Question

Pressing pills A drug manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The hardness of a sample from each batch of tablets produced is measured to control the compression process. The target value for the hardness is $\mu$=11.5. The hardness data for a random sample of 20 tablets are

11.627 11.613 11.493 11.602 11.360

11.374 11.592 11.458 11.552 11.463

11.383 11.715 11.485 11.509 11.429

11.477 11.570 11.623 11.472 11.531

Is there significant evidence at the 5% level that the mean hardness of the tablets differs from the target value? Carry out an appropriate test to support your answer.

Short answer

the mean hardness of the tablets does not differ from the target value.

Step-by-step solution

Given Information

The hardness data for a random sample of $20$tablets are

$$\begin{gathered} \begin{array}{l}11.627\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.613\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.493\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.602\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.360\end{array} \\ \begin{array}{l}11.374\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.592\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.458\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.552\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.463\end{array} \\ \begin{array}{l}11.383\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.715\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.485\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.509\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.429\\ 11.477\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.570\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.623\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.472\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11.531\end{array} \end{gathered}$$

Explanation

From the data given,

$\stackrel{-}{x}=11.516$

$\mu =11.5$

we calculate the null and alternative hypotheses,

$\begin{array}{r}{H}_0:\mu =11.5\\ H_a:\mu \ne 11.5\end{array}$

Using,

localid="1652942386796" $$\begin{gathered} t=\frac{\stackrel{-}{x}-\mu }{\frac{s}{\sqrt{n}}}=\frac{11.516-11.50}{\frac{0.0950}{\sqrt{20}}} \\ =0.77 \end{gathered}$$

The test statistic =$0.77$

Hence the mean hardness of the tablets does not differ from the target value $\mu =11.5$as the critical value$2.093$ is greater than the test statistic $0.77$.