Lesson 24: Review for Final Exam

Review:

• Complete a Course Evaluation
• Register for Spring Courses
  • BSAD 360 Advanced Statistics (online)
  • ECON 320 – World Economy
• Connect on LinkedIn

Presentation:

• Final Exam
  • Tue, Dec 16, 1:00-3:20p in HSB 111
  • Same format as midterm exams
  • Students are permitted a 1-page “cheat sheet” for formulas and notes (both sides)
  • Bring a Z-table, t-table and a calculator
• Topics/Problem Types:
  • Describe a small data set
    • Stemplot
    • Five-number summary with boxplot
    • Mean and Standard Deviation
  • Normal Curve
    • Calculate Z-scores
    • Find percentages under the normal curve (below, above, between)
    • Z-table
    • Empirical Rule
    • Reverse lookup, e.g., what value of x is greater than 90% of a normal distribution?
  • Estimation with Confidence Intervals
    • Point estimate
    • Margin of error (MoE) calculation
    • Point estimate +/- MoE = Confidence intervals
    • Large sample (Z table), n>30
    • Small sample (t table), n<=30
    • Proportion (p-hat) – unique standard error calculation
  • Significance Testing
    • 4 steps
    • Setup Ho and Ha
    • Calculate Z statistic
    • Calculate p-value
    • Interpret results (Reject or Fail to Reject)
  • Bivariate Analysis
    • Scatterplot and correlation
    • Sum of Squares
    • Linear regression
      • Slope
      • y-intercept
      • linear equation
    • Pearson correlation coefficient (r) and R-squared
    • Estimation/forecasting with linear regression
    • Categorical Predictor

Activity:

• Take last year’s Final Exam

Assignment:

• Study for the Final Exam

Lesson 23: Regression with a Categorical Predictor (Two-group case)

Review:

• Exam 3

• 

• Final Exam: Tue, Dec 16, 1:00-3:20p in HSB 111

Presentation:

• A/B Testing = comparing groups with simple regression
  • We can use a “Dummy Variable” to compare 2 groups
  • Each group is given a value of 0 or 1 so the category can be used numerically
• Demonstration:
  • Let’s say you want to run an experiment on a website:
    • Version A (control): old landing page
    • Version B (treatment): new landing page

You run each version on 3 days and record the number of sign-ups per 100 visitors.

Day	Version	Sign-ups per 100 visitors (Y)
1	A	5
2	A	6
3	A	4
4	B	8
5	B	9
6	B	7

To use regression, we have to first recode the category (group A or B) as a binary number, 0 or 1.

This allows us to compare A vs B using simple regression with a dummy variable.

Day	Version	X	Y
1	A	0	5
2	A	0	6
3	A	0	4
4	B	1	8
5	B	1	9
6	B	1	7

Now we can use our Sum of Squares approach to creating a regression equation.

Day	X	Y	(x^2)	(y^2)	(xy)
1	0	5	0	25	0
2	0	6	0	36	0
3	0	4	0	16	0
4	1	8	1	64	8
5	1	9	1	81	9
6	1	7	1	49	7
Σ	3	39	3	271	24

With our column sums we can calculate Sum of Squares.

• SSxx = 1.5
• SSyy = 17.5
• SSxy = 4.5

Then we can calculate the regression equation.

• b1 = SSxy/SSxx = 4.5/1.5 = 3
• b0 = Σy/n – b1*(Σx/n) = 39/6 – 3*(3/6) = 6.5 – 1.5 = 5
• y-hat = b1*x + b0 = 3*x + 5

To interpret we calculate y-hat for both x=0 and x=1.

• For x=0, y-hat = 5
• For x=1, y-hat = 8
• Our predicted conversion rate for Version A = 5 signups per 100 visitors
• Our predicted conversion rate for Version B = 8 signups per 100 visitors

We can also calculate r and r^2 to understand how much of the variance is explained by Version A vs B.

