Question

The owner of a factory that sells a particular bottled fruit juice claims the the average capacity of their product is 250 ml to the test claim a consumer group gets a sample of 100 such bottles calculates the capacity of each bottle and then finds the mean capacity to the be 248ml.The standard deviation is 5ml . Test the hypothesis that the population mean is less that 250 ml. Assumed that the population is normally distributed. Use level of significance level is 0.5 and the z value is 1.645

Answer 1

$H_0:\mu =250\\H_1:\mu <250\\Z=\sqrt{n}\frac{\bar{x}-\mu}{\sigma}=\sqrt{100}\frac{248-250}{5}=-4\\If\,\,we\,\,use\,\,z-value\,\,1.645, the\,\,null\,\,hypothesis\,\,is\,\,rejected,\\\sin ce\,\,-4<-1.645\\If\,\,we\,\,use\,\,0.5 level\,\,of\,\,significance:\\P\left( Z\leqslant -4 \right) =\varPhi \left( -4 \right) =3.17\cdot 10^{-5}<\alpha =0.5\Rightarrow the\,\,null\,\,hypothesis\,\,is\,\,rejected\\In\,\,both\,\,cases\,\,\mu <250\\$