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Chapter 7: Q45E (page 306)

Wire electrical-discharge machining (WEDM) is a process used to manufacture conductive hard metal components. It uses a continuously moving wire that serves as an electrode. Coating on the wire electrode allows for

cooling of the wire electrode core and provides an improved cutting performance. The article 鈥淗ighPerformance Wire Electrodes for Wire ElectricalDischarge Machining鈥擜 Review鈥� gave the following sample observations on total coating layer thickness (in mm) of eight wire electrodes used for

WEDM: \({\rm{21 16 29 35 42 24 24 25}}\)

Calculate a \({\rm{99\% }}\)CI for the standard deviation of the coating layer thickness distribution. Is this interval valid whatever the nature of the distribution? Explain.

Short Answer

Expert verified

The boundaries of the confidence interval for the standard deviation is \((4.8248,21.8461)\)

No, the population distribution is a normal distribution

Step by step solution

01

To Calculate a \({\rm{99\% }}\)CI for the standard deviation

Given:

\({\rm{c = 99\% = 0}}{\rm{.99}}\)

\(2116{\rm{ }}29{\rm{ }}35{\rm{ }}42{\rm{ }}24{\rm{ }}24{\rm{ }}25\)

The mean is the sum of all values divided by the number of values:

\({\rm{\bar x = }}\frac{{{\rm{21 + 16 + 29 + 35 + 42 + 24 + 24 + 25}}}}{{\rm{8}}}{\rm{ = }}\frac{{{\rm{216}}}}{{\rm{8}}}{\rm{ = 27}}\)

The variance is the sum of squared deviations from the mean divided by \({\rm{n - 1}}\).The standard deviation is the square root of the variance:

\({\rm{s = }}\sqrt {\frac{{{{{\rm{(21 - 27)}}}^{\rm{2}}}{\rm{ + \ldots }}{\rm{. + (25 - 27}}{{\rm{)}}^{\rm{2}}}}}{{{\rm{8 - 1}}}}} {\rm{\gg 8}}{\rm{.2115}}\)

02

To Determine the critical values

Determine the critical values using the chi-square table in the appendix, which are given in the row \({\rm{df = n - 1 = 8 - 1 = 7}}\)and in the columns of \(\frac{{{\rm{1 - c}}}}{{\rm{2}}}{\rm{ = 0}}{\rm{.005 and 1 - }}\frac{{{\rm{1 - c}}}}{{\rm{2}}}{\rm{ = 0}}{\rm{.995}}\) :

\(\begin{array}{l}{\rm{\chi }}_{{\rm{1 - 0}}{\rm{.005}}}^{\rm{2}}{\rm{ = \chi }}_{{\rm{0}}{\rm{.995}}}^{\rm{2}}{\rm{ = 0}}{\rm{.989 }}\\{\rm{\chi }}_{{\rm{0}}{\rm{.005}}}^{\rm{2}}{\rm{ = 20}}{\rm{.276}}\end{array}\)

The boundaries of the confidence interval for the standard deviation are then:

\(\begin{array}{l}\sqrt {\frac{{{\rm{n - 1}}}}{{{\rm{\chi }}_{{\rm{\alpha /2}}}^{\rm{2}}}}} {\rm{ \times s = }}\sqrt {\frac{{{\rm{8 - 1}}}}{{{\rm{20}}{\rm{.276}}}}} {\rm{ \times 8}}{\rm{.2115\gg 4}}{\rm{.8248}}\\\sqrt {\frac{{{\rm{n - 1}}}}{{{\rm{\chi }}_{{\rm{1 - \alpha /2}}}^{\rm{2}}}}} {\rm{ \times s = }}\sqrt {\frac{{{\rm{8 - 1}}}}{{{\rm{0}}{\rm{.989}}}}} {\rm{ \times 8}}{\rm{.2115\gg 21}}{\rm{.8461}}\end{array}\)

This interval is NOT valid whatever the nature of the distribution, because we require that the population distribution is a normal distribution.

