A design engineer wants to construct a sample mean chart for controlling the service life of a halogen headlamp his company produces. He knows from numerous previous samples that this service life is normally distributed with a mean of 500 hours and a standard deviation of 20 hours. On three recent production batches, he tested service life on random samples of four headlamps, with these results: $$\begin{array} { r }
{ \text { Service Life (hours) } }\
\begin{array} { l l l l l }
\text { Sample } & \
\hline 1 & 495 & 500 & 505 & 500 \
2 & 525 & 515 & 505 & 515 \
3 & 470 & 480 & 460 & 470
\end{array}\end{array}$$ If he uses upper and lower control limits of 520 and 480 hours, what is his risk (alpha) of concluding that service life is out of control when it is actually under control (Type I error)?
A) 0.0026
B) 0.0456
C) 0.3174
D) 0.6826
E) 0.9544

Question

A design engineer wants to construct a sample mean chart for controlling the service life of a halogen headlamp his company produces. He knows from numerous previous samples that this service life is normally distributed with a mean of 500 hours and a standard deviation of 20 hours. On three recent production batches, he tested service life on random samples of four headlamps, with these results: $$\begin{array} { r } 
 { \text { Service Life (hours) } }\
\begin{array} { l l l l l } 
\text { Sample } & \
\hline 1 & 495 & 500 & 505 & 500 \
2 & 525 & 515 & 505 & 515 \
3 & 470 & 480 & 460 & 470
\end{array}\end{array}$$ If he uses upper and lower control limits of 520 and 480 hours, what is his risk (alpha) of concluding that service life is out of control when it is actually under control (Type I error)?
A) 0.0026
B) 0.0456
C) 0.3174
D) 0.6826
E) 0.9544

Quizplus · Accepted Answer

The risk of a Type I error (alpha) is the probability that the sample mean falls outside the control limits when the process is actually in control. The control limits are set at 480 and 520 hours. The standard error of the mean for a sample size of 4 is 20/sqrt(4) = 10. The z-scores for the control limits are (480-500)/10 = -2 and (520-500)/10 = 2. The probability of the sample mean being within these limits is the area under the normal distribution between z = -2 and z = 2, which is approximately 0.9544. Therefore, the probability of being outside these limits (alpha) is 1 - 0.9544 = 0.0456.

A Design Engineer Wants to Construct a Sample Mean Chart