using sample to make quality control decisions


A beverage company fills its best-selling 300ml drinks by an automatic dispensing machine. The machine is set to dispense a mean of 310ml per bottle. Uncontrollable factors in the process can shift the mean away from 310ml and cause either underfill or overfill, both of which are undesirable. In such a case the dispensing machine is stopped and recalibrated. Regardless of the mean amount dispensed, the standard deviation of the amount dispensed always has value 22ml. A quality control engineer routinely selects 30 bottles from the production line to check the amounts filled. On one occasion, the sample mean is x = 320ml the sample standard deviation is s = 25ml ounce. Determine if there is sufficient evidence in the sample to indicate, at the 1% level of significance, that the machine should be recalibrated. 

solution: 
#inputs:
sample_mean = 320
sample_std = 25
alfa = 0.01
population_mean = 310
population_std = 22
sample_size = 30 
options(digits = 3)

#calculating statistic:
z_value = (sample_mean - population_mean) / (population_std / sqrt(sample_size))


#calculating rejection region
rejection_region = data.frame("lower" =seq(-Inf,qnormGC(alfa/2,mean = 0,sd= 1 ), by = 0.01), 
                             "upper" = seq(- qnormGC(alfa/2,mean = 0, sd= 1 ), Inf, by = 0.01 ))
conclusion: 
since the value is outside of the rejection region we accept H0 and decide not to recalibrated the machines. 

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