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Inferential Statistics - Single sample tests

Single sample z-test

What are you testing ? Population Mean
Know variance and unknown variance
Small sample size and large sample size
Example : Average phosphate in blood is less than 4.8 mg/dl , with a known standard deviation of 0.4 mg/dl
Data : 4.1, 3.9, 5.3, 4.7.................

Using the rubric for this examples:

 - Have a null and alternate hypothesis; H00≤ 4.8 and Halt0 > 4.8
 - Do some basic calculation / arithmetic on the data to create a single number called the "test statistic" ; zstat = ẍ-μ0 / 𝛔/√n
 - If we assume the null hypothesis to be true (and make some assumptions about the distribution of various variable), then the 'test statistic' should be no different than a single random draw from a specific probability distribution. This is the Z-distribution of N(0,12)
 - Test the probability that the "test statistic" you calculated belongs to this theoretical distribution. This is the p-value !; Use Z-tables, Excel, Matlab or R
 - Low enough p-value is grounds for rejecting the null hypothesis

* The p-value is the probability of seeing a test statistic as extreme as the calculated value if the null hypothesis is true.

* If Zstat  was computed to be 1.2 then

* P-value, Based on the standard null hypothesis:

* H0: μ ≤ 4.8

* If null hypothesis was 
* H0: μ ≥ 4.8

 * H0: μ = 4.8 (two tailed)

Examples and Formulas 

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