Data and Mathematical Modelling: Test Review

Everything for the Unit 6 test on one page: the key facts from all 4 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Data in Our Lives: Collection, Use and Consequences

Big data is data with huge volume, high velocity (arrives fast) and wide variety (many types).
Data moves through a lifecycle: collect, store, analyse, use. Each stage carries privacy or bias risks.
A sample is biased if it systematically differs from the population; a larger biased sample is still biased.
Meaningful consent means knowing what is collected, why, and who it is shared with, and being able to refuse.
On a truncated axis, compare the real values, not the bar heights: the real ratio is $value_{A} \div value_{B}$ .

Measures of Centre, Spread and Box Plots

Mean $= \frac{sum of values}{n}$ ; median = middle of the sorted list; mode = most frequent value.
Range = greatest value − least value. $IQR = Q_{3} - Q_{1}$ .
Quartile method: $Q_{1}$ and $Q_{3}$ are the medians of the lower and upper halves; when $n$ is odd, leave the median out of both halves.
Outlier fences: lower $= Q_{1} - 1.5 \times IQR$ , upper $= Q_{3} + 1.5 \times IQR$ . Values outside the fences are outliers.
Box plot: box from $Q_{1}$ to $Q_{3}$ , line at the median, whiskers to the least and greatest non-outliers, outliers as dots.

Scatter Plots, Correlation and Lines of Best Fit

Independent variable on the $x$ -axis, dependent variable on the $y$ -axis.
Describe correlation by direction (positive, negative, none), strength (strong, weak) and form (linear, non-linear).
The correlation coefficient $r$ is between $- 1$ and $1$ ; values near $\pm 1$ mean a strong linear pattern, values near $0$ mean little or no linear pattern.
Line of best fit: from two points on the line, $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ , then substitute one point into $y = m x + b$ to find $b$ .
Interpolation predicts inside the data's range; extrapolation predicts outside it and is less reliable.
Correlation alone does not prove that one variable causes the other.

The Mathematical Modelling Cycle

A mathematical model is a simplified mathematical description (equation, graph or table) of a real situation, built on assumptions.
The modelling cycle: ask a question, make assumptions and choose variables, plan and collect data, build the model, test it, report. Revise when the test fails.
Choose the model from the data's shape: constant change means linear; change that keeps shrinking or growing means a non-linear model.
Test a model with data that was not used to build it, and compare predicted and measured values.
Every report states the assumptions and the limits of the model, including where its predictions become impossible.

