SCI1020: Introduction to Statistical Reasoning

On completion of this workshop, you should be able to

1. Produce a scatterplot of quantitative data with appropriate explanatory and response axes

2. Recognise a linear pattern and the general formula for a straight line

3. Calculate a predicted value given the equation of the linear regression line

4. Add a linear line of best fit to data using MS EXCEL, and describe the regression line (equation and correlation)

5. Assess the closeness of fit using the least-squares criterion as reflected in the correlation coefficient

6. Obtain residual values and interpret their size and distribution about the line in the form of a residual plot

7. Find or calculate and interpret the squared correlation, r⊃2;

Preliminary Reading

D S Moore et al., Basic Practice of Statistics, Chapters 4–5.

These questions prepare you for the workshop. Complete them before class.

Q.1 State in your own words what is meant by each of the terms listed below. Be specific.

Q.2 What is the general equation of a straight line? Define all the terms in the equation.

Q.3 Do Q5.2 from Moore et al. text.

What is the regression line equation based on the description of the trend in this example?

Workshop Problem

Q.4 Demonstration of Correlation and Least Squares Regression Underscores are used instead of spaces

• Create a scatterplot of a linear trend (similar to Plot #1 below).

• Observe the correlation coefficient for different scatter patterns.

• Use “Draw your own line” to create a line of best fit.

• Change the intercept and slope to minimise the sum of squared residuals (“relative SS”).

• Compare your line to the “Show least-squares line” option.

(No written answers required—just observation.)

b) Describe the relationship in the x-y data plotted below:

Identify the association (positive/negative/none) and correlation (strong/moderate/weak/none).

REGRESSION ANALYSIS FOR LINEAR DATA

See also: “Introduction to Excel” Section 2.7, pp. 13–15.

Using Excel to produce scatterplot and linear line of best fit:

• Plot Response variable (y-axis) against Explanatory variable (x-axis)
  Excel: left-hand column = x

• Select chart layout with a line and fx to show the regression equation

• Note the correlation coefficient r and R⊃2;

• Identify regression coefficients (intercept and slope)

• Obtain full regression analysis using Data Analysis → Regression

  • Excel requires y-data first

  • Data must be in columns, not rows

• Check linearity using the residual plot

  • A random scatter of residuals above and below line indicates linearity is appropriate

Q.5 Do from Moore et al. and with the same data, 

• Download dataset “Sparrowhawk” from Moodle Week 4

• Produce scatterplot of the relationship

• Describe the association between New Adults arriving and the percentage of returning birds:
• What is the general strength of the association?
• EXTRA: What is the R-squared value?
• What does this R-squared value specifically tell us about this association?
• Apply linear regression analysis using Excel. Include the residual plot.

TUTOR CHECK OF PLOTS:

Scatterplot and Residual plot
If not done in class, attach printed plots.

Describe the residual plot.

Is there any trend—curve or pattern—or are points randomly scattered?
What does this tell you about fitting a linear model?

Moore Q5.39 (a)

What is the equation of the linear model for this relationship?
(Use descriptive variable names—NOT x and y.)

Moore Q5.39 (b)

Value of the slope = ___________

What does this slope indicate?
(Use the actual number to explain influence of explanatory variable on response.)

Use the model to predict new adult number if 60% of adults return:

EXTRA: Verify the residual for data point where x = 60.

Show full calculation.

Residual = (data y value − predicted y value)

Check against Excel output for x = 60.

Q.6 Appropriateness? Causation? Application?

Explain two cautions when interpreting x–y relationships using linear regression.
(Refer to Moore et al., Chapter 5 Summary.)

From this exercise, you should be able to:

• Draw a scatterplot

• Obtain a line of best fit and its equation

• Produce a residual plot

• Obtain the correlation coefficient

• Explain each item above and what it indicates about the data

Summary of Assessment Requirements

The assessment requires students to complete Worksheet 3 for SCI1020: Introduction to Statistical Reasoning, focusing on scatterplots, correlation, and linear regression. The worksheet assesses the student's ability to:

1. Create scatterplots with correct explanatory (x) and response (y) variables

2. Recognise linear patterns and understand the general straight-line equation

3. Use a regression equation to calculate predicted values

4. Apply Excel tools to add a line of best fit, generate regression outputs, and interpret correlation

5. Assess linear fit using the least-squares criterion and correlation coefficient

6. Interpret residuals and create residual plots

7. Calculate and interpret r⊃2;

8. Work with real datasets, including the “Sparrowhawk” dataset

9. Describe associations, interpret regression results, evaluate model appropriateness

10. Discuss cautions when interpreting x–y relationships

The worksheet includes preliminary reading, short-answer theory questions (Q1–Q3), workshop-based observational tasks (Q4), Excel-based regression analysis (Q5), and interpretation and statistical reasoning questions (Q6).

Key Pointers to Be Covered in the Assessment

The student must demonstrate:

1. Conceptual Knowledge

• Definitions of explanatory variable, response variable, correlation, regression line, residual

• Understanding the equation of a straight line (y = a + bx)

• Understanding the meaning of slope, intercept, r, and r⊃2;

2. Practical/Analytical Skills

• Producing scatterplots in Excel

• Adding a regression trendline with equation and R⊃2;

• Interpreting association strength and direction

• Calculating predicted values

• Generating and interpreting residual plots

• Performing full regression analysis using the Data Analysis Toolpak

3. Interpretation

• Assessing whether linear regression is appropriate

• Understanding what r⊃2; tells us about variability explained

• Interpreting slope in context

• Making predictions and checking residuals

• Identifying limitations of regression, including causation and extrapolation issues

4. Dataset Application 

1. Plotting scatterplot

2. Strength/type of association

3. Interpreting regression model

4. Predicting values

5. Calculating and interpreting residuals

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