# 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