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Chapter 16: Q44 SE (page 714)

An alternative to the \(p\) chart for the fraction defective is the np chart for number defective. This chart has \(UCL = n\bar p + 3\sqrt {n\bar p(1 - \bar p)} ,LCL = n\bar p - 3\sqrt {n\bar p(1 - \bar p)} \),

and the number of defectives from each sample is plotted on the chart. Construct such a chart for the data of Example 16.6. Will the use of an \(n p\) chart always give the same message as the use of a \(p\)chart (i.e., are the two charts equivalent)?

Short Answer

Expert verified

\({\rm{L C L = 0 ; U C L = 13}}{\rm{.25}}\)

Step by step solution

01

Step 1:To Find the process in-control

For the explained chart, and mentioned example, for\(\bar p = 0.0608\), and\(n = 100\), the limits are

\(\begin{array}{l}LCL = n\bar p - 3\sqrt {n\bar p(1 - \bar p)} = 6.08 - 3 \times \sqrt {6.08 \times 0.9392} \mathop = \limits^{(1)} 0\\UCL = n\bar p - 3\sqrt {n\bar p(1 - \bar p)} = 6.08 + 3 \times \sqrt {6.08 \times 0.9392} = 13.25\end{array}\)

(1) : it is less than zero, so set it to zero.

From the\(np\)chart below one can notice that the process is in-control (same as in in the\(p\)chart).

Two charts,\(p\)and\(np\)charts, give identical result This claim stands because of the fact that

\(\bar p - 3 \times \sqrt {\frac{{\bar p(1 - \bar p)}}{n}} < {\hat p_i} < \bar p + 3 \times \sqrt {\frac{{\bar p(1 - \bar p)}}{n}} \)

stand only when (if and only if)

\(n\bar p - 3 \times \sqrt {n\bar p(1 - \bar p)} < n{\hat p_i} = {x_i} < n\bar p - 3 \times \sqrt {n\bar p(1 - \bar p)} \)

which proves the claim.

02

Step 2:Final proof

\({\rm{L C L = 0 ; U C L = 13}}{\rm{.25}}\)

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Most popular questions from this chapter

Construct a control chart for the data of Exercise 25 by using the transformation suggested in the text.

Let \(\alpha \) be a number between 0 and 1 , and define a sequence \({W_1},{W_2},{W_3}, \ldots \) by \({W_0} = \mu \)and \({W_t} = \alpha {\bar X_t} + (1 - \alpha ){W_{t - 1}}\) for \(t = 1,2, \ldots .\)Substituting for \({W_{t - 1}}\) its representation in terms of \({\bar X_{t - 1}}\) and \({W_{t - 2}}\), then substituting for \({W_{t - 2}}\), and so on, results in

\(\begin{array}{l}{W_t} = \alpha {{\bar X}_t} + \alpha (1 - \alpha ){{\bar X}_{t - 1}} + \ldots \\ + \alpha {(1 - \alpha )^{t - 1}}{{\bar X}_1} + {(1 - \alpha )^t}\mu \end{array}\)

The fact that \({W_t}\) depends not only on \({\bar X_t}\) but also on averages for past time points, albeit with (exponentially) decreasing weights, suggests that changes in the process mean will be more quickly reflected in the \({W_t}\) 's than in the individual \({\bar X_t}\) 's.

a. Show that \(E\left( {{W_t}} \right) = \mu \).

b. Let \(\sigma _t^2 = V\left( {{W_t}} \right)\), and show that

\(\sigma _t^2 = \frac{{\alpha \left( {1 - {{(1 - \alpha )}^{2\eta }}} \right)}}{{2 - \alpha }} \times \frac{{{\sigma ^2}}}{n}\)

c. An exponentially weighted moving-average control chart plots the \({W_t}\)'s and uses control limits \({\mu _0} \pm 3{\sigma _t}\) (or \(\bar \bar x\) in place of \({\mu _0}\) ). Construct such a chart for the data of Example 16.9, using \({\mu _0} = 40\).

A sample of 50 items is to be selected from a batch consisting of 5000 items. The batch will be accepted if the sample contains at most one defective item. Calculate the probability of lot acceptance for \(p = .01,.02, \ldots ,10\), and sketch the OC curve.

Refer to Exercise\(11\). An assignable cause was found for the unusually high sample average refractive index on day\(22\). Recompute control limits after deleting the data from this day. What do you conclude?

If a process variable is normally distributed, in the long run virtually all observed values should be between饾潄-3饾潏and饾潄+3饾潏, giving a process spread of 6饾潏.

a.With LSL and USL denoting the lower and upper specification limits, one commonly used process capability indexis Cp = (USL 鈥� LSL)/6饾潏. The value Cp= 1 indicates a process that is only marginally capable of meeting specifications. Ideally, Cp should exceed 1.33 (a 鈥渧ery good鈥� process). Calculate the value of Cp for each of the cork production processes described in the previous exercise, and comment.

b. The Cp index described in (a) does not take into account process location. A capability measure that does involve the process mean is Cpk = min {(饾潏USL -饾潄)/3饾潏, (饾潄鈥� LSL)/3饾潏} Calculate the value of Cpk for each of the cork production processes described in the previous exercise, and comment. (Note: In practice, m and s have to be estimated from process data; we show how to do this in Section 16.2)

c. How do Cp and Cpk compare, and when are they equal?
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