Question

Aminah wish to perform the hypothesis testing H0: μ =1 versus H1: μ <1 versus with α=0.10. . The sample size 25 was obtained independently from a population with standard deviation 10. State the distribution of the sample mean given that null hypothesis is true and find the critical value, then calculate the values of sample mean if she reject the null hypothesis. Finally, compute the p-value, if the sample mean is -2.

Answer 1

H0: μ =1

H1: μ <1 [ left tail test ]

standard normal distribution will be used because population std dev is given .

significance level,   α =    0.1
critical value , Zα =       -1.2816 [from z distribution table]
          

hypothesis mean,   µo =    1
sample size,   n =   25
std dev,   σ = 10

std error of mean,   σx = σ/√n =    10 / √    25   = 2

We will reject the null hypothesis if we get a Z statistic < -1.2816
this Z-critical value corresponds to X critical value( X critical), such that                  
                  
(x̄ - µo)/σx ≤ Zα                  
x̄ ≤ Zα*σx + µo                  
x̄ ≤ -1.282   *   2 +   1
so, X = -1.5632   

so, value of sample mean is -1.5632 if rejects the null hypothesis

=---------------------------------

population std dev ,    σ =    10.0000                  
Sample Size ,   n =    25                  
Sample Mean,    x̅ =   -2.0000                  
                          
'   '   '                  
                          
Standard Error , SE = σ/√n =   10.0000   / √    25   =   2.0000      
Z-test statistic= (x̅ - µ )/SE = (   -2.000   -   1   ) /    2.000   =   -1.500
                          
  
p-Value   =   0.0668   [ Excel formula =NORMSDIST(z) ]