Unit 12
Unit 12:
General Linear Modeling

Unit 12: Assignment #1 (due before 11:59 pm Central on THU JUL 25):
 In this Unit, you’ll be introduced to the General Linear Model.
 First, read the remainder of Math Is Fun’s (no date) article, “Scatter Plots.” Note the new information, which is the following:
 On a Scatter Plot, we can draw a “Line of Best Fit.”
 A “Line of Best Fit” can also be called a “Trend Line.”
 The goal in drawing a “Line of Best Fit/Trend Line” is to fit the data as well as possible.
 We can use Least Squares estimation to find the “Line of Best Fit/Trend Line.”
 After finding our Line of Best Fit/Trend Line we can use interpolation to identify a potential value within the range of our data set.
 We can use extrapolation to identify a potential value outside of the range of our data set.
 Second, search the Internet for a tutorial or howto guide to teach you how to identify a Trend Line using your chosen data management platform.
 The howto guide you find can be in any format (e.g., video, written text, figures, or the like — or a combination of formats).
 However, the howto guide you find must be from the Internet and not from other sources (e.g., textbooks or friends).
 Third, using your classmates’ HeightFoot Length Data Set, which you analyzed in Unit 11 and for which you previously created a Scatter Plot (YourLastName_PSY210_Unit11_HeightFootLength_Data):
 Use your chosen data management platform to calculate the Line of Best Fit/Trend Line of your classmates’ Height and their Foot Length.
 If you are given options for your Line of Best Fit/Trend Line such as Linear, Logarithmic, Exponential, and the like, choose Linear.
 Make sure your Line of Best Fit/Trend Line also shows the line’s equation.
 If you’re using Microsoft Excel, your Scatter Plot with a Line of Best Fit/Trend Line will look something like this.
 If you’re using Google Sheets, your Scatter Plot with a Line of Best Fit/Trend Line will look something like this.
 If you’re using Apple Numbers, your Scatter Plot with a Line of Best Fit/Trend Line will look something like this.
 Take a screenshot of your Scatter Plot with a Line of Best Fit/Trend Line (not your entire screen) and save it as YourLastName_PSY210_Unit12_HeightFootLength_Trend.xxx (where xxx is the file type, for example, .jpg, .png, .jpeg, and the like).
 Interpolate one value either by visual inspection OR by using the Line of Best Fit/Trend Line’s equation.
 Extrapolate one value either by visual inspection OR by using the Line of Best Fit/Trend Line’s equation.
 Fourth, using the Daily Maximum Temperature and daily Coffee Sales data, which you analyzed in Unit 11 and for which you previously created a Scatter Plot (YourLastName_PSY210_Unit11_TempCoffeeJuice_Data):
 Use your chosen data management platform to calculate the Line of Best Fit/Trend Line of Daily Maximum Temperature and the daily Coffee Sales.
 If you are given options for your Line of Best Fit/Trend Line such as Linear, Logarithmic, Exponential, and the like, choose Linear.
 Make sure your Line of Best Fit/Trend Line also shows the line’s equation.
 Take a screenshot of your Scatter Plot with a Line of Best Fit/Trend Line (not your entire screen) and save it as YourLastName_PSY210_Unit12_TempCoffee_Trend.xxx (where xxx is the file type, for example, .jpg, .png, .jpeg, and the like).
 Interpolate one value either by visual inspection OR by using the Line of Best Fit/Trend Line’s equation.
 Extrapolate one value either by visual inspection OR by using the Line of Best Fit/Trend Line’s equation.
 Go to the Unit 12: Assignment #1 Discussion Board and make a new post in which you include the following:
 First, embed the screenshot of your classmates’ HeightFoot Length Data Scatter Plot with a Line of Best Fit/Trend Line (YourLastName_PSY210_Unit12_HeightFootLength_Trend.xxx).
 Second, identify the value you interpolated and the value you extrapolated from this Line of Best Fit/Trend Line.
