Chapter 8: Q.8 (page 523)

Engine parts Here are measurements (in millimeters) of a critical dimension on an SRS of $16$ of the more than $200$ auto engine crankshafts produced in one day:
$224.120$ , $224.001$ , $224.017$ , $223.982,$ $223.989,$ $223.960,$ $224.089,$ $223.987,$ $223.976,$ $223.902,$ $223.980,$ $224.098,$ $224.057,$ $223.913,$ $223.999$
(a) Construct and interpret a $95 %$ confidence interval for the process mean at the time these crankshafts were produced.
(b) The process mean is supposed to be $μ = 224$ but can drift away from this target during production. Does your interval from part (a) suggest that the process mean has drifted? Explain.
- Determine the sample size required to obtain a level $C$ confidence interval for a population mean with a specified margin of error.

Short Answer

Expert verified

(a) The sample is $35$ $35$

(b) Yes

Step by step solution

Part (a) Step 1: Given Information

Given in the question that,

$224.120$
$224.001$
$224.017$
$223.982$
$223.989$
$223.961$
$223.960$
$224.089$
$223.987$
$223.976$
$223.902$
$223.980$
$224.098$
$224.057$
$223.913$
$223.999$

we have to construct and interpret a $95 %$ confidence interval for the process mean at the time these crankshafts were produced.

Part (a) Step 2: Explanation

The formula to compute the confidence interval is:

$x - t \frac{α}{2} \times \frac{s}{\sqrt{n}} < μ < \bar{x} + t \frac{α}{2} \times \frac{s}{\sqrt{n}}$

The confidence interval using Ti-83calculator could be computed as:

Thus there is $95 %$ probability that the population mean is between $223.97 a n d 227.03$

Part (b) Step 1: Given Information

The process mean is supposed to be $μ = 224$ but can drift away from this target during production .we have to find that Does interval from part (a) suggest that the process mean has drifted .

Part (b) Step 2: Explanation

From the above part, it is known that the population mean process is between $223.97 a n d 224.03$ Since, the interval is containing $224$ it could be said that the process mean has drifted.