2.3: Data Is Trendy
- Page ID
- 148554
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)INTRODUCTION
You are going to look at data that is nearly linear and approximate with a linear equation. From there, you’ll be able to make predictions about what might happen in the future. This is a very common way to try to predict future events.
It is very important that you understand the concepts of slope and intercepts discussed in Preparation M.2. Please see the definitions below for a quick review:
Definition: The slope of a line is the ratio of the change in the vertical value (sometimes called output, y-value, dependent variable, or response variable) to the change in the horizontal value (sometimes called input, x-value, independent variable, or explanatory variable) as we move between any two points on the line. The slope of a line is therefore a rate of change. Often it is referred to as the ratio of the rise of the graph to the run of the graph, or “rise over run.” The slope can also be described as the rate at which the output values are changing (increasing or decreasing) as the input value increases by 1. It can be found by using any two points on the line. (Note: By convention, the input is on the horizontal axis and the output on the vertical axis.)
Definition: The vertical intercept of a line is the point at which the line intersects the vertical axis, where the input value is zero. This is also called the initial value if the input starts out at a value of zero.
Definition: The horizontal intercept of a line is the point at which the line intersects the horizontal axis, where the output is zero.
Definition: A linear model has a constant rate of change (slope). (This is not true of all models and their corresponding equations.)
It is also very important that you understand how to use technology to graph data and approximate the data with a linear equation. If necessary, please review any process with your instructor.
SPECIFIC OBJECTIVES
By the end of this collaboration, you should understand that
- linear models are used to model real-world data.
- linear trends can be observed and can be created for data with fluctuations.
- data that change in an approximately linear manner can be modeled by a trendline (line of best fit or piecewise lines of best fit).
- different interpretations of trends seen in data can be made.
By the end of this collaboration, you should be able to
- interpret the slope and y-intercept in context from an equation or a graph.
- understand when a linear model is appropriate as a model for data.
- use a spreadsheet to create a scatterplot and linear regression model from data.
PROBLEM SITUATION 1: WAGE GROWTH
In 2022, a co-worker created the following graph to represent her annual pay over the years.
(1) Find a linear equation that models the above situation. Use P to represent your co-worker’s pay in dollars and t to represent the number of years since 2012. If the trend continues, in what year would she expect to earn $36,000 a year? Take a minute to think about this on your own before sharing your ideas in your group.
It is important to note that your co-worker would be using extrapolation to predict her future salary, so her prediction is unlikely to be perfect and could even be quite a bit off. Extrapolation is the process of using a mathematical model to make predictions that are outside the range of the available data, based on the assumption that existing trends will continue. The problem with extrapolation is that the trend could change in the future, for any number of reasons, which would make the prediction unreliable. For example, your coworker’s company could become very successful, resulting in 10% raises for everyone in the company in 2020; or a recession could hit, resulting in no raises for any employees from 2017-2022. Our model from Question 1 would not account for these changes. People often use extrapolation to make predictions, but there is a risk to doing this, since it assumes that current trends will not change.
PROBLEM SITUATION 2: BABY GROWTH AND DEVELOPMENT
(2) The following table shows the average weight for infant boys:17
| Age | Average Weight for Boys (lbs) |
| Newborn | 7.16 |
| 1 month | 9.15 |
| 2 months | 10.91 |
| 3 months | 12.56 |
| 4 months | 14 |
| 5 months | 15.43 |
| 6 months | 16.53 |
| 7 months | 17.64 |
| 8 months | 18.74 |
| 9 months | 19.62 |
| 10 months | 20.28 |
A linear model could be created which could be used to fairly accurately calculate the average weight of an infant boy, given his age in months.
(a) Create a linear model equation for the weight of infant boys.
(b) Use your model to predict the weight of a 20-month-old boy. Round to one decimal place.
(c) Is your prediction in Question 2(b) reasonable? Explain.
(d) According to your model, what would a 25-year-old weigh? Is this reasonable?
PROBLEM SITUATION 3: VIEWS ON GLOBAL WARMING
Every year, the Berkeley Earth Surface Temperature (BEST) project releases an analysis of land-surface temperature records (some going back as far as 250 years).18 According to Berkeley Earth, “The analysis shows that the rise [or growth] in average world land temperature[is]... about 1.3 degrees Celsius.” BEST states that “global warming is real,” and attributes the temperature trend to a combination of volcanoes and CO2. There are others who are skeptical of the theory of “global warming” and hold a contrary view of weather data. These people believe that there are other explanations and reasons for the climate changes being experienced. Today, we are going to examine the Berkeley Earth project data and linear models for Global Surface Temperature Change to study the issue of global warming.