Where marks get lost

Thinking a huge sample must be accurate. The 1936 Literary Digest poll had about 2.4 million responses and still predicted the wrong winner because the sample was biased. (Data in Our Lives: Collection, Use and Consequences)
Reading bar heights instead of the axis values when the axis does not start at zero. (Data in Our Lives: Collection, Use and Consequences)
Assuming data that is not your name is not personal. Location history, purchases and photos can identify you. (Data in Our Lives: Collection, Use and Consequences)
Believing a computer's decision cannot be biased. A program that learns from biased past data can repeat that bias. (Data in Our Lives: Collection, Use and Consequences)
Calling any survey you disagree with biased. Bias is about how the sample was chosen or how questions were asked, not about the result. (Data in Our Lives: Collection, Use and Consequences)
Finding the median without sorting the data first. The middle of an unsorted list means nothing. (Measures of Centre, Spread and Box Plots)
Including the median in both halves when n is odd, which gives different quartiles from this course's method. Leave it out when n is odd. (Measures of Centre, Spread and Box Plots)
Drawing the whisker all the way to an outlier. The whisker stops at the greatest (or least) value that is not an outlier; the outlier is a separate dot. (Measures of Centre, Spread and Box Plots)
Calling a value an outlier because it looks big, without computing the fences. Use Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. (Measures of Centre, Spread and Box Plots)
Thinking a longer whisker or box section means more data values. Each of the four sections holds roughly a quarter of the data; a longer section means those values are more spread out. (Measures of Centre, Spread and Box Plots)
Putting the dependent variable on the x-axis. Ask which variable responds to the other; that one goes on the y-axis. (Scatter Plots, Correlation and Lines of Best Fit)
Forcing the line of best fit through the first and last data points, or joining the points dot to dot. The line follows the overall trend with points on both sides. (Scatter Plots, Correlation and Lines of Best Fit)
Using two data points that are not on your line to find its equation. Use points that lie on the line you drew. (Scatter Plots, Correlation and Lines of Best Fit)
Concluding that one variable causes the other just because they are strongly correlated. (Scatter Plots, Correlation and Lines of Best Fit)
Trusting an extrapolated prediction as much as an interpolated one. (Scatter Plots, Correlation and Lines of Best Fit)
Asking a vague question ("how long do batteries last?") that no measurement can answer. Make the question specific and measurable. (The Mathematical Modelling Cycle)
Not writing down assumptions, so nobody (including you) knows when the model applies. (The Mathematical Modelling Cycle)
Testing the model only on the same data used to build it. Test with new data. (The Mathematical Modelling Cycle)
Reporting a number with no units, no context and no limits, such as "4.7" instead of "about 4.7 hours of streaming under these conditions". (The Mathematical Modelling Cycle)
Keeping a model that gives impossible results (negative battery, tea colder than the room) instead of going back to revise the assumptions. (The Mathematical Modelling Cycle)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Data in Our Lives: Collection, Use and Consequences
1
A city wants to know how residents get to work. It surveys people waiting at subway stations. What is the main problem?
Data in Our Lives: Collection, Use and Consequences
2
A fitness band records your step count once every minute. How many readings does it make in one day?
readings
Data in Our Lives: Collection, Use and Consequences
3
Put the stages of the data lifecycle in order, starting with the first.
1. 1Store the data
2. 2Use the patterns to make a decision
3. 3Analyse the data for patterns
4. 4Collect the data
Data in Our Lives: Collection, Use and Consequences
4
Which request for data is most clearly more than an app needs?
Data in Our Lives: Collection, Use and Consequences
5
The whole group you want to learn about is the population. What is the part of the population you actually collect data from called?
Data in Our Lives: Collection, Use and Consequences
6
A line graph of a company's sales shows only the 3 months when sales rose, out of a 2-year period when sales mostly fell. What is misleading?
Measures of Centre, Spread and Box Plots
7
Find the mean of 4, 7, 7, 9, 13.
Measures of Centre, Spread and Box Plots
8
Using this course's method, find the first quartile Q1 of: 21, 25, 18, 30, 27, 24, 19, 35, 22.
Measures of Centre, Spread and Box Plots
9
A data set has Q1 = 46 and Q3 = 54. What is the upper fence for outliers?
Measures of Centre, Spread and Box Plots
10
About what fraction of the data in a box plot lies inside the box?
Measures of Centre, Spread and Box Plots
11
Match each term to its meaning.
Median
Mode
Range
Interquartile range
Measures of Centre, Spread and Box Plots
12
A very large value is added to a data set. What usually happens?
Scatter Plots, Correlation and Lines of Best Fit
13
You study how the number of hours a part-time worker works affects their weekly pay. Which variable belongs on the x-axis?
Scatter Plots, Correlation and Lines of Best Fit
14
Using the line of best fit $y = 5.8 x - 38$ for the snack stand, predict the drinks sold on a 25 °C day.
drinks
Scatter Plots, Correlation and Lines of Best Fit
15
Data on plant height was collected for days 1 to 20. Which prediction is an extrapolation?
Scatter Plots, Correlation and Lines of Best Fit
16
Across many towns, the number of fire trucks sent to a fire is positively correlated with the damage done. What is the best explanation?
Scatter Plots, Correlation and Lines of Best Fit
17
The model $v = 200 (1.44)^{d}$ gives the views of a video on day $d$ . Find the views on day 3, rounded to the nearest whole number.
views
Scatter Plots, Correlation and Lines of Best Fit
18
Put the steps for making a prediction from a scatter plot in order.
1. 1Substitute the new x-value and state whether it is interpolation or extrapolation
2. 2Choose two points on the line and find its equation
3. 3Draw a line of best fit through the middle of the points
4. 4Plot the data with the independent variable on the x-axis
5. 5Describe the direction, strength and form of the correlation
The Mathematical Modelling Cycle
19
Put the stages of the mathematical modelling cycle in order.
1. 1Test the model with new data
2. 2Build the model
3. 3Report the answer and the model's limits
4. 4Ask a specific question
5. 5Make assumptions and choose variables
6. 6Plan and collect data
The Mathematical Modelling Cycle
20
Using $B = 100 - 17 t$ , predict the battery level after 1.75 hours. Round to the nearest whole percent.
%
The Mathematical Modelling Cycle
21
Using $B = 100 - 17 t$ , after how many hours would the model predict 49% battery?
h
The Mathematical Modelling Cycle
22
A linear model for cooling tea predicts the tea will be −6 °C after 40 minutes in a 21 °C room. What should you do?
The Mathematical Modelling Cycle
23
Match each action to its stage of the modelling cycle.
Deciding to record the battery level every 30 minutes for 3 hours
Fitting the line B = 100 − 17t to the data
Checking the prediction for 4 hours against a new measurement of 30%
Writing that the model may fail when low-power mode turns on
The Mathematical Modelling Cycle
24
Why should a model be tested with data that was NOT used to build it?

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…