AP Statistics: Choosing the Correct Hypothesis Test

Data is Means Test Statistic is t

1 Sample? 2 Samples?

1 Sample t Test (t Test)

If Independent Use 2-Sample t Test

If paired Find differences,

use t-Test

Data is ProportionsTest Statistic is z

1 Sample? 1-Prop z Test

2 Samples? 2-Prop z Test

Categorical Data Test Statistic is χ2

1 Variable? χ2 GOF Test (Uses List)

Two-Way Table? χ2 Test of Association

(Uses Matrix)

AP Statistics – Hypothesis Test Statistics Name Formula Conditions or Assumptions*

One-sample z-test

xznμ

σ−

= (Normal distribution or n > 30) and σ known.

One-sample t-test

0xts n

μ−=

(Normal population or n > 30) and σ unknown df = n-1

Paired t-test

0

d

d dts n−

= (Normal population of differences or n > 30) and σ unknown df = n-1

One-proportion z-test

0

0 0(1 )p pz

p pn

−=

−

n .p > 10 and n (1 − p) > 10

Two-proportion z-test

1 2

1 2

1 1(1 )

p pz

p pn n

−=

⎛ ⎞− +⎜ ⎟

⎝ ⎠

1 2

1 2

x xpn n+

=+

n1.p1 > 5 AND n1(1 − p1) > 5 and

n2.p2 > 5 and n2(1 − p2) > 5 and

independent observations

Two-sample t-test

1 2

2 21 2

1 2

x xts sn n

−=

+

(Normal populations or n1+n2 > 30) and independent observations and σ1 and σ2 unknown

Chi-Square Test 2

2

1

( )ni i

i i

O EE

χ=

−=∑

All expected counts > 0 and no more than 20% are 5 or less df = n-1 for Goodness of Fit test df = (r-1)(c-1) for Test of Association

* Note that it is common to all tests that we require the sample to be an SRS Definition of Symbols Used

n = sample size

= sample mean

s = sample standard deviation

μ0 = population mean

σ = population standard deviation

t = t statistic

df = degrees of freedom

n1 = sample 1 size

n2 = sample 2 size

s1 = sample 1 std. deviation

s2 = sample 2 std. deviation

= sample mean of differences

d0 = population mean difference

sd = std. deviation of differences

p1 = proportion 1

p2 = proportion 2

μ1 = population 1 mean

μ2 = population 2 mean

O = observed count

E = expected count

AP Statistics – Confidence Interval Formulas Name Formula Conditions or Assumptions

One-sample t interval * sx t

n±

(Normal population or n > 30) and σ unknown df = n-1

One-proportion z interval

(1 )* p pp zn−

± n . p > 10 and n (1 − p ) > 10

Two-sample t interval

2 21 2

1 21 2

( ) * s sx x tn n

− ± + (Normal populations or n1+n2 > 40) and independent observations and σ1 and σ2 unknown

Two-proportion z interval

1 1 2 21 2

1 2

(1 ) (1 )( ) * p p p pp p zn n− −

− ± +

n1. p 1 > 5 AND n1(1 − p 1) > 5 and

n2. p 2 > 5 and n2(1 − p 2) > 5 and

independent observations

* Note that it is common to all intervals that we require the sample to be an SRS Definition of Symbols Used

n = sample size

= sample mean

s = sample standard deviation

σ = population standard deviation

t* = t-statistic critical value

z* = z-statistic critical value

df = degrees of freedom

1 = sample 1 mean

2 = sample 2 mean

n1 = sample 1 size

n2 = sample 2 size

s1 = sample 1 std. deviation

s2 = sample 2 std. deviation

p = sample proportion

p 1 = sample proportion 1

p 2 = sample proportion 2

Commonly used z* values: C-Level z* value

80% 1.282 90% 1.645 95% 1.960 98% 2.326 99% 2.576