Q.79 - The Practice of Statistics for AP

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.

Step-by-Step Solution

Answer

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

Step 1: Given Information

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

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

Step 2: 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,

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

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 $0.77$.