# Conditions for Inference

 # of Samples Sample – Independent or Dependent (matched / paired) Outcome Variable Type Sample Size (n), if relevant Measure(s) of location and spread of the sample Appropriate Statistical Test and/or procedure Conditions for Inference Typical HO Typical HA 1 NA Continuous >=30 Mean, standard error z-test ·         Variable must be quantitative ·         Measurements must be valid ·         Data must be derived by an SRS or similar method ·         The standard deviation of the population must be known prior to data collection ·         The sampling distribution of the sample mean is approximately normal μ=μ0 One-sided: μ> μ0 or μ< μ0   Two-sided: μ≠ μ0 1 NA Continuous <30 Mean, standard error t-test (student’s t-test) ·         Variable must be quantitative ·         Measurements must be valid ·         Data must be derived by an SRS or similar method ·         The sampling distribution of the sample mean is approximately normal μ=μ0 One-sided: μ> μ0 or μ< μ0   Two-sided: μ≠ μ0 2 Independent Continuous 2 Dependent Continuous 3+ Independent Continuous 1 NA Dichotomous Large 1 NA Dichotomous Small 2 Independent Dichotomous Large 2 Independent Dichotomous Small 2 Dependent Dichotomous 3+ Independent Dichotomous 1 NA Categorical or Ordinal (NOT dichotomous) 2+ Independent Categorical or Ordinal (NOT dichotomous)

Solution

# of Samples Sample – Independent or Dependent (matched / paired) Outcome Variable Type Sample Size (n), if relevant Measure(s) of location and spread of the sample Appropriate Statistical Test and/or procedure Conditions for Inference Typical HO Typical HA
1 NA Continuous >=30

Mean, standard error z-test ·         Variable must be quantitative

·         Measurements must be valid

·         Data must be derived by an SRS or similar method

·         The standard deviation of the population must be known prior to data collection

·         The sampling distribution of the sample mean is approximately normal

μ=μ0 One-sided:

μ> μ0 or μ< μ0

Two-sided:

μ≠ μ0

1 NA Continuous <30 Mean, standard error t-test (student’s t-test) ·         Variable must be quantitative

·         Measurements must be valid

·         Data must be derived by an SRS or similar method

·         The sampling distribution of the sample mean is approximately normal

μ=μ0 One-sided:

μ> μ0 or μ< μ0

Two-sided:

μ≠ μ0

2 Independent Continuous >30 Mean and standard error T test Variable must be quantitative

Measurements must be valid

Sample mean must follow the t distribution

Sample variance must be known

μ=μ0 One-sided:

μ> μ0 or μ< μ0

Two-sided:

μ≠ μ0

2 Dependent Continuous Not relevant Pooled variance, mean T test Measurement must be valid, individual group size must be known , pooled variance must be known and the mean follows the normal distribution d=u1-u2 Two tail test

H0: u1=u2,

H1: u1=! u2

3+ Independent Continuous <30 Mean , standard error Anova test Measurement must be valid , treatment means of group means must be known, error term may follow any ditribution U1=U2=…Un
 Two tail test H0: u1=un=u2, H1: u1=! u2=!un
1 NA Dichotomous Large Mode, median Logistic model True values must be known, odds ratio follows the logistic distribution , threshold for prediction p=!0 , where p is the proportion of values which are 1 One tail test , p>k , H1: p<k
1 NA Dichotomous Small Mode mode Simple threshold to be known for prediction, no particular distribution valid p=!0 , where p is the proportion of values which are 1 One tail test , p>k , H1: p<k
2 Independent Dichotomous Large Mode , Median , Binomial approximation , ratio proportion test Sample must have comparable number of observations, proportion can be approximated using the normal distribution P1=p2 Two tail test where  d=p1-p2 =0 , H1: d =!0
2 Independent Dichotomous Small Mode ratio proportion test There must be no outliers because the sample is small already, difference of the proportion must follow the normal distribution , or we can say the variance of the two proportion’s ratio follow the F distribution P1=p2 Two tail test where  d=p1-p2 =0 , H1: d =!0
2 Dependent Dichotomous Large Histogram, Median T test Measurement must be valid, individual group size must be known , pooled variance must be known and the mean follows the normal distribution P1=p2 Two tail test where  d=p1-p2 =0 , H1: d =!0
3+ Independent Dichotomous NA Mean Anova test Measurement must be valid , treatment means of group means must be known, error term may follow any ditribution P1=p2=p3…=pn Two tail test

H0: u1=un=u2,

H1: u1=! u2=!un

1 NA Categorical or Ordinal (NOT dichotomous) Large Mode One sample t test Measurement must be valid, individual group size must be known , variance or sample variance must be known P1=0 H0: p=1,

H1: p=!0

2+ Independent Categorical or Ordinal (NOT dichotomous) Small/ Large Median Paired sample t test Measurement must be valid, individual group size must be known , pooled variance must be known and the mean follows the normal distribution P1=p2 H0: p1-p2=0, H1: p1-p2 !=0