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 support your answer.

Step-by-Step Solution

Answer

We do not have enough evidence to conclude that the hardness of these tablets is something other than $11.5$.

Step 1: Given information

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

$11.627 11.613 11.493 11.602 11.360 11.374 11.592 11.458 11.552 11.463 11.383 11.715 11.485 11.509 11.429 11.477 11.570 11.623 11.472 11.531$

Step 2: Explanation

State: $H_0 : \mu = 11.5, H_a : \mu \ne 11.5$

Plan: Random: Random sample. 

Normal: The histogram indicates that the distribution is roughly symmetric with no outliers. 

Independent: There are more than $200$ tablets.

Do: $t = 0.772, P-value = 0.4494.$

Conclude: Since the P-value is greater than $0.05$, we fail to reject $H_0$.

We do not have enough evidence to conclude that the hardness of these tablets is something other than $11.5$.