Chapter 9.3, Problem 76E

To determine

To find: Whether there are sufficient evidence to show that bottles are being unfilled by the machine at 5% significance level.

Answer to Problem 76E

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

Explanation of Solution

Given:

  $\begin{array}{l}\text{Population mean}\left(\mu \right)=180\\ \text{Sample mean}\left(\overline{x}\right)=179.6\\ \text{Sample size}\left(n\right)=40\\ \text{Sample standard deviation}\left(s\right)=1.3\end{array}$

Formula used:

The test statistic formula is:

  $t=\frac{\overline{x}-{\mu }_o}{\frac{s}{\sqrt{n}}}$

Calculation:

The null and alternative hypotheses are:

  $\begin{array}{l}{H}_0:\mu =180\\ H_a:\mu <180\end{array}$

The alternative hypothesis represents that the test is left tailed.

The obtained Minitab output is:

  

The p -value is 0.029.

Here, $p\text{-value}\left(0.029\right)<\alpha \left(0.05\right)$ . The null hypothesis is rejected.

Conclusion:

At 5% level of significance there is sufficient evidence to show that bottles are being unfilled by the machine.