Statistical Inference Decision Tree

Statistical Inference Guideline Sample and Population Data

Inference for Means

Inference for Regression

Pages 2 & 3

Page 4

Inference for Proportions

Inference for Distributions

Page 5

Page 6

Known s

Slope, b

1 Proportion

Goodness of Fit

=> z

=> t

=> z

=> C2

s Unknown

Mean response, my

2 Proportions

=> t

=> t

=> z OR => C2 Independence

Independence/ Homogeneity

1 Observation, ŷ

>2 Proportions

=> t

=> C2 Independence

Quantitative Data AP Statistics – Godwin High School – Mr. Phillips

=> C2

Qualitative Data

Statistical Inference Guideline Inference for Means z - procedures

1 Sample/1 Mean

2 Samples/2 Means

Conditions: 1. s is known 2. SRS of population 3. Population is: ≈Normal OR n is large (≥30)

Confidence Interval:

 x  z*  s  n  TI-84+: STAT->TESTS->ZInterval

Conditions: 1. s1 & s2 are known 2. Independent SRS’s 3. Populations are: ≈Normal OR n1, n2 are large (≥30)

Test Statistic:

z



x  m0

s n

Confidence Interval:



TI-84+: STAT->TESTS->Z-Test

 x1  x2   z*

(s1 )2 (s 2 )2  n1 n2

TI-84+: STAT->TESTS->2-SampZInt

Test Statistic:

( x1  x2 )  ( m1  m2 ) z   (s 1 ) 2 (s 2 ) 2      n1 n2    TI-84+: STAT->TESTS->2-SampZTest

2 AP Statistics – Godwin High School – Mr. Phillips

Statistical Inference Guideline Inference for Means t - procedures

1 Sample/1 Mean

Matched Pairs

2 Samples/2 Means

df = n-1 (n is # observations)

df = n-1 (n is # pairs)

df = calculated (n1, n2 are # observations)

Conditions: 1. SRS of population 2. Population is ≈Normal OR large: n≥40 OR n≥15 – small skew/outliers OR n<15 – no skew/outliers

Confidence Interval:

 x  t *  s  n  TI-84+: STAT->TESTS->TInterval

Conditions: 1. Paired data from SRS’s 2. Distribution of pairs is: ≈Normal OR large: n≥40 OR n≥15 – small skew/outliers OR n<15 – no skew/outliers

Test Statistic:

t

x  m0

  s

n

TI-84+: STAT->TESTS->T-Test

Conditions: 1. Independent SRS’s 2. Populations are ≈Normal OR large: n1, n2≥ 40 OR n1,n2≥15 -small skew/outliers OR n1,n2<15 - no skew/outliers

Confidence Interval:

 x1  x2   t *

( s1 )2 ( s2 ) 2  n1 n2

TI-84+: STAT->TESTS->2-SampTInt

Test Statistic:

( x1  x2 )  ( m1  m2 ) t   ( s1 ) 2 ( s2 ) 2      n1 n2    TI-84+: STAT->TESTS->2-SampTTest

3 AP Statistics – Godwin High School – Mr. Phillips

Statistical Inference Guideline Inference for Regression t - procedures

Conditions: 1. y responses are independent 2. Variation of y responses ≈N 3. The true x,y relationship is linear 4. s of y is constant for all x, and is estimated by s

5.

s

1  ( y  yˆ )2 n2

Slope

Mean Response

Single Observation

b is unbiased estimator of b a is unbiased estimator of a

Estimate my when x = x* x* is a fixed, unknown value

Estimate y when x = x* Response, y, will vary for repeated observations

Confidence Interval:

Prediction Interval:

Confidence Interval:

b  t*

Notes: 1. Residual plot shows no pattern 2. Histogram of residuals has little skew/outliers 3. my=a+bx is true regression line where my is mean response and a, b are unknown parameters 4. df = n-2 where n is # observations

s  ( x  x )2

TI-84+: STAT->TESTS->LinRegTInt

Test Statistic:

t

bb s ( x   x )2

TI-84+: STAT->TESTS->LinRegTTest

 1 ( x *  x )2 yˆ  t *  s   n  ( x  x )2 

   



yˆ  t *  s 1 

 

1 ( x *  x )2  n  ( x  x )2

NOTE: less accurate than CI for my

4 AP Statistics – Godwin High School – Mr. Phillips

   

Statistical Inference Guideline Inference for Proportions z - procedures

2 Samples/2 Proportions

1 Sample/1 Proportion

Pooled ST: p 

Conditions: 1. SRS of population 2. Population ≥10n 3. CI: npˆ  10 & nqˆ  10

ST:

Test Statistic:

Confidence Interval:

z

ˆˆ pq n

TI-84+: STAT->TESTS->1-PropZInt

>2 Samples/>2 Proportions

Conditions: 1. Independent SRS’s 2. Pop1 ≥10n1; Pop2 ≥10n2 3. CI: n1 pˆ1; n1qˆ1  5 & n2 pˆ 2 ; n2qˆ2  5

ST: np0≥10 & nq0≥10

pˆ  z *

(n1 pˆ1  n2 pˆ 2 ) (n1  n2 )

n TI-84+: STAT->TESTS->1-PropZTest

Homogeneity

n1 p; n1q  5 & n2 p; n2 q  5

Test Statistic:

Confidence Interval:

pˆ  p0 p0 q0

C2 Test of

 pˆ1  pˆ 2   z*

( pˆ1qˆ1 ) n1



( pˆ 2 qˆ2 ) n2

TI-84+: STAT->TESTS->2-PropZInt

z 

ˆ1  p ˆ 2 )  ( p1  p2 ) (p  1 1  pq    n n  1 2 

TI-84+: STAT->TESTS->2-PropZTest

5 AP Statistics – Godwin High School – Mr. Phillips

Statistical Inference Guideline Inference for Distributions

C 2- procedures

Tests of Independence/Homogeneity

Goodness of Fit Test

I: 1SRS/2 Variables with r, c Categories H: ≥2 SRS’s/1 Variable with k Categories

1 SRS/1 Variable with k Categories df = k-1

Conditions: 1. SRS of population 2. All Expected counts ≥1 3. ≥80% Expected counts ≥5

df = (#rows-1)· (#columns-1)

Conditions: 1. I: SRS of population H: Independent SRS’s 2. All Expected counts ≥1 3. ≥80% Expected counts ≥5 ♦. For 2x2: All Expected counts ≥ 5

Ek  pk  n p k  hypothesized % n  total # observations

Eij 

ri  c j n

th

ri  i row; total th

c j  j column ; total n  total # observations

Test Statistic:

Test Statistic:

(O  E )2 C  E

(O  E )2 C  E

TI-84+: STAT->*TESTS->X2GOF-Test

TI-84+: STAT->TESTS->X2-Test

2

2

6 AP Statistics – Godwin High School – Mr. Phillips