AP Statistics: Unit 3 by Valeria

10/18/16-Least Squares Regression & Residual Intro:

valeriaalv0582 — 2016-10-18 12:25:56 UTC

Learn to interpret the slope and y-intercept of a least squares regression line
Learn to use the least-squares regression line to predict y for a given x
Explain the danger of extrapolation
Calculate and interpret residuals

Characteristics to interpret Scatter plots:

valeriaalv0582 — 2016-10-18 12:29:50 UTC

Strength- weak, moderate, strong (all of these are relative)
Form- linear or not
Outliers- the points that fall outside of the overall pattern
Direction- positive or negative

What is the correlation r?

valeriaalv0582 — 2016-10-18 15:24:05 UTC

Addresses direction & strength of a linear association
Only for 2 quantitative variables

Characteristics of the correlation:

valeriaalv0582 — 2016-10-18 15:26:43 UTC

-1< r < 1
r < 0 implies a negative association
r > 0 implies a positive association
r close to 0 implies weak association
r close to 1 implies strong association

10/17/16- Correlation:

valeriaalv0582 — 2016-10-18 15:31:52 UTC

Learn to interpret the correlation
Learn to understand the basic properties of correlation
Learn how the correlation is influenced by outliers

Variables:

valeriaalv0582 — 2016-10-18 15:40:06 UTC

Response Variable:

Measures an outcome of a study

Explanatory Variable:

May help explain/predict changes in a response variable

Scatterplots:

valeriaalv0582 — 2016-10-18 15:43:17 UTC

Show relationship between 2 quantitative variables measured on the same individuals.

Types of Correlations:

valeriaalv0582 — 2016-10-18 15:51:59 UTC

Example Problem:

valeriaalv0582 — 2016-10-20 12:07:11 UTC

Describe what the scatterplot reveals about the relationship between the sprint time & long jump distances for the students.

There is a moderate negative linear association between the sprint time & long jump distance

Residuals:

valeriaalv0582 — 2016-10-24 02:20:29 UTC

Definition: The difference between an observed value of the response variable & the value predicted by the regression line
When the plots are randomly scattered, form a somewhat linear line, then it is a linear model.

10/24/16: Interpreting Computer Output, Regression to the Mean

valeriaalv0582 — 2016-10-24 12:24:55 UTC

Determine the equation of a least-square regression line using computer output
Describe the slope, y-intercept, standard deviation of the residuals and r^2 are influenced by outliers
Find the slope and y-intercept of the least squares regression line from the means and standard deviation s or x and y and their correlation

Extrapolation:

valeriaalv0582 — 2016-10-24 13:09:08 UTC

The use of a regression line for prediction far outside the interval of values of the explanatory variable x used to obtain the line.
It isn't a good idea to extrapolate because few relationships are linear for all values of the explanatory variable.

Residual Equation:

valeriaalv0582 — 2016-10-24 16:20:49 UTC

Residual= y - y hat
y: the actual value
y-hat: the predicted value

Least-Squares Regression Line:

valeriaalv0582 — 2016-10-24 16:22:11 UTC

Definition: y on x is the line that makes the sum of the squared residuals as small as possible
The general form of a regression line is y hat= a + bx
- Y-hat is the predicted value of the response variable y for a given value of the explanatory value x
- B is the slope, the amount by which y is predicted to change when x increases by one unit
- A is the y-intercept, the predicted value of y when x=0

10/25/16: Putting it all together; Regression & Correlation

valeriaalv0582 — 2016-10-25 12:26:07 UTC

Explain how outliers affect the correlation, least squares regression line & standard deviation of residuals

X & Y Variables

valeriaalv0582 — 2016-10-25 12:40:34 UTC

Both variables matter for regression only

How to Interpret the Least Squares Regression Line:

valeriaalv0582 — 2016-10-26 01:06:01 UTC

Examples:

For each additional inch in height, we predict change in weight will be an average, 9.45 pounds.
For each additional row that goes back, the score is predicted to decrease by 1.1171 points.

