Mr. Rogers - AP Statistics Objectives
Syllabus 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter
11 t-Test- 12 Inf for Prop 13 Chi Test 14 Regression  
Unit Plan Practice Test
Latin

Latin/Greek Root Words

arch--------->ancient, example: archtype;         chrono------>time, example: chronology;             -dom----------->quantity/state, example: freedom               fer-------->carry, example: transfer;               gen--------->birth, example: generate;                 luc-------->light, example lucid;                 neo--------->new, example: neonatologist;                olig--------->few, example: oligarchy;              omni--------->all, omniscient;            sym--------->together, symbol;

(Statistics connection)

Chapter 14: Regression Significance Testing

AP Statistics Standards

V. Statistical Inference: Estimating population parameters and testing hypotheses (continued)

B. Tests of significance

7. Test for the slope of a least-squares regression line

Objectives

Essential Question: How can you express the uncertainty in the slope of a line?

Ch. 14.1 Inference for Regression

Relevance: Regression analysis (generally multiple regression analysis) is a very common form of data analysis found in technical journals--often the primary source of information for research papers. It is not possible to read and understand them without an understanding of inference for regression..

  1. State the 2 inferences drawn when using regression data. (slope & intercept)
  2. State the assumptions for regression inference.
  • for any x value, y-data is normally distributed (remember, Xs are perfect Ys are not.)
  • for any x value, the y-data's standard deviation is the same
  • the means of the y-data distribution at any value of x form a straight line relationship: my = a + bx
  1. Calculate the standard error of the least squares regression line.

s = [ S(y - ŷ)2 / (n-2) ]1/2

  1. Calculate the standard error of the slope.

SEb = s / [ (S(x - xbar)2 ]1/2

 

Homefun -- Read section 14.1, prob. 14.5, 14.7, 14.9

 

 

Activities

Lesson 1
Key Concept: How to evaluate the slope obtained in a regression analysis

Interactive Discussion: Why is the slope of the line is a big deal. It often has physical meaning

Famous slopes Mr. R has known:

  • electrical conductivity

  • COF

  • spring constant

  • plank's constant

  • density

  • g

  • the perfect gas law constant

  • etc. etc. etc

2-person teams: Using data from an Einstein's photoelectric experiment find plank's constant using regression analysis in Minitab. Calculate a confidence interval on the slope of the line (plank's constant). Compare this technique to IB error analysis techniques.

 

 

Essential Question: How can you spot a meaningless regression anaysis?

Evaluating Regression Results for the Slope

  1. Generate a confidence interval for the slope. (n-2 degrees of freedom)

b ± t*SEb

  1. Calculate the t-value for a hypothesis test of the slope Ho: b = 0. (n-2 degrees of freedom)

t = b / SEb

  1. Perform a significance test for the slope of a least squares regression line.

Ho: β = 0

Ha: β ≠ 0, β < 0, β > 0

Note: most computer programs return a p-value for Ha: β ≠ 0. For β < 0 or β > 0, simply cut the p-value in half

  1. Perform a significance test for the intercept of a least squares regression line.
Ho: a = 0     Note: often a does equal 0. A high
                    p-value indicates that the intercept
                    is essentially 0.

Ha: a ≠ 0, a < 0, a > 0

Note: most computer programs return a p-value for Ha: a ≠ 0. For a < 0 or a > 0, simply cut the p-value in half

 
  1. Correctly perform least squares regression using Minitab.
  2. Correctly interpret least squares regression computer output (such as from Minitab).

Homefun --  prob. 14.11, 14.17

 

Lesson 2
Key Concept: Spotting a meaningless regression analysis

Interactive Discussion: Objectives. While it's possible to spot a meaningless regression analysis, it's not possible to tell for certain if an analysis is meaningful.

2-person teams: (see the stats investigation below)

 

 

Essential Question: How can you account for variability in a predicted data point from a regression analysis?

Evaluating Regression Results for a Given y-hat

  1. Generate a confidence interval for the average y-hat. (n-2 degrees of freedom, see page 797)

y-hat ± t*SEμ-hat

  1. Generate a prediction interval for a y-hat at a specific value of x. (n-2 degrees of freedom, see page 797)

y-hat ± t*SEy-hat

 

 
 
Stats Investigation: How to Spot a Meaningless Regression Analysis - time approx 2 class periods (individual work)

Purpose: Determine if a regression analysis using random numbers that has a high r-square value can be detected with hypothesis tests on the slope and intercept.

Instructions: Remember the stats investigation you did earlier in which you determined that even random data can produce a high r-square value. Redo the regression/correlation analysis in Minitab on the 4 sets of data you saved  and interpret the results. Be sure to take all the recommended steps for producing a statistically significant regression analysis.

Questions /Conclusions:

  1. Based on your data, could you spot randomness with the hypothesis tests on the slope and intercept of the regression equation. Explain
  2. Outline all the steps which should be taken to produce a regression/correlation analysis with the best chance of being meaningful.
  3. Can a thorough statistical analysis of bivariate data, by itself fully establish that a regression result is meaningful? Explain
   
Essential Question: How can we evaluate or include possible lurking variable in a regression anaysis?

Ch. 14

  1. Perform multiple linear regression analysis using Minitab and correctly interpret the regression equation, R2, and the hypothesis tests.

  2. State how you would plot and interpret residuals for a multiple regression analysis. Remember, a residual = (yi - yhat).  Although there are several x-variables each with its own value, there is only one y-value and only one yhat. Hence, the residual is normally plotted against the y-value.

Classfun: Chapter 8, AMSCO book Review Exercise 1-6 p. 254, 1-7 p. 257, 1-6 259, 1-5 262

 

Lesson 3
Key Concept: Control of variables in a study

Interactive Discussion:

Review  the following from chap 5:

  1. Be as one with the three basic principles of experimental design.
  • Control - effects of lurking variables
  • Randomization - prevents sampling bias
  • Replication - collect numerous data points

2-person teams: Perform a multiple linear regression analysis on the SAT data used earlier in the year. Compare the results with those obtained earlier using 2 variable regression analysis.

 

  1. Plot H vs. W and the residual plot.
  2. Report the regression equation
  1. Report a 95% confidence interval for the slope. Also make a n appropriate drawing.
  2. Perform a hypothesis test on the slope
  1. Record a 90% prediction interval for a height calculated from your regression equation using a width of 60 inches.
  2. Write a conclusion
Mr

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