Q.81 - The Practice of Statistics for AP

Question

Refer to Exercise 79. Construct and interpret a $95\%$confidence interval for the population mean M. What additional information does the confidence interval provide .

Step-by-Step Solution

Answer

The confidence interval is $(11.561,11.472)$.

Step 1: Given information

From question 79, The hardness data for a random sample of $20$ tablets are

Step 2: Explanation

11.267
$11.613$
$11.493$
$11.602$
$11.36$
$11.374$
$11.592$
$11.458$
$11.552$
$11.463$
$11.383$
$11.715$
$11.485$
$11.509$
$11.429$
11.477
11.57
11.623
11.472
11.531

Step 3: Calculate the confidence interval

The confidence interval for population mean is:

$\overline{x}-t_{\alpha /2}\times \frac{s}{\sqrt{n}}<\mu <\overline{x}+t_{\alpha /2}\times \frac{s}{\sqrt{n}}$

The sample mean and standard deviation are $11.5164$ and $0.0950 .$

Degree of freedom:

  $$\begin{gathered} df=n-1 \\ =20-1 \\ =19 \end{gathered}$$

From critical value table, the t-value is $2.093 .$

The confidence interval is computed as:

$\overline{x}-t_{\alpha }/2\times \frac{s}{\sqrt{n}}<\mu <\overline{x}+t_{\alpha /2}\times \frac{s}{\sqrt{n}}$

$11.5164-2.096\times \frac{0.0950}{\sqrt{20}}<\mu <11.5164+2.096\times \frac{0.0950}{\sqrt{20}}$

$(11.561,11.472)$

The confidence interval is $(11.561,11.472)$.