Question

How much juice? One company’s bottles of grapefruit juice are filled by a machine that is set to dispense an average of 180 milliliters (ml) of liquid. The company has been getting negative feedback from customers about underfilled bottles. To investigate, a quality-control inspector takes a random sample of 40 bottles and measures the volume of liquid in each bottle. The mean amount of liquid in the bottles is 179.6 ml and the standard deviation is 1.3 ml. Do these data provide convincing evidence at the$\alpha =0.05$significance level that the machine is underfilling the bottles?

Short answer

The required answer is:

Yes, there is sufficient evidence to show that bottles are being unfilled by the machine.

Step-by-step solution

Given information

Population mean $\left(\mu \right)=180$

Sample mean $\left(\overline{x}\right)=179.6$

Sample size $\left(n\right)=40$

Sample standard deviation$\left(s\right)=1.3$

Formula used

We know,

The test statistic formula is:$t=\frac{\overline{x}-{\mu }_o}{\frac{s}{\sqrt{n}}}$

Calculation

The null and alternative hypotheses are as follows:

$$\begin{gathered} H_0:\mu =180 \\ {H}_a:\mu <180 \end{gathered}$$

The alternative hypothesis denotes that the test is left-handed.

The Minitab output is as follows:

The $p$-value is $0.029$

Here, $p$-value $(0.029)<\alpha (0.05)$The null hypothesis is rejected.

There is enough evidence to show that the machine is unfilling bottles at the $5\%$level of significance.