 Third, embed the screenshot of your Daily Maximum Temperature and daily Coffee Sales Scatter Plot with a Line of Best Fit/Trend Line (YourLastName_PSY210_Unit12_TempCoffee_Trend.xxx)
 Fourth, identify the value you interpolated and the value you extrapolated from this Line of Best Fit/Trend Line.
Unit 12: Assignment #2 (due before 11:59 pm Central on THU JUL 25):
 In this assignment, you’ll become more familiar with Lines of Best Fit, which are created by Linear Regression using Least Squares estimation.
 First, because we are moving from Correlations and Scatter Plots to Linear Regressions and Lines of Best Fit/Trend Lines, read Cavello’s (2020) article, “Correlation vs. Regression Made Easy: Which to Use + Why.” While reading this article, make sure you understand:
 Correlation and Regression are statistical analyses.
 One key difference between these two statistical analyses is that Correlation measures the degree of a relationship between two variables (x and y), whereas Regression estimates how one variable (x) affects another (y).
 Because Regression is used to predict, optimize, or explain how one variable (x) influences another variable (y), in Regression the x variable is considered the independent or predictor variable; the y variable is considered the dependent or criterion variable.
 Second, to understand the basic linear algebra that underlies Least Squares estimation, read Math Is Fun’s (2017) article, “Least Squares Regression.” While reading this article, make sure you understand:
 Although we could try to place a Line of Best Fit/Trend Line by hand, we can also calculate the best fit using Least Squares estimation.
 Least Squares estimation is based on linear algebra.
 The formula for a line is y = mx + b, where m is the slope, and b is the intercept.
 Therefore, Linear Regression created by using Least Squares estimation finds the best slope (m) and intercept (b) that fit the data.
 Third, to better understand practical applications of Linear Regression created by Least Squares estimation, read the Harvard Business Review’s (2015) article, “A Refresher on Regression Analysis.” While reading this article, make sure you understand:
 In Linear Regression, a dependent variable is the “factor you’re trying to understand or predict,” for example, monthly sales.
 In Linear Regression, one or more independent variables are the “factors you suspect have an impact on your dependent variable,” for example, rainy days.
 A Line of Best Fit/Trend Line such as y = 5x + 200 tells you “that if there is no x then y = 200 sales” (the intercept). And for every additional inch of rain, you made an additional five sales.”
 Now, to get some practice calculating and applying Linear Regression:
 First, go to MathBitsNotebook’s “Practice with Linear Regressions,” which our class is using with the author’s blessings.
 Second, complete each of the problems on that webpage.
 As you’ll notice, the webpage is neither interactive nor fillable. Therefore, you will need to record your answers in your own file, which you will subsequently convert to PDF.
 For completing these problems, you can use your chosen data management platform, or you can use a calculator or whatever tool you need; however, when you submit your answers to these problems, you must show the work that led you to select those answers.
 If you choose to use your chosen data management platform:
 First, type in the data presented in each table into your chosen data management platform, like this for Problem #1.
 Second, use your chosen data management platform to create a Scatter Plot, just like you did for Unit 12: Assignment #1.
 Third, use your chosen data management platform to calculate the Line of Best Fit/Trend Line for your Scatter Plot, just like you did for Unit 12: Assignment #1.
 Fourth, use the Line of Best Fit/Trend Line that you calculated to answer the problems!
 Remember that the formula for a line (including a Line of Best Fit/Trend Line) is y = mx + b, which means that m is the slope, and b is the intercept.
 For Problem #3, rather than using an exponential regression equation as the instructions state, use a linear regression equation (aka: a linear Line of Best Fit/Trend Line from your Scatter Plot, just like you did in Unit 12: Assignment #1).
 For Problem #3 and #4, when you are asked to predict a value outside the range of the data set, you will use extrapolation, just like you did in Unit 12: Assignment #1.
