Mr. Rogers - AP Statistics Objectives
Syllabus 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter
11 t-Test- Multi-Sample Testing Chi Test Regression Testing  
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)

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: Why is finding the slope of a straight line a big deal?

The Awesomeness of Slope

Famous slopes Mr. R has known:

    electrical conductivity
    COF
    spring constant
    Plank's constant
    density
    g
    the perfect gas law constant
    etc. etc. etc

 

Formative Assessment: 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 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 (b) & intercept (a), these are both parameters, hence, the Greek letters.
  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. See objective 3 below.
  • 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 12.1 Exercise 1, 3, 7, 15 pp. 759 to 762

 

 

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) Note, b is the estimate of slope from performing regression analysis. β is a parameter and is the actual slope.

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 - 0 ) / SEb     or   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 (see drawing at right).

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. It does not
necessarily mean that the intercept is meaningless.

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 -- Read section 12.2 Exercise 33, 35,37 pp. 786 to 788

 

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 give from averaging numerous y-values obtained at a specific value of x. (n-2 degrees of freedom)

y-hat t*SEμ-hat

SEμ-hat = s [ 1/n + ( x* - x-bar )^2 / (S(x - xbar)2) ]^0.5

 

  1. Generate a prediction interval for a y-hat from a single y value obtained at a specific value of x. (n-2 degrees of freedom)

y-hat t*SEy-hat

SEy-hat = s [ 1 + 1/n + ( x* - x-bar )^2 / (S(x - xbar)2) ]^0.5

 

 

 
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

Homefun --  Chapter 12 practice test (all parts) pp. 796 to 798

Cumulative AP Practice Test pp. 799 to 806

 

 

  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 an 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
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