8. Hypothesis Testing

8. Hypothesis Testing

• Based on the characteristics of the population, one needs to design the test for rejection or acceptance of H0 (Null hypothesis) and H1 (Alternative hypothesis)
• The outcome of the test will be based on the samples that are observed
• If H0 is accepted, then H1 is rejected, and vice versa
• H0 involves equality while H1 is a newly proposed hypothesis and it involves an inequality
• Critical Region
  • The decision to accept or reject H_0is based on an appropriate statistic computed from a given random sample
  • The set of values of the statistic favourable to the alternative is know as critical region or rejection region
• Testing the mean (Large Sample or variance is know ) is Z test
• Testing the mean (Small Sample or variance is unknown ) is T test

• Example - Steps taken to test the hypotheses under α=0.05:
  • Step 1: To determine the Null and Alternative Hypothesis
    • H0 = μ = 10kg vs H1 = μ != 10kg
  • Step 2: To identify the significant level
    • α=0.05
  • Step 3: To identify the Critical region
    • Critical region: z<-1.645 and z>1.645, where
    • z=(x ̅-μ0)/(σ/√n)
  • Step 4: To calculate the sample mean (x ̅) and compute Z
    • x ̅=(9.981+10.006+9.857+10.107+9.888+9.728+10.439+10.214+10.190+9.793) / 10 = 10.0203
    • n =10, and hence
    • z=(10.0203-10)/(0.2/√10) = 0.3209
  • Step 5: Decision – To accept H0 or reject H1, and vice versa
    • Accept H0 (Null Hypothesis) and conclude that the average calibration of a scale is equal to 10kg and therefore, the recalibration carried out is unnecessary.

• Hypothesis test in R-Programming (One Sample t-test):
  • H0: μ=3400, H1: μ<3400
  • Conclusion: Reject H0 at 0.05 level of significance. Therefore, population mean is less than 3400.

• Hypothesis test in R-Programming (Pair t-test):
  • H0: μ=0, H1 != 0
  • Conclusion Accept H0 at 0.05 level of significance.