 Third, save a document that contains
 your calculations (if you’ve used your chosen data management platform for these calculations, you can copy/paste screenshots of your Scatter Plots with Lines of Best Fit/Trend Lines into your document)
 and your answers
 in a PDF named YourLastName_PSY210_Unit12_LinearRegression_Practice.pdf.
 Fourth, learn how to test the size of your PDF by reading through this handout.
 Then, test the size of your PDF.
 If the size of your PDF is too large to email to yourself, reduce the size of your PDF by following the suggestions in this handout.
 Go to the Unit 12: Assignment #2 Discussion Board and make a new Discussion Board post to which you should attach — not embed, but attach — your PSY210_Unit12_LinearRegression_Practice.pdf.
 First, look underneath the textbox where you typically type (or paste into) the Discussion Board, and you will see the “Attach” tool; it is the word “Attach” preceded by a paperclip icon.
 Second, click on the “Attach” tool. Browse to the .pdf file on your computer and select your .pdf file.
 Third, upload your .pdf file.
 Fourth, click on “Post Reply.”
 Fifth, make sure that do not attach your .pdf file by using the “Files” menu option on the lefthand side of the Discussion Board. Instead, use only the “Attach” tool that is found underneath the Discussion Board text box.
 Sixth, make sure that the PDF you attached is named YourLastName_PSY210_Unit12_LinearRegression_Practice.pdf.
Unit 12: Assignment #3 (due before 11:59 pm Central on FRI JUL 26):
 In this assignment, you’re going to learn about residuals (in Linear Regression calculated by using Least Squares estimation) and how to interpret the percent of variance explained (or accounted for).
 First, to get an overview of the General Linear Model, of which Linear Regression is one type of model, read Poldrack’s (2020) Chapter 14, “The General Linear Model.” While reading this article, make sure you understand:
 Remember, as you’ve learned in previous Units, the basic assumption of statistical thinking is outcome = model + error.
 Remember, as you’ve learned in previous Units, in statistical thinking we use statistics to describe, define, and predict.
 As you’ve learned in this Unit, in Linear Regression, the dependent variable (usually referred to as y) is the outcome variable that our model aims to explain or predict.
 The independent variable (usually referred to as x) is the predictor variable that we want to use to explain or predict the dependent variable.
 In Linear Regression, our standard formula for a line, y = mx + b, where m is the slope and b is the intercept, is often renamed y = βx(x) + βo, where βx is called the betaweight of x, because it tells us how much to weight the x variable.
 However, y = βx(x) + βo is the same standard formula for a line.
 Second, to learn about residuals (in Linear Regression), read Statistics How To’s (2015) article, “Residual Values (Residuals) in Regression Analysis.” While reading this article, make sure you understand:
 Very rarely do all the data points lie directly on the Line of Best Fit/Trend Line.
 The vertical distance between a data point and the Line of Best Fit/Trend Line is that data point’s residual.
 Residuals are considered errors, not because they are wrong, but because they represent error in fitting the model (in this case, the Line of Best Fit/Trend Line) to the data.
 Third, to understand how Lines of Best Fit/Trend Lines are calculated, read Math Is Fun’s (2017) article, “Least Squares Regression – How Does It Work?” While reading this article, make sure you understand:
 Linear Regression (calculated by using Least Squares estimation) works by trying to make the squared errors as low as possible; hence the name Least Squares.
 Remember that the errors are the residuals (the vertical distances between each data point and the Line of Best Fit/Trend Line).
 The smaller the residuals, the less error in the model.
 Linear Regression is highly sensitive to data outliers.
 An outlying data value (an outlier) will pull the Line of Best Fit/Trend Line toward itself.
 Fourth, to get experience examining residuals, do the following:
 Look closely through this set of 14 Least Squares (Linear Regression) models produced by the Phet Least Squares simulator.
 Notice the variety in residuals across the different Least Squares (Linear Regression) models.
 Choose three Least Squares (Linear Regression) models that you think have the most interesting pattern of residuals (however you want to define interesting!).
