Answer to Question #271814 in Statistics and Probability for ella

Question

A water bottle company produces bottles with a net volume 400ml each. It is known that the standard deviation of the volume is 12ml. A sample of 50 bottles is tested.

If the sample yields mean volume of 420ml, construct a hypothesis test at the \alpha =0.01 level of significance for the null hypotheses H_(0):\mu =400ml against the alternative hypothesis H_(A):\mu !=400ml. State your conclusion and give a reason for your answer.

Answer 1

The hypotheses tested are,

$H_0:\mu= 400$

$Against$

$H_A:\mu\not=400$

We are given the following,

$\bar{x}=420,\space \sigma=12,\space n=50$

We will apply the standard normal distribution to perform this hypothesis test as follows.

$Z_c=(\bar{x}-\mu)/(\sigma/\sqrt{n})=(420-400)/(12/\sqrt{50})=20/1.6971=11.785113$

$Z_c$ is compared with the standard normal table value at $\alpha=0.01$ given as, $Z_{0.01/2}=Z_{0.005}=2.575$ and the null hypothesis is rejected if $Z_c\gt Z_{0.005}.$

Since $Z_c=11.785113\gt Z_{0.005}=2.575$, we reject the null hypothesis and we conclude that there is no sufficient evidence to show that the average net volume for bottles is 400 ml at 1% level of significance.