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Some variations in the Z-test – II

  • What if the sample had not been large enough?? For example, if Christos had met only 25 students, then what?
    • Conduct t-Test when sample size is small
    • Let the sample size, n = 25, X = $20,000, s = $8,000
    • From the t-table tc = 1.71 for = 0.05 and d.f. = 24
    • Decision rule: “Reject H0 if t > 1.7l.”
  • Points to observe:
    • You could not launch the course.. Why?
    • Hint: Is it because of T-Test?
  • NO!
    • Its because the sample size is small =>
    • Less value of n =>
    • Higher standard error =>
    • Lower confidence in rejecting the hypothesis =>
    • Almost akin to not taking a decision (hence not launching the product)

T- Test to be conducted, when sample size (n) is small (Typically<30) Degrees of freedom = (n-1)


  • Christos has surveyed the market and decided to launch the course. He has two markets in mind, where he can launch the course (and hence conducts the survey):
    • Chicago
      • Mean Income (Sample Size = 100): $19,500
      • Standard Deviation(s1): $300
    • New York:
      • Mean Income (Sample Size = 100): $18,500
      • Standard Deviation(s2): $400
    • What if Christos wants to launch the course in one of the markets?
    • What would be the decision criteria? What should be the testing strategy?
      • Use two means hypothesis: μc = μn
      • Which can also be reduced to μc – μn = 0
      • The only treatment to be made different is that the standard error has to be calculated as:

      • The rest of the treatment remains the same as one mean hypothesis.
    • Why is it possible to use the Z test for two means as well?
      • Hint: Linear combination of normal distributions is a normal distribution
  • If we call μc – μn as μh
    • Then the test can be appropriately modified as a two-tailed test on μh,
    • μh -> Population Mean
    • H0: μh  = $0
    • Ha: μh ≠ $0
  • Since we are checking for significance difference on both the ends, so it’s a two tailed test
  • This test can have another variant, if we check for a significant difference between two means as a particular value, in which the hypothesis would be modified as:
    • μh -> Population Mean
    • H0: μh  = $1,000
    • Ha: μh ≠ $1,000


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