Chapter 3, Problem 10P

i)

Summary Introduction

Interpretation: by using $\overline{x}$ chart find out when would the process stop.

Concept introduction:Mean(x-bar) charts:

X bar charts is a type of statistical process control chart which measures a central a shift in the central tendency of the process. In other words, X- bar charts monitor the mean or average of a product characteristics. The central line of the X-bar chart is the average of the samples and limits are defined as

Upper control limit $=\overline{\overline{x}}+A_2\overline{R}$

Lower control limit $=\overline{\overline{x}}-A_2\overline{R}$

( $A_2$ is a factor for mean chart)

Abbreviations:

  $\overline{\overline{x}}=$ Average of the sample means

  $\overline{R}=$ Average of the sample ranges

  $z=$ standard normal variable ( $2$ for $95.44\%$ confidence, $3$ for $99.74\%$ confidence)

  ${\sigma }_{\overline{x}}=$ standard deviation of the distribution of sample means, computed as $\frac{\sigma }{\sqrt{n}}$

  $\sigma =$ standard deviation

  $n=$ sample size (number of observations per sample)

R chart:

R-chart or range-chart is also a type of statistical process control chart, but it monitors the range of dispersion or variability of process. The central line of the r- chart is the average range and limits are:

  $\begin{array}{l}UCL=D_4\overline{R}\\ LCL=D_3\overline{R}\end{array}$

( $D_{4,}{D}_3$ are factors for range chart)

The data corresponding to sampling of the given firm is given below

Mean of the diameter $\left(\overline{\overline{x}}\right)$ is $0.015$ inch.

Average range $\left(\overline{R}\right)$ is $0.002$ inch.

Control limits are $3$ sigma.

ii)

Summary Introduction

Interpretation: by using $\overline{x}$ chart find out, when the process would be out of control.

Concept introduction:a process is said to be out control if one or more data points fall outside the control limits.

iii)

Summary Introduction

Interpretation: determine what trade off would consider by using result of a and b.

Concept introduction: trade off is a situational decision that involves diminishing or losing one quality, quantity or design in return for gains in other aspects