Q.8 - The Practice of Statistics for AP

Question

 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 $\mu =224$ 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.

Step-by-Step Solution

Answer

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

(b) Yes

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:

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

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

Thus there is $95\%$ probability that the population mean is between $223.97 and 227.03$

Part (b) Step 1: Given Information

The process mean is supposed to be $\mu =224$ 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 $223.97 and 224.03$Since, the interval is containing $224$ it could be said that the process mean has drifted.