• r = SSxy/Sqrt(SSxx*SSyy) = 4.5/Sqrt(1.5*17.5) = 0.878
• r^2 = (0.878)^2 = 0.771
• In other words, the group (Version A or B) explains about 77% of the variation in signups.

Why not just calculate the two means?

• Means only work for simple two-group comparisons.
• Regression handles:
  • more than two groups (multiple dummies → ANOVA)
  • continuous predictors (e.g., time on website)
  • multiple predictors at once (multiple regression)
• In industry, this is exactly what A/B test software does.

Activity:

The same company wants to test two email subject lines:

• Version A (control)
• Version B (new)

They record the number of clicks per 50 emails sent for each version across several small batches.

Batch	Version	Clicks per 50 emails (Y)
1	A	6
2	A	5
3	A	7
4	A	4
5	B	8
6	B	10
7	B	9
8	B	7

• Use a dummy variable for A/B, converting to 0 and 1 for X
• Then calculate the sum of squares, regression equation and r^2
• Which version works better?

Assignment:

• Complete a course evaluation: https://www.smartevals.com/login.aspx?s=csup&p=C1F7D22FF11900C7D8265E0DE68C25CE7B0DBC50&ebi=3602955&bid=7240469&spf=0&a2e=73/2250026/1767139200/dKsIfY8TyXrSKGMeEl6fA5QiJAc&bid=7240469

Lesson 22: Review for Exam 3

Review:

• Exam 3 on Thu Nov 20

Presentation:

• Review Topics:
  • Scatterplots
  • Correlation
    • Positive or Negative
    • Strong, Moderate, Weak
  • Pearson Correlation Coefficient
  • Least Squares Regression
    • Sum of Squares
    • Linear Regression
    • Calculate slope and y-intercept for linear equation
    • Calculate R-Squared
    • Forecasting with a regression equation (plug in x and solve for y)
  • Cautions about Regression and Correlation
    • Extrapolation
    • Spurious correlation
    • Correlation ≠ causation
    • Lurking variables
    • Influential observations
    • Sensitivity analysis
    • Demonstrating causation
      • Retrospective studies
      • Prospective studies

Activity:

• Complete this Practice Exam

No Class Today

Sorry for the late notice but there will be no class today.

The deadline for submitting Exam 2 corrections is extended until Tuesday, Nov 18.

Lesson 21: Cautions about Correlation and Regression

Review:

• Pearson Correlation Coefficient
• R-Squared
• Schedule
  • Finish Simple Regression this week
  • Review and Exam 3 next week
  • Thanksgiving

Presentation:

• Cautions about Correlation and Regression
  • Influential Observations
    • Outliers
  • Lurking Variables
    • Explanatory variables not included
    • Beyond scope
  • Extrapolation
    • Use of regression for prediction far outside the range of the explanatory variable
    • Predictions outside the range are unreliable
    • Further outside the range = less reliable predictions
  • The Question of Causation
    • Correlation does not imply causation
    • Spurious Correlations
    • Retrospective study = looking back to find possible causes for an established outcome among a sample population
    • Prospective studies = following a sample population over time and studying behaviors possibly linked to likelihood of an outcome
  • Video

Demonstration in Sheets:

Now, here’s some real data, gathered from USDA National Water and Climate Center and the USGS Water Data Center. The snowpack data (snow-water equivalent inches) is for the Freemont Pass location. The streamflow data (ft^3/SEC) is for the Arkansas River station in Pueblo.

Water Year	Snowpack (x)	Streamflow (y)
2018	20.4	577.5
2019	28.1	1599.3
2020	20.2	901.0
2021	15.5	659.4
2022	17.6	730.4
2023	17.1	1016.0
2024	22.2	1455.7

• We did this manually last time
  • Calculate Sum of Squares, the Linear Equation, the Pearson Correlation Coefficient and R^2.
  • Using the regression equation, estimate Streamflow for 2025 assuming Snowpack x=18.2.
• This time…
  • Use Sheets to produce the regression equation and point estimate for 2025.
  • Check your answer from last week.

