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 H_{O}  Typical H_{A} 
1  NA  Continuous  >=30

Mean, standard error  ztest  · 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=μ_{0}  Onesided:
μ> μ_{0} or μ< μ_{0}
Twosided: μ≠ μ_{0} 
1  NA  Continuous  <30  Mean, standard error  ttest (student’s ttest)  · 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=μ_{0}  Onesided:
μ> μ_{0} or μ< μ_{0}
Twosided: μ≠ μ_{0} 
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 H_{O}  Typical H_{A}  
1  NA  Continuous  >=30

Mean, standard error  ztest  · 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=μ_{0}  Onesided:
μ> μ_{0} or μ< μ_{0}
Twosided: μ≠ μ_{0} 

1  NA  Continuous  <30  Mean, standard error  ttest (student’s ttest)  · 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=μ_{0}  Onesided:
μ> μ_{0} or μ< μ_{0}
Twosided: μ≠ μ_{0} 

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=μ_{0}  Onesided:
μ> μ_{0} or μ< μ_{0}
Twosided: μ≠ μ_{0} 

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=u1u2  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 


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=p1p2 =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=p1p2 =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=p1p2 =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: p1p2=0, H1: p1p2 !=0 