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Chapter 8: Q151E (page 452)

Gouges on a spindle. A tool-and-die machine shop produces extremely high-tolerance spindles. The spindles are 18-inch slender rods used in a variety of military equipment. A piece of equipment used in the manufacture of the spindles malfunctions on occasion and places a single gouge somewhere on the spindle. However, if the spindle can be cut so that it has 14 consecutive inches without a gouge, then the spindle can be salvaged for other purposes. Assuming that the location of the gouge along the spindle is random, what is the probability that a defective spindle can be salvaged?

Short Answer

Expert verified

The probability that a defective spindle can be salvaged is 0.4444.

Step by step solution

01

Given information

The spindle is an 18-inches slender rod. The defective spindle can be cut so that there are 14 consecutive inches without the gouge.

Let X represents the location of the gouge on the slender rod. Therefore, X is uniformly distributed between 0 inches and 18 inches.

02

Computing the probability of the defective spindle

The probability density function of X is:

fx=118;0⩽x⩽18.

One can 14 consecutive inches without the gouge if it is present within 4 inches of either side on the slender rod.

The probability that a defective spindle can be salvaged is obtained as:

PX<4+PX>14=P0<X<4+P14<X<18=4-018+18-1418=418+418=818=0.4444.

For the uniform distribution: P(a<X<b)=b-ad-c;c⩽a<b⩽d.

Thus, the required probability is 0.4444.

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Most popular questions from this chapter

Drug content assessment. Scientists at GlaxoSmithKlineMedicines Research Center used high-performance liquidchromatography (HPLC) to determine the amountof drug in a tablet produced by the company (Analytical

Chemistry, Dec. 15, 2009). Drug concentrations (measuredas a percentage) for 50 randomly selected tablets are listedin the table below and saved in the accompanying file.

a. Descriptive statistics for the drug concentrations areshown at the top of the XLSTAT printout on the nextpage. Use this information to assess whether the dataare approximately normal.

b. An XLSTAT normal probability plot follows. Use thisinformation to assess whether the data are approximatelynormal.

91.28 92.83 89.35 91.90 82.85 94.83 89.83 89.00 84.62

86.96 88.32 91.17 83.86 89.74 92.24 92.59 84.21 89.36

90.96 92.85 89.39 89.82 89.91 92.16 88.67 89.35 86.51

89.04 91.82 93.02 88.32 88.76 89.26 90.36 87.16 91.74

86.12 92.10 83.33 87.61 88.20 92.78 86.35 93.84 91.20

93.44 86.77 83.77 93.19 81.79

Descriptive statistics(Quantitative data)

Statistic

Content

Nbr.of Observation

50

Minimum

81.79

Maximum

94.83

1st Quartile

87.2725

Median

89.375

3rd Quartile

91.88

Mean

89.2906

Variance(n-1)

10.1343

Standard deviation(n-1)

3.1834

Traffic sign maintenance. Refer to the Journal of Transportation Engineering (June 2013) study of traffic sign maintenance in North Carolina, Exercise 8.54 (p. 489). Recall that the proportion of signs on NCDOT-maintained roads that fail minimum requirements was compared to the corresponding proportion for signs on county-owned roads. How many signs should be sampled from each maintainer to estimate the difference between the proportions to within .03 using a 90% confidence interval? Assume the same number of signs will be sampled from NCDOT-maintained roads and county-owned roads

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Find the following probabilities for the standard normal random variable z:

a.P(0<z<2.25)b.P(-2.25<z<0)b.P(-2.25<z<1.25)d.P(-2.50<z<1.50)e.P(z<-2.33orz>2.33)

Corporate sustainability of CPA firms. Refer to the Business and Society (March 2011) study on the sustainability behaviors of CPA corporations, Exercise 2.23 (p. 83). Recall that the level of support for corporate sustainability (measured on a quantitative scale ranging from 0 to 160 points) was obtained for each of 992 senior managers at CPA firms. The accompanying Minitab printout gives the mean and standard deviation for the level of support variable. It can be shown that level of support is approximately normally distributed.

a. Find the probability that the level of support for corporate sustainability of a randomly selected senior manager is less than 40 points.

b. Find the probability that the level of support for corporate sustainability of a randomly selected senior manager is between 40 and 120 points.

c. Find the probability that the level of support for corporate sustainability of a randomly selected senior manager is greater than 120 points.

d. One-fourth of the 992 senior managers indicated a level of support for corporate sustainability below what value?

Descriptive Statistics: Support

Variables

N

Mean

StDev

Variance

Minimum

Maximum

Range

Support

992

67.755

26.871

722.036

0.000

155.000

155.000
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