Activity/Assignment:

• Use Sheets (or Excel) to solve the problems from last week (see below)
• Use Sheets to find the regression equation for this larger real estate data set

Problem 1. A small coffee shop tracks daily high temperature (°F) and the number of iced coffees sold. Calculate Sum of Squares and the equation of the regression line.

Day	Temp (x)	Iced Coffees Sold (y)
1	60	42
2	65	48
3	70	52
4	75	61
5	80	65

Problem 2. A local brewery tracks weekly social media ad spending and kegs sold. Calculate Sum of Squares and the equation of the regression line.

Week	Ad Spend (x, $100’s)	Kegs Sold (y)
1	2	11
2	3	15
3	4	17
4	5	20
5	7	26
6	9	30

Problem 3. A realtor wants to estimate the relationship between square footage (x) and listing price (y in $1000s). Calculate Sum of Squares and the equation of the regression line.

House	Sq Ft (x)	Listing Price (y)
A	1,600	315
B	1,850	344
C	2,000	365
D	2,300	399
E	2,500	425

Lesson 20: Pearson Correlation Coefficient and R-Squared

Review:

• Linear Regression
• Exam 2 corrections
  • Only for test takers who took Exam 2 on the scheduled date.
  • You must complete this entire exam.
  • Your exam score will increase up to 10 points or up to 50% of points deducted, whichever is less. For example, if you received an 88 on the exam you can only earn up to 6 points.
  • Your actual points awarded will be based on your exam score. For example, if you get an 80% on the new exam, you will earn 8/10 points (or .8 x 50% of points deducted).

Presentation:

• Pearson Correlation Coefficient
  • r = SSxy/√(SSxx*SSyy)
  • -1 < r < 1
• R-Squared
  • Calculate r and then square the result, i.e., =r^2
  • 0 < r^2 < 1
• Let’s solve this problem from last time, including calculating r and r^2, from beginning to end.

Problem 2. A local brewery tracks weekly social media ad spending and kegs sold. Calculate Sum of Squares and the equation of the regression line.

Week	Ad Spend (x, $100’s)	Kegs Sold (y)
1	2	11
2	3	15
3	4	17
4	5	20
5	7	26
6	9	30

Let’s also estimate kegs sold when the ad spend is $650 (x=6.5)

Video: Fitting Lines to Data

Activity:

Now, here’s some real data, gathered from USDA National Water and Climate Center and the USGS Water Data Center. The snowpack data (snow-water equivalent inches) is for the Freemont Pass location. The streamflow data (ft^3/SEC) is for the Arkansas River station in Pueblo.

Water Year	Snowpack (x)	Streamflow (y)
2018	20.4	577.5
2019	28.1	1599.3
2020	20.2	901.0
2021	15.5	659.4
2022	17.6	730.4
2023	17.1	1016.0
2024	22.2	1455.7

Calculate Sum of Squares, the Linear Equation, the Pearson Correlation Coefficient and R^2.

Using the regression equation, estimate Streamflow for 2025 assuming Snowpack was 18.2.

Assignment:

• Complete the corrections exam or enjoy a pleasant weekend.

Lesson 19: Linear Regression

Review:

• Exam 2 review
• Scatterplots
• Sum of Squares

Presentation:

• Linear regression (simple, bivariate)
  • Calculate equation of the regression line
    • y-hat = b1*x + bo
      • b1 = “slope” of the line
      • b0 = “y-intercept”
    • Calculate Sum of Squares: SSxx, SSyy, SSxy
    • b1 = SSxy/SSxx
    • b0 = (∑y/n) – b1*(∑x/n)
• Example 1: Beer Sales by Temp
  • Use temperature to predict kegs sold
  • Historical Data (Temp degrees F, # of kegs sold)
    • Day 1: 60°, 12
    • Day 2: 70°, 14
    • Day 3: 80°, 16
    • Day 4: 90°, 18
    • Day 5: 100°, 20
  • Scatter Plot
  • Table setup
• Sum of Squares calculations
  • SSxx = ∑x² – (∑x)²/n
  • SSyy = ∑y² – (∑y)²/n
  • SSxy = ∑xy – (∑x*∑y)/n
• Calculate Slope of regression line
  • b1 = SSxy/SSxx
• Calculate the y-intercept
  • b0 = (∑y/n) – b1*(∑x/n)
• Video: Fitting Lines to Data

