The P-value of a one-sided test. The P-value is a way to proceed with a hypothes

Question

The P-value of a one-sided test. The P-value is a way to proceed with a hypothesis test without fixing the lpha level in advance. For example, consider the above setup where X1,...,Xn are i.i.d. N(mu, sigma^2) with sigma^2=9 and n=100; we want to do a one-sided test of H0: mu=10 vs. H1: mu<10. The test rejects H0 if the sample mean arX is less than C where C is some value associated with the lpha level. Instead of fixing the lpha level, we compute the P-value as follows. Suppose you observe arX = 9.5 (say). The P-value is the probability (under H0) that we might have observed a value more extreme--in this case smaller--than what we did observe (which was 9.5). Hence, P-value = P( arX < 9.5 | H0). Calculate this P-value, and argue that with our observed data we could reject H0 for all levels lpha greater (or equal) than the P-value.

Explanation / Answer

Formulating the null and alternative hypotheses,              
              
Ho:   u   >=   10  
Ha:    u   <   10  
              
As we can see, this is a    left   tailed test.      
              
              
Getting the test statistic, as              
              
X = sample mean =    9.5          
uo = hypothesized mean =    10          
n = sample size =    100          
s = standard deviation =    3          
              
Thus, z = (X - uo) * sqrt(n) / s =    -1.666666667          
              
Also, the p value is              
              
p =    0.047790352   [ANSWER, P VALUE]         
              
Hence, for any alpha greater than this P (say, 0.05), we can reject Ho.

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