# 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 t
_{c}= 1.71 for ∈ = 0.05 and d.f. = 24 - Decision rule: “Reject H
_{0}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(s
_{1}): $300

- New York:
- Mean Income (Sample Size = 100): $18,500
- Standard Deviation(s
_{2}): $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: μ
_{c}= μ_{n} - Which can also be reduced to μ
_{c}– μ_{n}= 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.

- Use two means hypothesis: μ
- Why is it possible to use the Z test for two means as well?
- Hint: Linear combination of normal distributions is a normal distribution

- Chicago
- If we call μ
_{c}– μ_{n}as μ_{h}- Then the test can be appropriately modified as a two-tailed test on μ
_{h}, - μ
_{h}-> Population Mean - H
_{0}: μ_{h}= $0 - H
_{a}: μ_{h}≠ $0

- Then the test can be appropriately modified as a two-tailed test on μ
- 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:
- μ
_{h}-> Population Mean - H
_{0}: μ_{h}= $1,000 - H
_{a}: μ_{h}≠ $1,000

- μ