Activity:

• Example 2: Beer Sales by Price
  • Historical Data (Price $ # of kegs sold)
    • Day 1: $90, 22
    • Day 2: $100, 20
    • Day 3: $110, 18
    • Day 4: $120, 16 
    • Day 5: $130, 14
  • Calculate Sum of Squares
  • Calculate the slope and y-intercept
  • Write the linear equation

Assignment:

First, go back to the problems assigned last week. Use your Sum of Squares calculations to find the equation of the regression line for each problem.

1. Advertising vs. Sales (n = 8)
   • SSxx = 55.5
   • SSyy = 1,385.875
   • SSxy = 275.75
2. Training Hours vs. Productivity (n = 8)
   • SSxx = 42
   • SSyy = 327.5
   • SSxy = 117
3. Price vs. Units Sold (n = 8)
   • SSxx = 1050
   • SSyy = 29,400
   • SSxy = –5,500

Problem 1. A small coffee shop tracks daily high temperature (°F) and the number of iced coffees sold. Calculate Sum of Squares and the equation of the regression line.

Day	Temp (x)	Iced Coffees Sold (y)
1	60	42
2	65	48
3	70	52
4	75	61
5	80	65

Problem 2. A local brewery tracks weekly social media ad spending and kegs sold. Calculate Sum of Squares and the equation of the regression line.

Week	Ad Spend (x, $100’s)	Kegs Sold (y)
1	2	11
2	3	15
3	4	17
4	5	20
5	7	26
6	9	30

Problem 3. A realtor wants to estimate the relationship between square footage (x) and listing price (y in $1000s). Calculate Sum of Squares and the equation of the regression line.

House	Sq Ft (x)	Listing Price (y)
A	1,600	315
B	1,850	344
C	2,000	365
D	2,300	399
E	2,500	425

Lesson 18: Least Squares Regression

Review:

• Scatterplots

Presentation:

• Least Squares Regression
  • Find a line of best fit for a set of points
  • Least Squares Regression Proof: MethodLeastSquares
  • Example 1: Beer Sales by Temp
    • Use temperature to predict kegs sold
    • Historical Data (Temp degrees F, # of kegs sold)
      • Day 1: 60°, 12
      • Day 2: 70°, 14
      • Day 3: 80°, 16
      • Day 4: 90°, 18 
      • Day 5: 100°, 20
    • Scatter Plot
    • Table setup
  • Sum of Squares calculations
    • SSxx = ∑x² – (∑x)²/n
    • SSyy = ∑y² – (∑y)²/n
    • SSxy = ∑xy – (∑x*∑y)/n
  • Example 2: Beer Sales by Price
    • Experiment with price to predict demand
    • Historical Data (Price $ # of kegs sold)
      • Day 1: $90, 22
      • Day 2: $100, 20
      • Day 3: $110, 18
      • Day 4: $120, 16 
      • Day 5: $130, 14
    • Students complete calculations
  • We will use these Sum of Squares calculations (next time) to determine
    • linear equation
    • Pearson correlation coefficient
    • R-squared

Activity:

Problem 1 – Advertising and Sales (n = 8)

A retail chain studied the link between weekly advertising spend and sales (in $1,000s).

Week	Advertising (x)	Sales (y)
1	6	64
2	8	77
3	10	85
4	4	55
5	7	72
6	9	82
7	3	50
8	11	88

Task:
Calculate SSxx, SSyy and SSxy

Problem 2 – Training Hours and Productivity (n = 8)

An HR analyst examined whether more employee training hours lead to higher productivity (daily packages processed per worker).

