stor 155, section 2, last time 2-way tables –sliced populations in 2 different ways –look for...

Stor 155, Section 2, Last Time• 2-way Tables

– Sliced populations in 2 different ways

– Look for independence of factors

– Chi Square Hypothesis test

• Simpson’s Paradox

– Aggregating can give opposite impression

• Inference for Regression

– Sampling Distributions – TDIST & TINV

Reading In Textbook

Approximate Reading for Today’s Material:

Pages 634-667 & Review

Approximate Reading for Next Class:

Pages 634-667 & Review

Inference for RegressionChapter 10

Recall:

• Scatterplots

• Fitting Lines to Data

Now study statistical inference associated with fit lines

E.g. When is slope statistically significant?

Recall Scatterplot

For data (x,y)

View by plot:

(1,2)

(3,1)

(-1,0)

(2,-1)

Toy Scatterplot, Separate Points

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2 -1 0 1 2 3 4

x

y

Recall Linear Regression

Idea:

Fit a line to data in a scatterplot

• To learn about “basic structure”

• To “model data”

• To provide “prediction of new values”


Given a line, , “indexed” by

Define “residuals” = “data Y” – “Y on line”

=

Now choose to make these “small”

),( 11 yx

abxy

)( abxy ii

),( 22 yx

),( 33 yx

ab&

ab&


Make Residuals > 0, by squaring

Least Squares: adjust to

Minimize the “Sum of Squared Errors”

ab&

21

)(

n

iii abxySSE

Least Squares in Excel

Computation:

1. INTERCEPT (computes y-intercept a)

2. SLOPE (computes slope b)

Revisit Class Example 14http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg14.xls

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg14.xls

Inference for Regression

Idea: do statistical inference on:

– Slope a

– Intercept b

Model:

Assume: are random, independent

and

iii ebaXY

ie

eN ,0


Viewpoint: Data generated as:

y = ax + b

Yi chosen from

Xi

Note: a and b are “parameters”


Parameters and determine the

underlying model (distribution)

Estimate with the Least Squares Estimates:

and

(Using SLOPE and INTERCEPT in Excel,

based on data)

a b

a b


Distributions of and ?

Under the above assumptions, the sampling

distributions are:

• Centerpoints are right (unbiased)

• Spreads are more complicated

a b

aaNa ,~ˆ

bbNb ,~ˆ

Inference for RegressionFormula for SD of :

• Big (small) for big (small, resp.)– Accurate data Accurate est. of slope

• Small for x’s more spread out– Data more spread More accurate

• Small for more data– More data More accuracy

a

n

ii

ea

xxaSD

1

2ˆ

e

Inference for RegressionFormula for SD of :

• Big (small) for big (small, resp.)– Accurate data Accur’te est. of intercept

• Smaller for – Centered data More accurate intercept

• Smaller for more data– More data More accuracy

b

n

ii

eb

xx

xn

bSD

1

2

21ˆ

e

0x

Inference for RegressionOne more detail:

Need to estimate using data

For this use:

• Similar to earlier sd estimate,

• Except variation is about fit line

• is similar to from before

e

2

ˆˆ1

2

n

bxays

n

iii

e

s

2n 1n


Now for Probability Distributions,

Since are estimating by

Use TDIST and TINV

With degrees of freedom =

e es

2n

Inference for RegressionConvenient Packaged Analysis in Excel:

Tools Data Analysis Regression

Illustrate application using:

Class Example 32,

Old Text Problem 10.12

Inference for RegressionClass Example 32,

Old Text Problem 10.12Utility companies estimate energy used by

their customers. Natural gas consumption depends on temperature. A study recorded average daily gas consumption y (100s of cubic feet) for each month. The explanatory variable x is the average number of heating degree days for that month. Data for October through June are:

Inference for RegressionData for October through June are:

Month X = Deg. Days Y = Gas Cons’n

Oct 15.6 5.2

Nov 26.8 6.1

Dec 37.8 8.7

Jan 36.4 8.5

Feb 35.5 8.8

Mar 18.6 4.9

Apr 15.3 4.5

May 7.9 2.5

Jun 0 1.1


Old Text Problem 10.12

Excel Analysis:http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg32.xls

Good News:

Lots of things done automatically

Bad News:

Different language,

so need careful interpretation


Inference for RegressionExcel Glossary:

Excel Stor 155

R2 r2 = Prop’n of Sum of Squares

Explained by Line

intercept Intercept b

X Variable Slope a

Coefficient Estimates & .a b


Excel Stor 155

Standard Errors

Estimates of & .

