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Q2. The data below represents the number of hours each week that 40 teenagers spent on household chores. Construct a frequency table and draw the corresponding histogram and polygon.

Example The data below show the distribution of masses (in kilograms) of 60 students in Year 7 at Northwood Secondary College.

First construct a frequency table. The lowest data value is 34.2 and the highest is 62.3. Divide the data into class intervals. If we started the first class interval at, say, 30 kg and ended the last class interval at 65 kg, we would have a range of 35. If each interval was 5 kg, we would then have 7 intervals which is a reasonable number of class intervals. The resulting frequency table is shown below. A Tally column would certainly be necessary but has been omitted here for convenience. The corresponding histogram is also shown.

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Q3. The number of words per sentence in a magazine article were counted:

Using class intervals 0−9, 10−19, etc., draw up a frequency table and add a cumulative frequency column. Draw the corresponding histogram. (Use a ruler)

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Dot Plots A dot plot is a simplified type of histogram and is a good way of presenting a small amount of data. It is easy to see where clusters of scores occur as well as to identify any outliers (numbers that seem out of line with the other scores). Each score is represented by a symbol, usually a dot. Example This dot plot represents the body temperatures (in °C) of hospital patients, where normal body temperature is about 37°C:

The graphical representation allows us to readily see the following points.

• There were 10 patients

• Most had a temperature of 38˚C

• Most temperatures were between 37˚C and 39˚C

• The ‘outlier’ was 42˚C (possibly a very ill patient). Exercise 4 Q1. A Year 11 Maths class was surveyed to find out how many hours each student spent on Maths each week. The results are shown below.

a) Draw a dot plot for this data. b) Are there any outliers? Comment on possible reasons for these.

c) Are there any clusters or gaps?

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Q2. The number of hours that students spent playing computer games in a week were found to be:

a) Draw a dot plot.

b) Comment on any unusual features. Q3. Here are two dot plots produced in a factory making pins. Machines A and B produce the pins, which must have a diameter of 8 cm ± 0.01 cm or they are rejected. The dot plots show 50 pins from each machine and have the same scale.

a) Are there any clusters?

b) How many pins from each machine would be rejected?

c) Which machine is better? Why?

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Bar Chart A bar chart is similar to a histogram. However, it consists of bars of equal width separated by small, equal spaces and may be arranged either horizontally or vertically.

In bar charts the frequency is graphed against a variable as shown in both figures above. The variable may or may not be numerical. However, at this point we consider only numerical variables. The numerical variable should take discrete values ie whole numbers. Exercise 5 Q1. Construct a bar chart for the data shown below. This represents a score out of 10 in a Maths quiz. 3 4 4 5 5 6 7 7 7 8 8 9 9 10 10 1

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Interpreting Graphs

Example These column graphs represent the 896 international students who enrolled in Australian tertiary institutions in 1994. List some statistical facts that can be interpreted from the graphs.

Solution

• About a third of the students came from Hong Kong.

• Business Office Studies was the most popular area of study with

approximately 45% of students enrolled in this area.

• Hotel and Tourism courses were the second most popular with about 15%

of students enrolling in these.

• There were twice as many students from Hong Kong as there were from

Indonesia.

• Pakistan had the least number of students.

• Rural Studies and Manufacturing were the least popular courses.

Exercise 6 Q1. This graph shows how Pete’s Pizza Parlour spends the money from pizza sales.

a) Apart from rent, give three overheads that Pete may have.

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b) How much of a large pizza, sold for $16, is spent on: (i) overheads (ii) labour? c) Pete’s profit for selling pizzas over a week is $900. How much did he earn from pizza sales in that week? d) If Pete’s profit increases to $1200 next week, how much will he spend on advertising? e) What are the advantages of using a sector graph to display this information? Q2. The line graph shows the sales of compact discs over a 6-month period.

a) How many CDs were sold in January? b) What could be a reasons for the most discs being sold in January? c) In which month was the least number of CDs sold? d) In which month were 250 000 CDs sold? (e) Between which 2 months did the greatest drop in sales occur?

