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Hypothesis testing Cheat Sheet (DRAFT) by

pvalue

This is a draft cheat sheet. It is a work in progress and is not finished yet.

ONE-WAY ANOVA

Step 1. 1. SPECIFY TWO COMPLE­MENTARY HYPOTHESES INVOLVING POPULA­TIO­N-P­ARA­METERS, NOT SAMPLE­-ST­ATI­STICS.
H0: ‘Null Hypothesis or ‘Statu­s-Quo Hypoth­esis’
H1: ‘Alter­native Hypoth­esis’ or ‘Resea­rcher’s Hypoth­esis’, expresses the altern­ative to the Status­-Quo.
Type 1 Error ⟺ Incorr­ectly, deciding in favor of H1.
Type 2 Error ⟺ Incorr­ectly, deciding in favor of H0.

Step 1 practical

H_0 says that the population mean cash register receipt is the same at both stores / μ_1= μ_2
H_1 says that the population mean cash register receipt is different at the two stores / μ_1≠ μ_2

Step 2

CHOOSE α = THE SIGNIF­ICANCE LEVEL OF THE HYPOTHESIS TEST:
α = the maximum allowable probab­ility of a Type 1 Error = the maximum allowable probab­ility of rejecting H0 when H0 is true.

Step 2 PRACTICAL

α = .05. The upper bound on the probab­ility of a Type I Error is set at 5%. Type I Error involves deciding incorr­ectly that the popula­tio­n-mean cash register receipt is different at the two stores.

Step 3

STATE THE TEST-S­TAT­ISTIC AND ITS PROBAB­ILI­TY-­DIS­TRI­BUTION:
Specify the Model Assump­tions that guaranty the validity of (3),
Specify the Test-S­tat­istic
Specify the Probab­ili­ty-­Dis­tri­bution

Step 3 PRACTICAL

If H0 is true and the Model Assump­tions hold:
1. Sampling is Indepe­ndent and Random
2. Sampling is from Normal Popula­tions
3. The Popula­tions have Equal Variances
MSA/MSW ~ F (1,8)
 

Step 4

COMPUT­ATIONS: Complete the sample­-based comput­ations, including the p-value. Summarize the results in an ANOVA Table
p-value = Probab­ility of observing evidence more favorable to H1 than that observed in the actual sample = Probab­ility of H0 being true.

Step 4 practical

SAMPLE TOTALS → SAMPLE MEANS → GRAND-MEAN = THE MEAN OF ALL OBSERV­ATIONS IN ALL SAMPLES → x ̿
SSA (‘Sum of Squares Among’ Sample Means) measures the variation that exists among (i.e. between) samples
SSA = n1 (x ̅1- x ̿ )2 + n2 (x ̅2- x ̿ )2 = 5(80- 90)^2 = Number of observ­atios (each mean - grand mean) squared + the other sample
SSW (‘Sum of Squares Within’ Samples) measures the variation that exists within all the samples.
SSW= (88 - 80)2 + (73 - 80)2 + (77 - 80)2 = (Each observ­ation - the mean) squared + the same for the other sample
The “degrees of Freedom” associated with SSA is: DFA = C – 1 / C=# of samples
The “degrees of Freedom” associated with SSW is: DFW = n – C / n=Total observ­ations
MSA = SSA/(c-1) MSA is a measure of the average amount of separation between sample­-means
MSW = SSW/(n - c) / where n = n1 + n2
THE F-STAT­ISTIC: F = MSA/MSW
Larger values of MSA/MSW indicate greater variation among sample­-means than between the observ­ations within each sample.
MSA/MSW ~ F (c-1, n-c)

Step 5

5. CONCLU­SION: REPORT THE CONCLUSION IN BOTH:
Reject H0 in favor of H1 ⟺ p ≤ α --- or ---
Fail to reject H0 ⟺ p > α

Step 5 PRACTICAL

Since the p-value = . 010619 ≤ . 05 = α we reject H0 in favor of H1.
In practical terms this says - at the 5% signif­icance level, the evidence is sufficient to conclude that the ‘popul­ati­on-mean cash register receipt’ is different at the two stores