On the website skepticalscience.com, the data from the Berkeley Earth project has been used as the source for two different analyses of the same information. The first one is labeled “How Realists View Global Warming” and the other is “How Contrarians View Global Warming.” The two infographics are reproduced in this collaboration. You will see that each includes a line or line segments along with the actual data. Use these infographics to answer the following questions.
Part I: The Realist View
Below is an infographic showing the data gathered by BEST showing the monthly “Global Temperature Anomaly” (that is, the change in average Global Temperature, in degrees Celsius) for the years 1970–2022. Shown on the graph is a trendline, also known as a line of best fit, that is a linear model representing the trend of the data. It is stated on the project’s website that the baseline period is the 1850-1900 mean.19 This means that the data from 1970–2022 is compared with the data from 1850-1900.
Use the infographic to answer Questions 3–10.
(3) What are the units on the horizontal axis? What is the variable?
(4) What are the units on the vertical axis? What is the variable?
(5) What does one point on the graph represent?
(6) Now look at the trendline that is drawn on the graph. What does this line represent?
(7) Do you think the trendline is reasonable? Does it represent the trends in the data?
(8) (a) Estimate the value where the trendline crosses the vertical axis (the value in 1970). Take a minute to think about this on your own before sharing your ideas in your group. Describe the meaning of the point where the trendline crosses the vertical axis.
(b) Estimate the slope of this line in degrees Celsius per year (although the original data is monthly, the slope will be per year). Write an interpretation of the slope in this context. Be sure to use units with your values, and use complete sentences.
Finding exact points on this line is difficult. Just do your best to estimate two points and use them to find the slope.
(c) Using your own estimates, estimate the equation of the line. Take a minute to try this on your own before sharing in your group.
(9) What is the model saying about the general pattern of the global temperature. Is it rising or falling?
(10) Using your linear equation from above, predict the temperature change at the end of 2030. Also predict the temperature change at the end of 2050. Do you think this is a reliable prediction for 2050? Why or why not? Explain using complete sentences.
Part II: The Contrarian View
The infographic below contains the same data points for the Global Temperature Change (the change in average Global Temperature in the years from 1970–2022) that were included in the infographic showing “How Realists View Global Warming.” Also shown is a piecewise model, which is a model defined by multiple pieces, each of which has its own trendline that represents the trend of the data within a certain interval. This infographic shows a linear piecewise model that consists of 6 line segments—each of which can be considered a trendline for that restricted time period. This infographic is meant to depict a common view stated by contrarians that global warming is not real since there have been several periods over the last 50 years when global temperatures have declined on average. Note that the units on both axes and the data are identical to what was in the Realist View. The only difference is the trendlines that are used to analyze the data. Use this infographic to answer the questions below.
(11) Look at the line segments that are drawn on the graph. What do these lines represent as a whole?
(12) Now look at just the segment of the piecewise “best fit” model from 2010 to 2015 (the line segment second farthest to the right).
(a) Estimate where the line segment would cross the vertical axis (in year 1970). (Hint: It might be useful to use a straight-edged item to extend the line to the vertical axis. You can see the line is very slightly slanted.)
(b) What is the slope value of this line segment? Give an interpretation of the slope value.
(c) Estimate the equation of the 2010 to 2015 line segment.
(13) Do you think we could use the equation you found in Question 12(c) to predict the mean Global Surface Temperature Change in 2035? Why or why not?
(14) Explain what the Contrarian Model is saying about the general pattern of global temperature. Is it rising or falling? Think about this on your own for a minute before sharing your ideas in your group.
(15) If you look at the overall trend of the data in the infographic, it is increasing. However, each line segment in the piecewise model is decreasing (ever so slightly). What does this mean? Explain what the model is saying about the general pattern of global temperature change.
(16) Come up with an example of a similar piecewise linear model. Note that models can be represented in different ways – tables, graphs, written descriptions, or equations.
MAKING CONNECTIONS
Record the important mathematical ideas from the discussion.
________________________________