 Fifth, to get a handson feel for residuals, do the following:
 Go to the Phet Least Squares simulator, which was created by and is hosted by the University of Colorado at Boulder.
 Click (to open) the “Best Fit Line” panel in the upper lefthand corner.
 Then, check the box that’s also labeled “Best Fit Line.”
 Also, check the box that’s labeled “Residuals.”
 Do not check the box that’s labeled “Squared Residuals.”
 Although the squared residuals are interesting to look at (and remember that Least Squares estimation works to reduce the squared residuals), their visual representation on the simulator makes the simulation a bit too busy for our present purposes.
 Click (to open) the “Correlation Coefficient” panel, which is also on the lefthand side.
 Now, from the bowl of data points in the lower lefthand corner, select one data point and drag it into the chart area.
 Select another data point and drag it into the chart area.
 Now that you have two data points, a Least Squares estimation (and its Line of Best Fit/Trend Line) can be calculated.
 Continue selecting data points and dragging them into the chart area until you have TEN total data points.
 Play around with creating different Least Squares models and examining each model’s Line of Best Fit/Trend Line and residuals.
 To clear the chart and start again, you can either refresh the webpage OR click on the orange reset button in the lower righthand corner.
 After you’ve gotten familiar with creating different Least Squares models, refresh the webpage or click the orange reset button.
 Fifth, use the Phet Least Squares simulator to demonstrate the A, B, C, and D scenarios illustrated in this handout.
 A: Create a Least Squares (Linear Regression) model for which most of the 10 data points lie on or near the Line of Best Fit/Trend Line.
 Notice the high correlation coefficient.
 Notice how small (vertically short) the residuals are; that means you have less error in fitting the model.
 Take a screenshot of this simulation (not your entire screen) and save it as YourLastName_PSY210_Unit12_Residuals_A.xxx (where xxx is the file type, for example, .jpg, .png, .jpeg, and the like).
 B: Using the “A” regression you just created, increase the residual of only one data point by dragging it as far away from the Line of Best Fit/Trend Line as you can.
 Notice that the correlation coefficient drops in magnitude and the other residuals also increase, even though you didn’t move them yourself.
 Take a screenshot of this simulation (not your entire screen) and save it as YourLastName_PSY210_Unit12_Residuals_B.xxx (where xxx is the file type, for example, .jpg, .png, .jpeg, and the like).
 C: Begin a new simulation by refreshing the webpage/clicking the orange reset button and set up a new simulation, just like you did before (i.e., click to open the “Best Fit Line” panel; check the box for “Best Fit Line,” check the box for “Residuals,” click to open the “Correlation Coefficient” panel).
 Create a Least Squares (Linear Regression) model for which most of the 10 data points do NOT lie on or near the Line of Best Fit/Trend Line.
 Notice the lower correlation coefficient.
 Notice how much larger (vertically longer) the residuals are, which means you have more error in that model.
 Take a screenshot of this simulation (not your entire screen) and save it as YourLastName_PSY210_Unit12_Residuals_C.xxx (where xxx is the file type, for example, .jpg, .png, .jpeg, and the like).
 D: Using the “C” regression you just created, increase the residual of only one data point by dragging it as far away from the Line of Best Fit/Trend Line as you can.
 Notice that the correlation coefficient changes in magnitude.
 If you keep dragging that one data point, you might even be able to change the direction of the correlation from positive to negative or vice versa!
 Notice, however, you are increasing the error.
 Take a screenshot of this simulation (not your entire screen) and save it as YourLastName_PSY210_Unit12_Residuals_D.xxx (where xxx is the file type, for example, .jpg, .png, .jpeg, and the like).
 Next, to learn about the percent of variance explained (or accounted for), do the following:
 First, read Poldrack’s (2020) Chapter 14 “Quantifying Goodness of Fit of the Model.” While reading this article, make sure you understand:
 We often want to quantify how well our model fits our data.
