Q 590. Both 1 sample T-test and 1 sample Z-test are used to compare a population mean against a target value. What are the key differences between the two tests, their usage and their application? Include relevant examples to illustrate your response.

 

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Both the test are for comparison of Sample Mean Vs. Known Mean. Following is the Key differences between 1 Sample T test and 1 Sample Z Test.

 

1. 1 Sample T test assumes that standard deviation is Unknown vs. Sample Z Test assume standard deviation is known.

2. 1 Sample T test is suitable for small population sample size Vs. 1 Sample Z test is for large population size.

 

Usage of 1 Sample T test can be use for survey of small group with unknown standard deviation. Where as 1 Sample Z test can be use in Clinical trial in pharmaceutical industry where standard deviation is known. for example a company claim than mean score of niraparib anticancer drug trial is greater than 82 with standard deviation of 20. If a sample of 81 patients was selected with a mean of scores of 90 then check if there is enough evidence to support this claim at a 0.05 significance level. 

1 Sample Test

 

A one sample test is a hypothesis test which generally used to determine whether our sample’s characteristics are significantly stands different from a known or hypothesized parameter.

Also, it generally used for single population group or sample.

 

There are 2 most common tests are available which used to compare the sample:

·       1 sample T-test

·       1 sample z-Test

 

	1 sample T-test	1 sample z-Test
Sample size	It is more robust for sample size <30	More suitable for large sample size, sample size >30
Population standard deviation	Used when population standard deviation is unknown	Used when population standard deviation is known
Application	Used when we want to determine whether a sample mean significantly differ from a hypothesized mean or sample in small sample size	Used when we want to determine whether a sample mean significantly differ from a hypothesized mean or sample in large sample size
Usage	We use when we generally want to compare a sample mean with population mean	We generally used sample standard deviation instead of population standard deviation.
Assumption	Both tests assume data is normally distributed
Applicability	More relevant as it can handle unknown population standard deviation with small sample size	Relevant only if we have large sample size with known standard deviation
Robustness	More robust with small sample size	Less robust with small sample. Used only for large sample
Calculation Method	Involves degree of freedom	No degree of freedom adjusted
Software Availability	Available with all the statistical software	Less available with all

 

Below are some real-world examples:

1 sample T-test – Let suppose a battery company want to check the charging life of newly battery which just developed. So, company will select some battery randomly and check for their usage time.  After getting usage time they will check whether it is significantly different from population mean.

 

1 sample Z-Test – Let suppose a researcher want to check whether a newly developed drug can lower the blood pressure or not. So, take a known mean which is 120 mmHg. So, researcher will take sample which would be generally more than 30 to determine whether this drug can lower the blood pressure with known mean.

 

Pradeep Shukla has correctly highlighted the differences, assumptions and applicability of the two tests. Hence his answer has been selected as the winning answer.