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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猢�x猢�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猢�a<b猢�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.

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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

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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

The 鈥渓ast name鈥� effect in purchasing. The Journal of Consumer Research (August 2011) published a study demonstrating the 鈥渓ast name鈥� effect鈥攊.e., the tendency for consumers with last names that begin with a later letter of the alphabet to purchase an item before consumers with last names that begin with earlier letters. To facilitate the analysis, the researchers assigned a number, x, to each consumer based on the first letter of the consumer鈥檚 last name. For example, last names beginning with 鈥淎鈥� were assigned x = 1; last names beginning with 鈥淏鈥� were assigned x = 2; and last names beginning with 鈥淶鈥� were assigned x = 26.

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The gender diversity of a large corporation鈥檚 board of directors was studied in Accounting & Finance (December 2015). In particular, the researchers wanted to know whether firms with a nominating committee would appoint more female directors than firms without a nominating committee. One of the key variables measured at each corporation was the percentage of female board directors. In a sample of 491firms with a nominating committee, the mean percentage was 7.5%; in an independent sample of 501firms without a nominating committee, the mean percentage was role="math" localid="1652702402701" 4.3% .

a. To answer the research question, the researchers compared the mean percentage of female board directors at firms with a nominating committee with the corresponding percentage at firms without a nominating committee using an independent samples test. Set up the null and alternative hypotheses for this test.

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What is the line of means?

Independent random samples from normal populations produced the results shown in the next table.

Sample 1


Sample 2

1.23.11.72.83.0

 4.22.73.63.9

a. Calculate the pooled estimate of 蟽2.

b. Do the data provide sufficient evidence to indicate that 渭2&驳迟;渭1? Test using 伪=.10.

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d. Which of the two inferential procedures, the test of hypothesis in part b or the confidence interval in part c, provides more information about (渭1鈭�渭2)?
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