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Chapter 16: Q26E (page 700)

For what\(\bar x\)values will the LCL in a\(c\)chart be negative?

Short Answer

Expert verified

\(\bar x < 9\)

Step by step solution

01

Find the value of \(\bar x\)

The \(c\) chart for the number of defectives in a unit has lower and upper control limits given by

\(\begin{aligned}{l}LCL = \bar x - 3\sqrt {\bar x} \\UCL = \bar x + 3\sqrt {\bar x} \end{aligned}\)

and the center line is\(\bar x\). When \(LCL \le 0\) it is set to zero.

From

\(LCL = \bar x - 3\sqrt {\bar x} \)

it follows that

\(\bar x - 3\sqrt {\bar x} < 0\)

if and only if

\({\bar x^2} < 9\bar x\)

which results in

\(\bar x < 9\)

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Most popular questions from this chapter

Resistance observations (ohms) for subgroups of a certain type of register gave the following summary quantities:

\(\begin{array}{*{20}{c}}i&{{n_i}}&{{{\overline x }_i}}&{{s_i}}&i&{{n_i}}&{{{\overline x }_i}}&{{s_i}}\\1&4&{430.0}&{22.5}&{11}&4&{445.2}&{27.3}\\2&4&{418.2}&{20.6}&{12}&4&{430.1}&{22.2}\\3&3&{435.5}&{25.1}&{13}&4&{427.2}&{24.0}\\4&4&{427.6}&{22.3}&{14}&4&{439.6}&{23.3}\\5&4&{444.0}&{21.5}&{15}&3&{415.9}&{31.2}\\6&3&{431.4}&{28.9}&{16}&4&{419.8}&{27.5}\\7&4&{420.8}&{25.4}&{17}&3&{447.0}&{19.8}\\8&4&{431.4}&{24.0}&{18}&4&{434.4}&{23.7}\\9&4&{428.7}&{21.2}&{19}&4&{422.2}&{25.1}\\{10}&4&{440.1}&{25.8}&{20}&4&{425.7}&{24.4}\\{}&{}&{}&{}&{}&{}&{}&{}\end{array}\)

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.

Consider the double-sampling plan for which both sample sizes are 50 . The lot is accepted after the first sample if the number of defectives is at most 1 , rejected if the number of defectives is at least 4 , and rejected after the second sample if the total number of defectives is 6 or more. Calculate the probability of accepting the lot when \(p = .01,.05\), and \(.10\).

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?

Consider a\(3\)-sigma control chart with a center line at\({\mu _0}\)and based on\(n = 5\). Assuming normality, calculate the probability that a single point will fall outside the control limits when the actual process mean is

a. \({\mu _0} + .5\sigma \)

b. \({\mu _0} - \sigma \)

c.\({\mu _0} + 2\sigma \)
See all solutions

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