Answer to Question #346284 in Statistics and Probability for Sario

Question

soft drink dispenser is designed to dispense 250 mL of drink per cup. However, there have

been recent reports to the managements of a fast-food restaurant of both underfilled and

overfilled cups. Seeking to investigate this matter, they fill up 40 cups using this machine. If the mean amount of drink in the 40 cups is 245 mL with standard deviation of 10 mL, is there a

cause for concern for the management? Use a level of significance of 0.01

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=250$

$H_1:\mu\not=250$

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is $\alpha = 0.01,$ $df=n-1=39$ and the critical value for a two-tailed test is $t_c =2.707913.$

The rejection region for this two-tailed test is $R = \{t:|t|>2.707913\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{245-250}{10/\sqrt{40}}=-3.1623$$

Since it is observed that $|t|=3.1623>2.707913=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for two-tailed, $df=39$ degrees of freedom, $t=-3.1623$ is $p= 0.003027,$ and since $p= 0.003027<0.01=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$

is different than 250, at the $\alpha = 0.01$ significance level.