Hence the boundaries of the confidence interval for the standard deviation is \((4.8248,21.8461)\)

No, the population distribution is a normal distribution

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Most popular questions from this chapter

TV advertising agencies face increasing challenges in reaching audience members because viewing TV programs via digital streaming is gaining in popularity. The Harris poll reported on November 13, 2012, that 53% of 2343 American adults surveyed said they have watched digitally streamed TV programming on some type of device.

a. Calculate and interpret a confidence interval at the 99% confidence level for the proportion of all adult Americans who watched streamed programming up to that point in time.

b. What sample size would be required for the width of a 99% CI to be at most .05 irrespective of the value of p 藛?

Example 1.11 introduced the accompanying observations on bond strength.

11.5 12.1 9.9 9.3 7.8 6.2 6.6 7.0

13.4 17.1 9.3 5.6 5.7 5.4 5.2 5.1

4.9 10.7 15.2 8.5 4.2 4.0 3.9 3.8

3.6 3.4 20.6 25.5 13.8 12.6 13.1 8.9

8.2 10.7 14.2 7.6 5.2 5.5 5.1 5.0

5.2 4.8 4.1 3.8 3.7 3.6 3.6 3.6

a.Estimate true average bond strength in a way that conveys information about precision and reliability.

(Hint: \(\sum {{{\bf{x}}_{\bf{i}}}} {\bf{ = 387}}{\bf{.8}}\) and \(\sum {{{\bf{x}}^{\bf{2}}}_{\bf{i}}} {\bf{ = 4247}}{\bf{.08}}\).)

b. Calculate a 95% CI for the proportion of all such bonds whose strength values would exceed 10.

A sample of 14 joint specimens of a particular type gave a sample mean proportional limit stress of \({\rm{8}}{\rm{.48}}\)MPa and a sample standard deviation of . \({\rm{79}}\)MPa a. Calculate and interpret a \({\rm{95\% }}\)lower confidence bound for the true average proportional limit stress of all such joints. What, if any, assumptions did you make about the distribution of proportional limit stress?

b. Calculate and interpret a \({\rm{95\% }}\)lower prediction bound for the proportional limit stress of a single joint of this type.

A state legislator wishes to survey residents of her district to see what proportion of the electorate is aware of her position on using state funds to pay for abortions.

a. What sample size is necessary if the \({\rm{95\% }}\)CI for p is to have a width of at most . \({\rm{10}}\)irrespective of p?

b. If the legislator has strong reason to believe that at least \({\rm{2/3}}\)of the electorate know of her position, how large a sample size would you recommend?

The alternating current (AC) breakdown voltage of an insulating liquid indicates its dielectric strength. The article 鈥淭esting Practices for the AC Breakdown Voltage Testing of Insulation Liquids'' (IEEE)gave the accompanying sample observations on breakdown voltage (kV) of a particular circuit under certain conditions.

\(\begin{array}{l}{\rm{62\;50\;53\;57\;41\;53\;55\;61\;59\;64\;50\;53\;64\;62}}\\{\rm{\;50\;68 54\;55\;57\;50\;55\;50\;56\;55\;46\;55\;53\;54\;}}\\{\rm{52\;47\;47\;55 57\;48\;63\;57\;57\;55\;53\;59\;53\;52\;}}\\{\rm{50\;55\;60\;50\;56\;58}}\end{array}\)

a. Construct a boxplot of the data and comment on interesting features.

b. Calculate and interpret a \({\rm{95\% }}\)CI for true average breakdown voltage m. Does it appear that m has been precisely estimated? Explain.

c. Suppose the investigator believes that virtually all values of breakdown voltage are between \({\rm{40 and 70}}\). What sample size would be appropriate for the \({\rm{95\% }}\)CI to have a width of 2 kV (so that m is estimated to within 1 kV with \({\rm{95\% }}\)confidence)
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