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f) What were the total sales for the 6 months? g) What percentage of the CD sales were in November? Q3.

a) What is the graph about? b) What was the budget for Die Hard, and how much did it make at the US box office? d) Which movie grossed the most money at the US box office? e) Which movies were flops? Explain how you know this. (f) Which movie made the greatest percentage profit of takings at the box office compared to the budget for making the movie?

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Q4. This graph from the Bureau of Meteorology gives rainfall information for Alice Springs.

a) Which month had the largest number of raindays? b) Which month had the highest rainfall? c) What was the rainfall for: (i) September (ii) April? (d) How many raindays were in: (i) February (ii) October? (e) When would be the best time to travel to Alice Springs? Why? (f) Without using numbers this graph clearly shows a relationship between the number of rainy days in a month and the amount of rainfall. What is this relationship?

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Describing the shape of stem plots and histograms Symmetric distributions The data shown in the histogram below can be described as symmetric.

There is a single peak and the data trail off on both sides of this peak in roughly the same fashion. Similarly in the stem plot below, the distribution of the data could be described as symmetric.

The single peak for these data occur at the stem 3. On either side of the peak, the number of observations reduces in approximately matching fashion. Skewed distributions Each of the histograms below show examples of skewed distributions.

The figure on the left shows data which are negatively skewed. The data in this case peak to the right and trail off to the left. The figure on the right shows positively skewed data. The data in this case peak to the left and trail off to the right. Exercise 7 Q1. For each of the following stem plots, describe the shape of the distribution of the data and comment on the existence of any outliers. a) b) c)

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Q2. For each of the following histograms, describe the shape of the distribution of the data and comment on the existence of any outliers.

a) b) c) Q3. Multiple Choice. The distribution of the data shown in this histogram could be described as:

A negatively skewed B negatively skewed with one outlier C positively skewed D positively skewed with one outlier E symmetric. Q4. The number of hours of exercise completed each week by a group of employees at a company is shown on the stem plot below.

a) Describe the shape of the distribution of these data and comment on the existence of any outliers. b) What does this tell us about the number of hours of exercise completed weekly by the employees in this company?

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2020 MA2

INVESTIGATION WEEK 1/2

Olympic Records Below are ten graphs showing how Olympic records have changed over time in ten athletic events.

Can you deduce which event each graph represents?

The list of the ten events is given, in no particular order, to help you deduce which event belongs to each graph.

1. Men’s High Jump

2. Women’s 10,000 metres

3. Men's 100 metres

4. Women's High Jump

5. Men's Long Jump

6. Men's Decathlon

7. Men's 4 x 100 metres relay

8. Women's 100 metres,

9. Men's Javelin

10. Women's 1500 metres

Determine what the units might be on the vertical axis for each

graph?

Some graphs show a decreasing trend and some an increasing

trend? Identify those graphs and describe the trend.

Are there any unusual features on any of the graphs? What might

be a plausible explanation for them?

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Be sure to write up your solutions next to each graph listing the

event, labelling the vertical axis with the appropriate units, and discussing any increasing or decreasing trends and any unusual

features (with explanations).

Event:

Event:

Event:

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Event:

Event:

Event:

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Event:

Event:

Event:

Event:

CRITERIA EXPECTATIONS POSS MULT GIVEN TOTAL

Practical Student completes practical work, including exercises and Mathspace task, of the brief to an acceptable standard set by the teacher.

2 3

/6

Investigation Task

Student completes the investigation task of the week to an acceptable standard set by the teacher.

2 2

/4

Communication Student responses are accurate and appropriate in presentation of mathematical ideas, with clear and logical working out shown.

4 - /4

Knowledge and Application

Student submitted work selects and applies appropriate mathematical techniques to solve practical problems and demonstrates proficiency in the use of mathematical facts, techniques and formulae.

4 -

/4

Submission Guidelines

Timeliness Student submits the exercises, Mathspace/online task and investigation by the set deadline. See scoring guidelines for specific details.

2 -

/2

FINAL /20

Student Reflection: How did you go with this week’s work? What was interesting? What did you find easy? What do you need to work on? Mathspace task title: Mathspace score: