Question

Investigating a complaint from a buyer that there is short-weight selling, a manufacturer takes a random sample of twenty-five 32 g cans of coffee from a large shipment and finds that the mean weight is 31 g with a standard deviation of 0.6 g. Is there evidence of short-weighing at the 0.01 level significance?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\ge32$

$H_a:\mu<32$

This corresponds to a left-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 $\alpha = 0.01,$ $df=n-1=34$ degrees of freedom, and the critical value for a left-tailed test is $t_c = -2.44115.$

The rejection region for this left-tailed test is $R = \{t: t<-2.44115\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{31-32}{0.6/\sqrt{35}}\approx-9.8601$$

Since it is observed that $t=-9.8601< -2.44115=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for left-tailed, $df=34$ degrees of freedom, $t=-9.8601$ is $p=0,$ and since $p=0<0.01=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is less than 32, at the $\alpha = 0.01$ significance level.