Dept	Training Hours (x)	Productivity (y)
A	5	46
B	8	54
C	4	42
D	6	49
E	10	59
F	3	40
G	9	57
H	7	51

Task:
Calculate SSxx, SSyy and SSxy

Problem 3 – Price and Units Sold (n = 8)

A manufacturer tested different price points to estimate demand.

Trial	Price (x)	Units Sold (y)
1	40	210
2	35	250
3	45	190
4	50	160
5	30	280
6	55	150
7	25	310
8	60	130

Task:
Calculate SSxx, SSyy and SSxy

Lesson 17: Scatterplots

Review:

• Exam 2 Results

• 

Presentation:

• Scatterplots
  • Visualize relationship between 2 variables
    • independent (x) and dependent (y)
    • horizontal axis (x) and vertical axis (y)
  • Visual Correlation
    • Direction: Positive or Negative
    • Strength: Strong, Moderate, Weak
  • Examples
    • Ice Cream Sales
    • Colorado TCAP Scores and % Free-Reduced Lunch
    • Gender Wage Gap
  • Alternative Uses
    • Predicted vs Actual
    • Hot/Crazy
  • Unit 10 Scatterplots Video

Activity:

Complete the problems below twice, once by hand and once in Sheets or Excel.

Problem 1. A study was conducted on mercury (Hg) concentrations in fish taken from Lake Natoma in California. The researchers were concerned that mercury concentration levels in sample fish tissue might differ depending on the lab testing the fish. Fish tissue samples from 10 largemouth bass were sent to two labs, Columbia Environmental Research Center (CERC) and University of California, Davis (UC Davis), for inter-laboratory comparison. Mercury concentration is measured in micrograms of mercury per gram of fish tissue (dry weight). Use the mercury concentration data below to create a Scatterplot and characterize the nature of any correlation.

Problem 2. Satellites are one of the many tools used for predicting flash floods, heavy rainfall, and large amounts of snow. Geostationary Operational Environmental Satellites (GOES) collect data on cloud top brightness temperatures (measured in degrees Kelvin (°K)). It turns out that colder cloud temperatures are associated with higher and thicker clouds, which could be associated with heavier precipitation. Data consisting of cloud top temperature measured by a GOES satellite and rainfall rate measured by ground radar appear in the table below. Because ground radar can be limited by location and obstructions, having an alternative for predicting the rainfall rates can be useful. Use the temperature and radar rain rate data below to create a Scatterplot and characterize the nature of any correlation.

Problems copied from the Against All Odds video series guide for Unit 10.

Lesson 16: Review for Exam 2

Review:

• Significance Testing
• Exam 2
  • Thu, Oct 23
  • Covers Lessons 8–15
• Review with Rylee tomorrow (Wed) 1:30-4p in HSB 2nd floor conference

Exam 2 Topics:

• Census
• Sampling
• Sampling Distributions
• Central Limit Theorem
• Estimation
  • Confidence Intervals = Point estimate +/- Margin of error
  • Z-table
  • t-distribution for small samples
  • Inference for Proportions
• Significance Testing
  • Step 1. Set up Ho and Ha
  • Step 2. Calculate Z statistic
  • Step 3. Calculate p-value
  • Step 4. Compare p-value to alpha (0.05) and decide to Reject Ho or Fail to Reject Ho
• Problem Types
  • Estimation with confidence intervals
    • Mean estimation with large sample (n>30)
    • Mean estimation with small sample (n<30)
    • Proportion estimation
  • Significance Testing
    • One-tail
      • Left tail (negative Z, simple table lookup)
      • Right tail (positive Z, 1 – table lookup)
    • Two-tail
      • Left tail (negative Z, simple table lookup), multiply x2
      • Right tail (positive Z, 1 – table lookup), multiply x2

Activity:

• Practice Exam