 One way to make this quantification is by calculating the amount of variance explained.
 If there is only one x variable, we can calculate the percentage of variance explained by squaring the correlation coefficient and converting that value to a percentage.
 For example, if the correlation coefficient is .500, the square is .250, which means that the model can explain 25.000% of the variance.
 Second, read Statistics How To (2015) “What is Explained Variance?” While reading this article, make sure you understand:
 The percentage of variance explained is how much variance can be explained by our model.
 A higher percentage of variance explained indicates a better fitting model.
 A better fitting model allows us to make better predictions from our model.
 Even if the correlation coefficient is statistically significant, the regression might not be explaining very much of the variance. (Statistical significance is not practical significance.)
 Third, from the set of 14 Least Squares (Linear Regression) models you looked at previously and by remembering that the percentage of variance explained can be calculated by squaring the correlation coefficient and converting that value to a percentage:
 Identify the Least Squares (Linear Regression) model that explains the greatest percentage of variance (of the set of 14).
 Identify the Least Squares (Linear Regression) model that explains the lowest percentage of variance (of the set of 14).
 Go to the Unit 12: Assignment #3 Discussion Board.
 First, identify by topic (e.g., Height vs Shoe Size) the three Least Squares (Linear Regression) models you thought had the most interesting pattern of residuals.
 Second, briefly explain why you found each pattern of residuals interesting.
 Third, embed each of the four screenshots you made from manipulating the simulator (YourLastName_PSY210_Unit12_Residuals_A.xxx; YourLastName_PSY210_Unit12_Residuals_B.xxx; YourLastName_PSY210_Unit12_Residuals_C.xxx; YourLastName_PSY210_Unit12_Residuals_D.xxx)
 Fourth, identify by topic (e.g., Height vs Shoe Size) the Least Squares (Linear Regression) model that explains the greatest percentage of variance.
 Fifth, state the greatest percentage of variance explained.
 Sixth, identify by topic (e.g., Height vs Shoe Size) the Least Squares (Linear Regression) model that explains the lowest percentage of variance.
 Seventh, state the lowest percentage of variance explained.
Unit 12: Assignment #4 (due before 11:59 pm Central on FRI JUL 26):
 In this assignment, you will learn about another family member in the General Linear Model family, Analysis of Variance (ANOVA).
 First, to understand the difference between ANOVA and Linear Regression, look through Professor Gernsbacher’s handout. While looking through this handout, make sure you understand the following:
 Both Linear Regression and ANOVA are members of the General Linear Model family (because they both draw upon general linear modeling for their calculations).
 In Linear Regression, both the y variable and the x variable are continuous measurements.
 But in ANOVA, although the y variable is continuous, the x variable is discrete (categorical).
 In both Linear Regression and ANOVA, the y variable is always the DEPENDENT variable (also called the outcome variable), and the x variable is always the INDEPENDENT variable (also called the predictor variable).
 Second, to get a basic introduction to ANOVA, read Bevans’s (2020) article, “An Introduction to the OneWay ANOVA.” While reading this article, make sure you understand the following:
 Use a oneway ANOVA when you are analyzing data with one categorical (DISCRETE) independent variable and one CONTINUOUS dependent variable.
 In ANOVA, the null hypothesis is that the dependent variable is NOT affected by (does
not depend on) the categories of the independent variable.
 In ANOVA, the alternate hypothesis is that the dependent variable IS affected by (does depend on) the categories of the independent variable.
 ANOVA calculates an Ftest to assess the null hypothesis.
 The Ftest compares the amount of explained to unexplained variance.
 The Ftest is the following ratio:
variance in the dependent variable that’s explained by the independent variable TO
variance in the dependent variable that’s unexplained by the independent variable
 The more of the dependent variable’s variance that’s explained by the discrete categories (of the independent variable), the higher the Ftest ratio will be.
 A higher ratio (a larger Ftest statistic) means that more of the dependent variable’s
variance can be explained by the independent variable’s categories.
