Hypothesis

• Statistical hypothesis


  A guess or conjecture about the numerical value of some unknown population parameters
• Null hypothesis (H0)


  The hypothesis that is assumed to be true until evidence indicates otherwise
• Null hypothesis


  • H0: μ = 120
  • H0: p = 0.5
  • H0: μ ≤ 50

  
• Simple hypothesis


  A hypothesis that expresses a single value for the unknown parameter
• Composite hypothesis


  A hypothesis that expresses a range of values for the unknown parameter
• Alternative hypothesis (H1)


  • H1: μ < 120
  • H1: μ > 120
  • H1: μ ≠ 120

  
• In deciding whether to reject a hypothesis or not, two types of errors are generally committed: type I and type II errors
• Type I error


  The error of rejecting a true hypothesis
• Type II error


  The error of accepting a false hypothesis
• Alpha (α)


  The probability of committing a type I error
• Beta (β)


  The probability of committing a type II error
• Type I error


  • Maria insists she is 30 years old when she is actually 32
• Type II error


  • Planning to hunt the Philippine monkey-eating eagle, which is against the law
• Critical region


  The part of the set of all possible values of a test statistic for which H0 is rejected
• One-sided test


  A test where the alternative hypothesis specifies a direction (left-tailed or right-tailed)

  
• Two-sided test


  A test where the alternative hypothesis is non-directional
• Parameter


  A numerical characteristic of a population, as distinct from a statistic of a sample
• Central limit theorem


  If the sample is large, the normal curve can be used as a model or if the population is normally distributed

  
• Steps in hypothesis testing

  1. Identify the parameter/given

  2. Formulate the hypotheses

  3. State the significance level

  4. Select the appropriate test statistic

  5. State the critical region

  6. Computation

  7. Statistical decision

  8. Conclusion
• Formulas for testing a population mean


  • Case 1: If σ is known

  Case 2: If σ is unknown and n ≥ 30

  Case 3: If σ is unknown and n < 30
• Frozen food company testing mean length of corn

  • Given: n = 20, x̄ = 8.8 inches, σ = 1.5 inches, μ0 = 9.0 inches, α = 0.05

  Hypotheses: H0: μ = 9.0, H1: μ ≠ 9.0

  Test statistic: z = (x̄ - μ0) / (σ/√n)

  Critical region: z < -z0.025 or z > z0.025

  Conclusion: Do not reject H0, not enough evidence that mean length is not 9.0 inches
• Testing mean weight of sacks of rice


  • Given: n = 100, x̄ = 48.54 kg, s = 20 kg, α = 0.01

  Hypotheses: H0: μ = 50 kg, H1: μ < 50 kg

  Test statistic: z = (x̄ - μ0) / (s/√n)

  Critical region: z < -zα

  Conclusion: Reject H0, evidence that mean weight is less than 50 kg
• The average length of corn is not 9.0 inches
• Average weight of 100 randomly selected sacks of rice

  48.54 kilos with a standard deviation of 20 kilos
• Test the hypothesis at 0.01 level of significance that the true mean weight is less than 50 kilos


  1. Step 1. The parameter of interest is the μ where the sample comes from

  2. Step 2. Formulate the hypotheses

  3. Step 3. State the significance level

  4. Step 4. Select the appropriate test statistic

  5. Step 5. State the critical region

  6. Step 6. Computation

  7. Step 7. Statistical decision

  8. Step 8. Conclusion

  
• Assumed that the population is normally distributed
• Central limit theorem is to be used
• Hypotheses


  H0: μ = 50

  H1: μ < 50
• Test statistic


  z = (x̄ - μ0) / (s/√n)

  
• Reject H0 if z < -z0.01
• z = -0.73 is not in the critical region, so H0 is not rejected

  
• The test result does not provide sufficient evidence to indicate that the true mean weight of sack of rice is less than 50 kilos
• Average length of time for students to have their subjects controlled

  30 minutes
• New controlling procedure using modern computing machines

  Average controlling time of 22 minutes with a standard deviation of 11.9 minutes
• Test the hypothesis that the average length of time to control student's subjects is less than 30 minutes


  1. Step 1. The parameter of interest is the μ where the sample comes from

  2. Step 2. Formulate the hypotheses

  3. Step 3. State the significance level

  4. Step 4. Select the appropriate test statistic

  5. Step 5. State the critical region

  6. Step 6. Computation

  7. Step 7. Statistical decision

  8. Step 8. Conclusion
• Use level of significance of 0.05 and assume the population of controlling times to be normally distributed
• Hypotheses

  H0: μ = 30

  H1: μ < 30
• Test statistic

  t = (x̄ - μ0) / (s/√n)
• Reject H0 if t < -t0.05,11
• t = -2.326 is in the critical region, so H0 is rejected