Answer to Question #339923 in Statistics and Probability for zeus

Question

Determine the appropriate test statistic to be used in the given statements. And then, make an inference by following the steps in testing hypotheses.

Coca-Cola produced 1.5 liters of soda. The production department reported that the standard deviation of the bottle is 0.7 liter. The quality control department conducted a random checking on the content of the bottles and obtained 1.45 liters from 100,1.5 liters bottles. Test if there's enough evidence that the average amount in bottles is different from the standard 1.5 liters. Use a 5% level of significance.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=1.5$

$H_a:\mu\not=1.5$

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

Based on the information provided, the significance level is $\alpha = 0.05,$ and the critical value for a two-tailed test is z_c = 1.96$\alpha = 0.05.$

The rejection region for this two-tailed test is $R = \{z: |z| > 1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{1.45-1.5}{0.7/\sqrt{100}}=-0.7143$$

Since it is observed that $|z| = 0.714 3\le 1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is $p=2P(Z<-0.7143)=0.475042,$ and since $p=0.475042>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$  is different than 1.5, at the $\alpha = 0.05$ significance level.