10/20/16: LSRL on TI & Residual Plots

valeriaalv0582 — 2016-10-26 01:14:42 UTC

Explains the concept of least squares
Determine the equation of a least-squares regression line using technology
Construct & interpret residual plots to assess if a linear model is appropriate

How to Interpret the slope & y-intercept of the regression line:

valeriaalv0582 — 2016-10-26 01:58:16 UTC

Examples:

The slope, b= 2.63 ml/s, is the predicted amount of additional soda we expect to remain in the can for each additional second of tapping.
The y-intercept, a= 248.6 ml, is the amount of soda predicted to remain in the can if it's not tapped at all.

Regression Equation Used in Examples: soda=248.6mL +(2.63 mL/s) Tapping time

Correlation r:

valeriaalv0582 — 2016-10-26 02:50:24 UTC

r can be both positive & negative
- positive slope= r
- negative slope= -r

Relationship between r & r^2 & s:

valeriaalv0582 — 2016-10-26 02:55:29 UTC

r^2 & s both measure how well the least-squares regression line models the data (how much scatter there is from the least-squares regression line)
s is measured in the units of the response variable, r^2 is on a standard scale
neither address form
To find r from r^2 you must take into account the direction of the association

Important Equations:

valeriaalv0582 — 2016-10-26 03:18:10 UTC

1. b=r(Sy/Sx)
2. a=mean-of-y - b(mean-of-x)
3. y hat= a + bx
When finding the answer from the first equation (b) substitute b from the second equation to the answer from the first one.
Substitute both answers from each equation and put them in the third equation.
r= Correlation Coefficient
r^2= Coefficient of determination
s= Standard deviation of residual
Mean-of-x & Mean-of-y are on the line.

Over & Under Prediction:

valeriaalv0582 — 2016-10-26 03:25:47 UTC

Over prediction is when the residual is negative
Under prediction is when the residual is positive for that x value
Example:
- The model overpredicts the fat content by about 7.9g
Question from the example:
- Calculate & interpret the residual for the Big Mac, with 45g of carbs & 29g of fat
- 2nd>Calc>x=45>36.877709; 29-36.877709=-7.8777

How to Interpret r^2 and s

valeriaalv0582 — 2016-10-26 03:57:47 UTC

Examples:

r^2=.047; There is 4.7% in variation in test scores that is accounted for by the linear model relating test score to row assignment.
s=10.0673; On average, the predicted score will vary from the actual score by 10.0673 points.
s=13.8; On average, the predicted weight will vary from the actual weight by 13.8 pounds

Review Questions Answers:

valeriaalv0582 — 2016-10-27 12:39:26 UTC

A sociologist studying the relationship between early childhood nutrition & academic achievement in middle school among children in a certain city finds that the correlation between these 2 variables is .86. What conclusions can he draw from the study?

Children in this city who have a healthy diet in early childhood tend to do better in middle school.
Reasoning: It is only about the children in the specific city.

Which of the following quantities is minimized by the least squares regression line?

The sum of the squared differences between observed values of the response variable & values of the response variable predicted by the model.
Reasoning: The least squares line is based on distances between observed values & the regression line, but not the distances described in this choice.

Which of the following statements about the slope of the least-squares regression line is true?

It has the same sign as the correlation coefficient r.

One of the following is a correct statement involving correlation. Which one is correct?

The correlation between amount of fertilizer & yield of tomatoes was found to be r=.33.
Reasoning:Think about the response & explanatory variables within the statement.

Which of the following is an equation of least squares regression line for these data?
Option 1 [height=117.99+1.878(foot length)]
Y intercept (between Coef & Constant); Slope (Coef & X Variable)

Outliers:

valeriaalv0582 — 2016-10-28 02:26:32 UTC

If an outlier is in the same direction as the rest of the scatterplot, it will strengthen the relationship
If an outlier is not in the same direction as the rest of the scatterplot, it will weaken the relationship
If an outlier is going to the side of the rest of the scatter plot, it will make it smaller (possibly positive or negative depending on the graph)

Residual Plot

valeriaalv0582 — 2016-10-28 12:09:16 UTC

Definition: A scatterplot of the residuals against the explanatory variable.
Helps assess whether a linear model is appropriate.