Author(s): Brenda Gunderson, Ph.D., 2012 Unless otherwise ...

Author(s): Brenda Gunderson, Ph.D., 2012

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Statistics 350 Help Card

Summary Measures

Sample Mean x x1 x2 xn n

? xi

n

Sample Standard Deviation

s

? (xi x)2

n 1

? xi 2 nx 2

n 1

Normal Random Variables

x

z score

observation mean standard deviation

x P V

x Percentile: x zV P

x If X has the N (P,V ) distribution, then the variable

Z

X P V

has the

N (0,1)

distribution.

Probability Rules

x Complement rule P( AC ) 1 P( A)

x Addition rule General: P( A or B) P( A) P(B) P( A and B)

For independent events: P( A or B) P( A) P(B) P( A)P(B)

For mutually exclusive events: P( A or B) P( A) P(B) x Multiplication rule

General: P( A and B) P( A)P(B | A)

For independent events: P( A and B) P( A)P(B)

For mutually exclusive events: P( A and B) 0 x Conditional Probability

P( A and B) General: P( A | B)

P(B)

For independent events: P( A | B) P( A)

For mutually exclusive events: P( A | B) 0

Discrete Random Variables

Mean

? E(X ) P

xi pi x1 p1 x2 p2 xk pk

Standard Deviation

? s.d.(X ) V

(xi P)2 pi

?  xi2 pi P 2

Binomial Random Variables

P( X

k)

????

n k

????

p

k

(1

p) nk

where

????

n k

????

n! k!(n k)!

Mean E( X ) P X np

Standard Deviation

s.d.( X ) V X np(1 p)

Normal Approximation to the

Binomial Distribution

If X has the B(n, p) distribution and the sample size n is large enough (namely np t 10 and n(1 p) t 10 ),

 then X is approximately N np, np(1 p) .

Sample Proportions

p^ x n

Mean

Ep^  P p^ p

Standard Deviation

s.d.( p^ ) V p^

p(1 p) n

Sampling Distribution of p^

If the sample size n is large enough (namely, np t 10 and n(1 p) t 10 )

then p^ is approximately N???? p,

p(1 n

p)

????

.

Sample Means

Mean E X

P X

P

Standard Deviation

s.d.( X )

V X

V n

Sampling Distribution of X If X has the N (P,V ) distribution, then X is

 N

P X

,V

X

N???? P,

V n

???? .

If X follows any distribution with mean P and standard deviation V and n is large,

then

X

is approximately

N???? P,

V n

????

.

This last result is Central Limit Theorem .

Page 1

Population Proportion

Parameter p

Statistic

p^

Standard Error

s.e.( p^ )

p^ (1 p^ ) n

Confidence Interval p^ r z*s.e.( p^ )

Conservative Confidence Interval

p^ r z* 2n

Large-Sample z-Test

z

p^ p0

p0 (1 p0 )

n

Sample Size

n

????

z* 2m

????

2

Two Population Proportions

Parameter Statistic

p1 p2 p^1 p^ 2

Standard Error

s.e.( p^1 p^ 2 )

p^1(1 p^1) p^ 2 (1 p^ 2 )

n1

n2

Confidence Interval

p^1 p^ 2 r z *s.e. p^1 p^ 2

Large-Sample z-Test

z

p^1 p^2

p^ (1

p^ )????

1 n1

1 n2

????

where p^

n1 p^1 n2 p^2 n1 n2

Population Mean

Parameter

P

Statistic

x

Standard Error

s.e.(x) s n

Confidence Interval x r t *s.e.(x)

df = n ? 1

Paired Confidence Interval d r t *s.e.(d )

One-Sample t-Test

t x P0 x P0 s.e.(x) s n

df = n ? 1 df = n ? 1

Paired t-Test t d 0 s.e.(d )

d sd n

df = n ? 1

Two Population Means

General

Parameter

P1 P2

Statistic

x1 x2

Standard Error

Pooled

Parameter

P1 P2

Statistic

x1 x2

Standard Error

s.e.x1 x2

s12 s22 n1 n2

pooled s.e.x1 x2

sp

1 1 n1 n2

where s p

(n1

1) s12

(n2

1)

s

2 2

n1 n2 2

Confidence Interval

x1 x2 r t*s.e.(x1 x2 )

Two-Sample t-Test

t

x1 x2 0 s.e.(x1 x2 )

x1 x2 s12 s22 n1 n2

df = min(n1 1, n2 1) df = min(n1 1, n2 1)

Confidence Interval

x1 x2 r t* pooled s.e.(x1 x2 )

Pooled Two-Sample t-Test

t

x1 x2 0 pooled s.e.(x1 x2 )

x1 x2

sp

11 n1 n2

df = n1 n2 2 df = n1 n2 2

One-Way ANOVA

? SS Groups = SSG = ni (xi x) 2 groups

MS

Groups

=

MSG

=

SSG k 1

ANOVA Table

? SS Error = SSE = (ni 1) si 2 groups

?  SS Total = SSTO = xij x 2

MS

Error

=

MSE

=

s

2 p

MS Groups F

SSE N k

Source SS

DF MS

F

Groups SS Groups k ? 1 MS Groups F Error SS Error N ? k MS Error

Total

SSTO

N ? 1

values

MS Error

Confidence Interval

xi r t* s p ni

df = N ? k

Under H0, the F statistic follows an F(k ? 1, N ? k) distribution.

Page 2

Linear Regression Model

Population Version:

Mean:

PY x E(Y ) E 0 E1x

Individual: yi E 0 E1 xi H i

where H i is N (0,V )

Sample Version:

Mean:

y^ b0 b1x

Individual: yi b0 b1xi ei

Parameter Estimators

b1

S XY S XX

? x xy y ? x x2

?x xy ? x x2

b0 y b1 x

Residuals e y y^ = observed y ? predicted y

Correlation and its square

r

S XY

S XX SYY

r 2 SSTO SSE SSREG

SSTO

SSTO

? where SSTO SYY

y y2

Regression

Standard Error of the Sample Slope

s.e.(b1 )

s

S XX

s

? x x2

Confidence Interval for E1 b1 r t*s.e.(b1)

df = n ? 2

t-Test for E1 To test H 0 : E1 0 t b1 0 s.e.(b1 )

df = n ? 2

MSREG or F

MSE

df = 1, n ? 2

Confidence Interval for the Mean Response

y^ r t *s.e.(fit)

df = n ? 2

where s.e.(fit) s 1 (x x) 2 n S XX

Prediction Interval for an Individual Response

y^ r t *s.e.(pred)

df = n ? 2

where s.e.(pred) s 2 s.e.(fit)2

Standard Error of the Sample Intercept

s.e.(b0 )

s 1 x2 n SXX

Confidence Interval for E 0 b0 r t*s.e.(b0 )

df = n ? 2

Estimate of V s MSE

? ? SSE

n2

where SSE

y y^ 2

e2

t-Test for E 0 To test H 0 : E 0 0 t b0 0 s.e.(b0 )

df = n ? 2

Chi-Square Tests

Test of Independence & Test of Homogeneity Test for Goodness of Fit

Expected Count E expected row totalu column total total n

Expected Count Ei expected npi0

Test Statistic

? & 2

O E2

E

df = (r ? 1)(c ? 1)

? (observed expected) 2 expected

Test Statistic

? & 2

O E2

E

df = k ? 1

? (observed expected) 2 expected

If Y follows a F 2 df  distribution, then E(Y) = df and Var(Y) = 2(df).

Page 3

From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition, 2012. Used

with permission.

From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition, 2012. Used

with permission.

From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition, 2012. Used

with permission.

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