Inference Formulas & Conditions

Inference Formulas & Conditions Test

Conditions

One-Sample z-test (for a population mean)

1.)  known 2.) SRS 3.) population is normal, or large sample size (n ≥ 30) 4.) Independence 1.)  1 and  2 are known 2.) SRS 3.) population is normal, or large sample size ( n1 , n2 ≥ 30) 4.) Samples are Independent 1.) SRS 2.) population is normal, or large sample size (n ≥ 30) 3.) Independence 1.) SRS 2.) population is normal, or large sample size ( n1 , n2 ≥ 30) 3.) Samples are independent

Two-Sample z-test (for a difference in 2 population means)

One-Sample t-test (for a population mean) Two-Sample t-test (for a difference in 2 population means)

Matched Pairs t-test

1.) samples are not independent (they are matched) 2.) SRS 3.) differences are normal or large sample size (n ≥ 30)

Null Hypothesis

Ho: µ = µo

Ho: µ1 = µ2

Test Statistic

Z

normalcdf (lower limit, upper limit)

x  o



Confidence Interval

x  z*

 n

n

z

x1  x2

 12 n1



 22

x  o s Ho: µ = µo n with n – 1 df x –x H 0 : 1  2 t  12 2 2 s1 s2 or  H 0 : 1  2  0 n1 n2 With smaller of n1  1 or n2  1 df x  o s n with n – 1 df t

normalcdf (lower limit, upper limit)

x1  x2  z

*

 12 n1



 22 n2

n2

t

H 0 : d  0

p-value

tcdf((lower limit, upper limit, degrees of freedom tcdf((lower limit, upper limit, degrees of freedom

tcdf((lower limit, upper limit, degrees of freedom

* x  tn1

s n

(x1 – x 2 )  t *

* x  tn1

s n

s12 s22  n1 n2

Test One Proportion ztest

Conditions 1.) SRS 2. ) Independence: pop > 10 times sample 3. ) Normality np  10 n(1  p)  10

Null Hypothesis

H 0 : p  p0

Test Statistic pˆ  p0 z p0 (1  p0 ) n

p-value Confidence Interval normalcdf (lower pˆ (1  pˆ ) * limit, upper pˆ  z n limit) Note: the condition for CI is

npˆ  10 and n(1  pˆ )  10

Two Proportion ztest

Chi Square Goodness of Fit Test Chi Square test for independence t-test for the slope of a linear regression equation

1.) SRS 2. ) Independence: pop > 10 times sample 3. ) Normality n1 pˆ c  5 n1 (1  pˆ c )  5 n2 pˆ c  5 n2 (1  pˆ c )  5 1.) SRS or data is from an entire population. 2. ) all expected counts are ≥5 1.) all expected counts ≥5

1.) y responses are independent. 2.) y responses are normal. 3.) standard deviation about the regression line remains constant. 4.) variables are linearly related

H 0 : p1  p2 or H 0 : p1  p2  0

pˆ1  pˆ 2

z

1 1 pˆ c (1  pˆ c )    n1 n2  x x Note: pˆ c  1 2 n1  n2

normalcdf (lower limit, upper limit)

H 0 : pa  p1, pb  p2 ,...

(obsv.  exp .) 2   exp . df=# of categories - 1

χ²cdf(χ² statistic, 100000,df)

H 0 : there is no association between the variables (independent)

(obsv.  exp .) 2   exp . df= (# rows – 1)(# columns – 1)

χ²cdf(χ² statistic, 100000,df)

2

2

t H0 :   0

b SEb

Degrees of freedom = n2

 pˆ1  pˆ 2  z*

tcdf((lower limit, upper limit, degrees of freedom

pˆ1 (1  pˆ1 ) pˆ 2 (1  pˆ 2 )  n1 n2

b  t *SE b`

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