(recall from Sampling Dist’ns)

T – Stat. (Est. – mean) / SE, i.e. put

on scale of T – distribution

P-value For 2-sided test of:

a b

0:.0:0

b

aHvs

b

aH A


Excel Stor 155

Lower 95%

Upper 95%

Ends of 95% Confidence

Interval for a and b

(since chose 0.95 for Confidence level)

Predicted . Points on line at ,

i.e. .iXiY

baX i


Excel Stor 155

Residual for .

Recall: gave useful information about quality of fit

(useful to plot)

Standard Residuals:

on standardized scale

e

ii bXaY

ˆˆ

iX bXaY iiˆˆ

Inference for RegressionSome useful variations:

Class Example 33,

Text Problems 10.23 - 10.25

Excel Analysis:http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg33.xls


Inference for RegressionClass Example 33, (10.23 – 10.25)

Engineers made measurements of the Leaning Tower of Pisa over the years 1975 – 1987. “Lean” is the difference between a points position if the tower were straight, and its actual position, in tenths of a meter, in excess of 2.9 meters. The data are:


(10.23 – 10.25)

The data are:

Year Lean

75 642

76 644

77 656

78 667

79 673

80 688

81 696

82 698

83 713

84 717

85 725

86 742

87 757

Inference for RegressionClass Example 33, (10.23 – 10.25) :

(a) Plot the data, does the trend in lean over time appear to be linear?

(b) What is the equation of the least squares fit line?

(c) Give a 95% confidence interval for the average rate of change of the lean.




HW:

10.17 b,c

10.26 (using log base 10, for part c:

Est’d slope: 0.194

Est'd intercept: -379

95% CI for slope: [0.186, 0.202])

And Now for Something Completely Different

Graphical Displays:

• Important Topic in Statistics

• Has large impact

• Need to think carefully to do this

• Watch for attempts to fool you


Graphical Displays: Interesting Article:

“How to Display Data Badly”

Howard Wainer

The American Statistician, 38, 137-147.

Internet Available:

http://links.jstor.org


Main Idea:

• Point out 12 types of bad displays

• With reasons behind

• Here are some favorites…


Hiding the data in the scale


The eye perceives

areas as “size”:


Change of

Scales in

Mid-Axis

Really trust

the

Post???

Review Slippery Issues

Major Confusion:

Population Quantities

Vs.

Sample Quantities


Population Quantities:• Parameters• Will never know• But can think about

Sample Quantities:• Estimates (of parameters)• Numbers we work with• Contain info about parameters


Population Mathematical Notation:

(fixed & unknown)

Sample Mathematical Notation :

(summaries of data, have numbers)

p,,

psX ˆ,,


Sampling Distributions:

Measurement Error:

Counting / Proportions:

nNX

,~

n

pppNpnBip

)1(,,~ˆ


Confidence Intervals: Based on margin of error:

Measurement Error:

brackets 95% of time

Counting / Proportions:

brackets 95% of time

],[ mXmX

]ˆ,ˆ[ mpmp

m

p


Hypothesis Testing:

Statement of Hypotheses:

Actual Test:

P-value = P{What saw or m.c. | Bdry}

AHH ,0

Hypothesis Testing from 3/22

Other views of hypothesis testing:

View 2: Z-scores

Idea: instead of reporting p-value (to assess statistical significance)

Report the Z-score

A different way of measuring significance

Hypothesis Testing – Z scores

E.g. Fast Food Menus:

Test

Using

P-value = P{what saw or m.c.| H0 & HA bd’ry}

000,20$:0 H

000,20$: AH

10,400,2$,000,21$ nsX

Hypothesis Testing – Z scores

P-value = P{what saw or or m.c.| H0 & HA bd’ry}

rybdXP '|000,21$

000,20$|000,21$ XP

102400$

000,20$000,21$

nsX

P

317.1 ZP

stor 155, section 2, last time 2-way tables –sliced populations in 2 different ways –look for...

Documents

data slide

small slide

data y y

data x

model data

regression distributions

review slide

accuracy slide