Question

Pressing pills Refer to Exercise 77.

a. Construct and interpret a 95% confidence interval for the true hardness μ of the tablets in this batch. Assume that the conditions for inference are met.

b. Explain why the interval in part (a) is consistent with the result of the test in Exercise 77.

Short answer

Part (a) $(11.4719,11.5609)$

Part (b) Confidence interval contains $11.5$

Step-by-step solution

Part (a) Step 1: Given information

n $=10$

c $=95\%=0.95$

Part (a) Step 2: Concept

The formula used:$E=t_{a/2}\times \frac{s}{\sqrt{n}}$

Part (a) Step 3: Calculation

Conditions:

Random and independent are the three requirements.

Normal/Large sample ($10$percent condition).

Because the sample is a random sample, I'm satisfied.

Because the sample of $20$tables represents less than $10\%$of the total number of tables, independent: fulfilled.

Because the pattern in the normal quantile plot is generally linear, the normal/large sample is satisfied.

Because all of the prerequisites have been met, it is appropriate to calculate the population's confidence interval.

The mean is

$\overline{x}=\frac{230.328}{20}$

$\overline{x}=11.5164$

The variance is

$s=0.0950$

In the table of the Students T distribution, look in the row starting with degrees of freedom $df=n-1=20-1=19$and in the column with $c=95$percent to find the $t-$value:

$t_{a/2}=2.093$

The margin of error is

$$\begin{gathered} E=t_{a/2}\times \frac{s}{\sqrt{n}} \\ =2.093\times \frac{0.0950}{\sqrt{20}} \\ =0.0445 \end{gathered}$$

The boundaries of the confidence interval are

$$\begin{gathered} \overline{x}-E=11.5164-0.0445=11.4719 \\ \overline{x}+E=11.5164+0.0445=11.5609 \end{gathered}$$

There are $95\%$confident that the true hardness of the tablets in this batch is between $11.4719$and $11.5609$

Part (b) Step 1: Explanation

From the Result part (a):

$(11.4719,11.5609)$

It is noted that the confidence interval contains the value $11.5$indicating that the mean hardness is likely to be $11.5$As a result, the claim that the mean is $11.5$is not rejected. There is insufficient evidence to support the claim that the genuine hardness of the tablets in this batch is lower than $11.5$

We then realized that we had arrived at the same conclusion as we had in the prior experiment.