Hypothesis Tests

 

Now, we are clear that the determination of rejection regions depends on the alternative hypothesis.


H1: μ >μwhich determines the rejection region falls under the right tail

H1: μ <μwhich determines the rejection region falls under the left tail

H1: μ ≠ μwhich determines the rejection region falls under both tails


Moreover, the level of significance α tells us the size of the rejection region. As mentioned before, the value of  the level of significance α is predetermined, analyst or scientist just set it up before sampling or doing experiments and we know that that is the probability of Type II Error which is rejection the existing accepted value when it is still true. It gives damages to the producers and it is even called producer’s risk.


We usually set the level of significance α a small value like 0.01 or 1%, 0.05 or 5% 0.10 or 10% and not so big.


β depends on the values stated in the alternative hypothesis. As for population mean as the parameter in this case,


H1: μ >μ0

H1: μ <μ0

H1: μ ≠ μ


This population mean μ must have a specific value in order to determine β value which is the probability of making Type II Error. Once we figure out the specific value, then you can use a formula to find β. But remember, the parameter population mean and the population is required to be such that the probability distribution of the test statistics is normal. Test statics values are usually if in a question given in the question, if in the real life situation from the test statistics data. Test statistics are population mean μt, population standard deviation σ and size of the sample n. To be specific, the population standard deviation must be written as σμt stating this value is the standard deviation corresponding to the population mean of the test statistics.


Let’s call the specific value to be μ1, and the test statistic value to be μt, and the population mean stated in the null hypothesis to be μ0, they are all population mean but reported at different situation and time.


The value of β = P(Type II Error) can be obtained using the following formula, here population mean is used as a parameter.


If ,    H1: μ >μ0

Use,  

        β = P(Type II Error) = P( Z < { (μ0 -  μ1)/σμt  } + zα )

     

Note: Getting standard deviation σμt is different depending on the size of the population



If,       H1: μ <μ0

Use,

           β = P(Type II Error)   = P( Z < { (μ0 -  μ1)/σμt  } - zα )  



Note: Getting standard deviation σμt is different depending on the size of the population



If,      H1: μ ≠ μ

Use,

         β = P(Type II Error) = P(  { (μ0 -  μ1)/σμt  } - zα/2     <   Z   <    { (μ0 -  μ1)/σμt  } + zα/2    )



Note: Getting standard deviation σμt is different depending on the size of the population

for example if n is large (n ≥ 30), then σμt = (σ/√n) etc.



Now, let’s see how to test hypothesis


First Situation

Population mean value of a population

with a known standard deviation and large sample size


Next

DNA Pot (c) 2009 - Current

Review Concepts before testing: How to find β