 Now, you’re going to get some practice calculating oneway ANOVAs on two different data sets. Each of the (discrete) independent variables in these ANOVAs are three categories, which are also called levels (i.e., a oneway ANOVA with three levels).
 If your Lastname comes FIRST alphabetically in your NEW Chat Group, you will analyze the FOOD TRUCKS Data Set AND the SUMMER JOBS Data Set.
 If your Lastname comes LAST alphabetically in your NEW Chat Group, you will analyze the INTERNET PROVIDERS Data Set AND the SICK LEAVE Data Set.
 If your Lastname comes NEITHER first nor last, you will analyze the FOOD TRUCKS Data Set AND the SICK LEAVE Data Set.
 For EACH data set you’re analyzing, do the following:
 First, download each data set.
 For downloading these data, you will need to use ONLY the browsers Chrome, Firefox, or Edge.
 For downloading these data, you can’t use the browser Safari.
 When prompted, save the file in your PSY210_Summer2021_Unit12 folder.
 Second, import each data set into a blank spreadsheet in your chosen data management platform.
 Remember to follow Andrews’ (2020) Import Data howto article.
 Save each new spreadsheet naming the file, YourLastName_PSY210_Unit12_ZZZ_Data, where ZZZ is the name of the data set.
 Using your chosen data management platform, calculate the Mean, the Standard Deviation, and the N (the number of data values in each category) of each of the three categories.
 Name the screenshot YourLastName_PSY210_Unit12_YYY_Screenshot.xxx (where YYY is the name of the data set and xxx is the filetype, for example, .jpg, .png, .jpeg and the like).
 Your screenshot should include only the part of your spreadsheet that contains the Means, Standard Deviations, and the Ns for the three categories, NOT your entire screen.
 Choose ONE of these online OneWay ANOVA calculators:
 After calculating each OneWay ANOVA (one ANOVA on each data set), do the following:
 First, check the Means, Standard Deviations, and Ns provided by the online calculator to make sure they match the Means, Standard Deviations, and Ns you calculated yourself.
 Second, take a screenshot of the Fratio and the pvalue (not your entire screen).
 Name the screenshot YourLastName_PSY210_Unit12_YYY_ANOVA_Screenshot.xxx (where YYY is the name of the data set and xxx is the filetype, for example, .jpg, .png, .jpeg and the like).
 Third, decide whether you can reject the null hypothesis that the dependent variable is NOT affected by (does not depend on) the categories of the independent variable.
 Remember from Unit 8, if the pvalue is low enough (e.g., p < .050), you can reject the null hypothesis that the dependent variable is NOT affected by (does not depend on) the categories of the independent variable.
 If the pvalue is not low enough (e.g., p ≥ .050), you cannot reject the null hypothesis that the dependent variable is NOT affected by (does not depend on) the categories of the independent variable.
 Now, you’re going to get more practice calculating oneway ANOVAs on two additional data sets. Each of the (discrete) independent variables in these ANOVAs are FOUR categories, which are also called levels (i.e., a oneway ANOVA with FOUR levels).
 If your Lastname comes FIRST alphabetically in your NEW Chat Group, you will analyze the DOG OBEDIENCE Data Set AND the TEXTBOOK COSTS Data Set.
 If your Lastname comes LAST alphabetically in your NEW Chat Group, you will analyze the FAMILY VACATIONS Data Set AND the WEDDING GIFTS Data Set.
 If your Lastname comes NEITHER first nor last, you will analyze the DOG OBEDIENCE Data Set AND the WEDDING GIFTS Data Set.
 For EACH of these data sets, do the following:
 First, download each data set (and save it in your PSY210_Summer2021_Unit12 folder).
 Again, for downloading these data sets, you will need to use ONLY the browsers Chrome, Firefox, or Edge. You can’t use the browser Safari to download these data sets.
