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Some Important notes on use of Student’s T-test and Z-test for Hypothesis Testing:

The t-statistic was introduced in 1908 by William Sealy                  Gosset,     a   chemist   working   for
the Guinness brewery in Dublin, Ireland ("Student" was his pen name).

Basic Underlying Assumptions in T-test are:

             1) Samples are independent and randomly drawn from a normal population.

             2) Sample size is small. (n<=30)

Below are the seven situations which give an idea about when to use T-test or Z-test:

             •   Situation 1 : Samples are independent and randomly drawn from normal population whose mean
                 (µ) and standard deviation (σ) are known

        Sample size is large i.e. n>=30

        Test to be used for Hypothesis testing: Z-test



             •   Situation 2 : Samples are independent and randomly drawn from normal population whose mean
                 (µ) and standard deviation (σ) are known

        Sample size is small i.e. n<=30

        Test to be used for Hypothesis testing: Z-test or T-test



             •   Situation 3 : Samples are independent and randomly drawn from a normal population whose
                 mean (µ) is known but standard deviation (σ) is not known

        Population size is large

        If standard deviation of sample is known then use sample standard deviation as best estimate for
        population standard deviation.

        Test to be used for Hypothesis testing: Z-test



             •   Situation 4: Samples are independent and randomly drawn from a Normal population whose
                 mean (µ) is known but standard deviation (σ) is not known

        Population size is small

        If standard deviation of sample is known then use sample standard deviation as best estimate for
        population standard deviation.

        Test to be used for Hypothesis testing: T-test
•    Situation 5: Samples are independent and randomly drawn from any population whose standard
         deviation (σ) is known and whose sample size is large.

Test to be used for Hypothesis testing: Z-test because of central limit theorem



    •    Situation 6: Samples are independent and randomly drawn from any population whose standard
         deviation (σ) is unknown and whose sample size is large.

Use sample standard deviation to approximate to population standard deviation and use Z-test.



    •    Situation 7: Samples are independent and randomly drawn from any population whose standard
         deviation (σ) is unknown and sample size is small.

In this situation, other test like Wilkokson’s Test is used

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Statistics -- Important notes

  • 1. Some Important notes on use of Student’s T-test and Z-test for Hypothesis Testing: The t-statistic was introduced in 1908 by William Sealy Gosset, a chemist working for the Guinness brewery in Dublin, Ireland ("Student" was his pen name). Basic Underlying Assumptions in T-test are: 1) Samples are independent and randomly drawn from a normal population. 2) Sample size is small. (n<=30) Below are the seven situations which give an idea about when to use T-test or Z-test: • Situation 1 : Samples are independent and randomly drawn from normal population whose mean (µ) and standard deviation (σ) are known Sample size is large i.e. n>=30 Test to be used for Hypothesis testing: Z-test • Situation 2 : Samples are independent and randomly drawn from normal population whose mean (µ) and standard deviation (σ) are known Sample size is small i.e. n<=30 Test to be used for Hypothesis testing: Z-test or T-test • Situation 3 : Samples are independent and randomly drawn from a normal population whose mean (µ) is known but standard deviation (σ) is not known Population size is large If standard deviation of sample is known then use sample standard deviation as best estimate for population standard deviation. Test to be used for Hypothesis testing: Z-test • Situation 4: Samples are independent and randomly drawn from a Normal population whose mean (µ) is known but standard deviation (σ) is not known Population size is small If standard deviation of sample is known then use sample standard deviation as best estimate for population standard deviation. Test to be used for Hypothesis testing: T-test
  • 2. Situation 5: Samples are independent and randomly drawn from any population whose standard deviation (σ) is known and whose sample size is large. Test to be used for Hypothesis testing: Z-test because of central limit theorem • Situation 6: Samples are independent and randomly drawn from any population whose standard deviation (σ) is unknown and whose sample size is large. Use sample standard deviation to approximate to population standard deviation and use Z-test. • Situation 7: Samples are independent and randomly drawn from any population whose standard deviation (σ) is unknown and sample size is small. In this situation, other test like Wilkokson’s Test is used