 Second, import the data set into a blank spreadsheet in your chosen data management platform, and save the new spreadsheet as YourLastName_PSY210_Unit12_ZZZ_Data, where ZZZ is the name of the data set.
 Third, using your chosen data management platform, calculate the Mean, the Standard Deviation, and the N (the number of data values in each category) of each of the FOUR categories.
 Name the screenshot YourLastName_PSY210_Unit12_YYY_Screenshot.xxx (where YYY is the name of the data set and xxx is the filetype, for example, .jpg, .png, .jpeg and the like).
 Your screenshot should include only the part of your spreadsheet that contains the Means, Standard Deviations, and the Ns for the three categories, NOT your entire screen.
 Fourth, choose a different online OneWay ANOVA calculator than you used before and calculate a OneWay ANOVA.
 Fifth, check the Means, Standard Deviations, and Ns provided by the online calculator to make sure they match the Means, Standard Deviations, and Ns you calculated yourself.
 Sixth, take a screenshot of the Fratio and the pvalue (not your entire screen).
 Name the screenshot YourLastName_PSY210_Unit12_YYY_ANOVA_Screenshot.xxx (where YYY is the name of the data set and xxx is the filetype, for example, .jpg, .png, .jpeg and the like).
 Seventh, decide whether you can reject the null hypothesis that the dependent variable is NOT affected by (does not depend on) the categories of the independent variable.
 Remember from Unit 8, if the pvalue is low enough (e.g., p < .050), you can reject the null hypothesis that the dependent variable is NOT affected by (does not depend on) the categories of the independent variable.
 If the pvalue is not low enough (e.g., p ≥ .050), you cannot reject the null hypothesis that the dependent variable is NOT affected by (does not depend on) the categories of the independent variable.
 Go to the Unit 12: Assignment #4 Discussion Board and make a new post in which you do the following, for EACH data set you analyzed:
 First, embed the screenshot of the Means, the Standard Deviations, and the Ns you calculated for each category using your chosen data management platform.
 Second, embed the screenshot of the Fratio and the pvalue (not your entire screen).
 Third, write the sentence [filling the blanks]: I ___ [can or cannot] reject the null hypothesis that ___ [the dependent variable] is not affected by (does not depend on) ___ [categories of the independent variable].
 For example: I can reject the null hypothesis that apartments’ rental prices are not affected by (depend on) apartments’ location (Near Campus, West of Campus, Downtown).
 Or for example: I cannot reject the null hypothesis that apartments’ rental prices are not affected by (depend on) apartments’ location (Near Campus, West of Campus, Downtown).
Unit 12: Assignment #5 (due before 11:59 pm Central on SUN JUL 28):
 Meet online with your NEW Chat Group (which you formed during Unit 8) for a onehour textbased Group Chat at a time/date that your Chat Group previously arranged.
 BEFORE YOU MEET WITH YOUR CHAT GROUP, each member of the Chat Group must do ALL of the following:
 First, learn what “Regression to the Mean” is by reading Farnam Street’s (2015) article, “Regression Toward the Mean: An Introduction with Examples.”
 While reading this article, write down as many examples that you can of Regression to the Mean that are presented in the article.
 Second, to make sure you understand what “Regression to the Mean” is, read Rational Wiki’s (2017) article, “Regression to the Mean.”
 While reading this article, write down as many examples that you can of Regression to the Mean that are presented in the article.
 Third, think of (and write down) FIVE examples of Regression to the Mean that you can think of and that are NOT mentioned in either article.
 Fourth, to better understand our human propensity to discern patterns that might not be valid, read a segment of Poldrack’s (2020) Chapter 8, “Randomness in Statistics.”
 Fifth, to continue to understand our human propensity to discern patterns that might not be valid, look through this handout of “Faces In Things.”
 Sixth, think of (and write down or capture in images) THREE OTHER examples of patterns you were tempted to discern that were not valid.
 NOTE: None of your three examples need be visual patterns; they can be visual — but they can also be any pattern, including behavioral patterns.
 DURING YOUR ONEHOUR GROUP CHAT:
 First, begin your chat with each Chat Group member indicating ONE of the nine “How Are You Feeling at the START of Today’s Group Chat?” images. More than one Chat Group member can indicate the same image if that’s how they are feeling, and please refer to each image by its number.
 Second, begin your Group Chat by making sure that ALL the Chat Group members understand Regression to the Mean.
 Third, EACH Chat Group member needs to report the examples of Regression to the Mean that they read in Farnam Street’s (2015) article.
 Fourth, EACH Chat Group member needs to report the examples of Regression to the Mean that they read in Rational Wiki’s (2017) article.
 Fifth, EACH Chat Group member needs to report the FIVE examples of Regression to the Mean that they previously wrote down (that were not mentioned in either article).
 Sixth, EACH Chat Group member needs to report their favorite Faces In Thing image.
 Seventh, EACH Chat Group member needs to report the THREE OTHER examples of patterns they were tempted to discern that were not valid.
 AT THE END OF YOUR ONEHOUR GROUP CHAT do the following:
 Each Chat Group member needs to indicate ONE of the nine “How Are You Feeling at the END of Today’s Group Chat?” images. More than one Chat Group member can indicate the same image if that’s how they are feeling, and please refer to each image by its number.
 NOTE: The “How Are You Feeling at the END of Today’s Group Chat” grid of images differs from the “How Are You Feeling at the START of Today’s Group Chat” grid of images.
 Nominate one member of your Chat Group (who participated in the Chat) to make a post on the Unit 12: Assignment #5 Discussion Board that summarizes your Group Chat in at least 200 words. This Chat Group member should not post their 200word summary of your Group Chat until they have completed their Course Journal for the current Unit.
 This Chat Group member needs to be sure to include the FIVE examples of Regression to the Mean that each Chat Group member brought to the Group Chat (that were not mentioned in either article).
 This Chat Group member also needs to be sure to include the THREE examples of patterns each Chat Group member was tempted to discern that were not valid.
 At the end of the 200word summary, this member needs to write this sentence filling the blanks: “I have completed my Course Journal for the current Unit. It contains ___ words and the two things I learned during this Unit about which I journaled about are ___ and ___.”
 Nominate a second member of your Chat Group (who participated in the Group Chat using the browser Chrome on their laptop, rather than on their mobile device) to save the Chat transcript, as described in the Course How To (under the topic, “How To Save and Attach a Chat Transcript”).
 This Chat Group member needs to make a post on the Unit 12: Assignment #5 Discussion Board and attach the Chat transcript, saved as a PDF, to that Discussion Board post. This Chat Group member should not post the transcript of your Group Chat until they have completed their Course Journal for the current Unit.
 In their Discussion Board post, this member needs to write this sentence filling the blanks: “I have completed my Course Journal for the current Unit. It contains ___ words and the two things I learned during this Unit about which I journaled about are ___ and ___.”
 Remember to attach the Chat transcript by clicking on the word “Attach.” (Do not click on the sidebar menu “Files.”).
 Nominate a third member of your Chat Group (who also participated in the Chat) to make another post on the Unit 12: Assignment #5 Discussion Board that states the name of your Chat Group, the names of the Chat Group members who participated in the Chat, the date of your Chat, and the start and stop time of your Group Chat. This Chat Group member should not post the names, date, and times of your Group Chat until they have completed their Course Journal for the current Unit.
 In their Discussion Board post, this member needs to write this sentence filling the blanks: “I have completed my Course Journal for the current Unit. It contains ___ words and the two things I learned during this Unit about which I journaled about are ___ and ___.”
 If only two students participated in the Group Chat, then one of those two students needs to do two of the above three tasks.
 Before ending the Group Chat, arrange the date and time for the Group Chat you will need to hold during the next Unit (Unit 13: Assignment #5).
Congratulations, you have finished Unit 12! Onward to Unit 13!

