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Chapter 9: Q46 E (page 391)

Example\(7.11\)gave data on the modulus of elasticity obtained\(1\)minute after loading in a certain configuration. The cited article also gave the values of modulus of elasticity obtained\(4\)weeks after loading for the same lumber specimens. The data is presented here. \(1\begin{array}{*{20}{c}}{Observation}&{1 min}&{4 weeks}&{Difference}\\1&{10,490}&{9,110}&{1380}\\2&{16,620}&{13,250}&{3370}\\3&{17,300}&{14,720}&{2580}\\4&{15,480}&{12,740}&{2740}\\5&{12,970}&{10,120}&{2850}\\6&{17,260}&{14,570}&{2690}\\7&{13,400}&{11,220}&{2180}\\8&{13,900}&{11,100}&{2800}\\9&{13,630}&{11,420}&{2210}\\{10}&{13,260}&{10,910}&{2350}\\{11}&{14,370}&{12,110}&{2260}\\{12}&{11,700}&{8,620}&{3080}\\{13}&{15,470}&{12,590}&{2880}\\{14}&{17,840}&{15,090}&{2750}\\{15}&{14,070}&{10,550}&{3520}\\{16}&{14,760}&{12,230}&{2530}\end{array}\)

Calculate and interpret an upper confidence bound for the true average difference between\(1\)-minute modulus and\(4\)-week modulus; first check the plausibility of any necessary assumptions.

Short Answer

Expert verified

\(95\% \)upper confidence bound: \(2858.5386\)

Step by step solution

01

Determine the upper confidence bound

Given:

\(n = 16\)

I will calculate a\(95\% \)upper confidence bound. Other confidence bounds can be obtained similarly.

\(c = 95\% = 0.95\)Determine the sample mean of the differences. The mean is the sum of all values divided by the number of values.

\(\bar d = \frac{{1380 + 3370 + 2580 + \ldots + 2750 + 3520 + 2530}}{{16}} \approx 2635.625\)

Determine the sample standard deviation of the differences:

\({s_d} = \sqrt {\frac{{{{(1380 - 2635.625)}^2} + \ldots + {{(2530 - 2635.625)}^2}}}{{16 - 1}}} \approx 508.6448\)

Determine the\({t_\alpha }\)using the Student's T distribution table in the appendix with\(df = n - 1 = 16 - 1 = 15\):

\({t_\alpha } = {t_{1 - c}} = {t_{0.05}} = 1.753\)

The margin of error is then:

\(E = {t_{\alpha /2}} \cdot \frac{{{s_d}}}{{\sqrt n }} = 1.753 \cdot \frac{{508.6448}}{{\sqrt {16} }} \approx 222.9136\)

The upper confidence bound for\({\mu _d}\)is then:

\(\bar d + E = 2635.625 + 222.9136 = 2858.5386\)

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Most popular questions from this chapter

The invasive diatom species Didymosphenia geminata has the potential to inflict substantial ecological and economic damage in rivers. The article "Substrate Characteristics Affect Colonization by the BloomForming Diatom Didymosphenia geminata" (Acquatic Ecology, 2010:\(33 - 40\)) described an investigation of colonization behavior. One aspect of particular interest was whether the roughness of stones impacted the degree of colonization. The authors of the cited article kindly provided the accompanying data on roughness ratio (dimensionless) for specimens of sandstone and shale.

\(\begin{array}{*{20}{l}}{ Sandstone: }&{5.74}&{2.07}&{3.29}&{0.75}&{1.23}\\{}&{2.95}&{1.58}&{1.83}&{1.61}&{1.12}\\{}&{2.91}&{3.22}&{2.84}&{1.97}&{2.48}\\{}&{3.45}&{2.17}&{0.77}&{1.44}&{3.79}\end{array}\)

\(\begin{array}{*{20}{l}}{ Shale: }&{.56}&{.84}&{.40}&{.55}&{.36}&{.72}\\{}&{.29}&{.47}&{.66}&{.48}&{.28}&{}\\{}&{.72}&{.31}&{.35}&{.32}&{.37}&{.43}\\{}&{.60}&{.54}&{.43}&{.51}&{}&{}\end{array}\)

Normal probability plots of both samples show a reasonably linear pattern. Estimate the difference between true average roughness for sandstone and that for shale in a way that provides information about reliability and precision, and interpret your estimate. Does it appear that true average roughness differs for the two types of rocks (a formal test of this was reported in the article)? (Note: The investigators concluded that more diatoms colonized the rougher surface than the smoother surface.)

Is there any systematic tendency for part-time college faculty to hold their students to different standards than do full-time faculty? The article “Are There Instructional Differences Between Full-Time and Part-Time Faculty?” (College Teaching, \(2009: 23 - 26)\) reported that for a sample of \(125 \) courses taught by fulltime faculty, the mean course \(GPA\) was \(2.7186\) and the standard deviation was \(.63342\), whereas for a sample of \(88\) courses taught by part-timers, the mean and standard deviation were \(2.8639\) and \(.49241,\) respectively. Does it appear that true average course \(GPA\) for part-time faculty differs from that for faculty teaching full-time? Test the appropriate hypotheses at significance level \( .01\).

Researchers sent 5000 resumes in response to job ads that appeared in the Boston Globe and Chicago Tribune. The resumes were identical except that 2500 of them had "white sounding" first names, such as Brett and Emily, whereas the other 2500 had "black sounding" names such as Tamika and Rasheed. The resumes of the first type elicited 250 responses and the resumes of the second type only 167 responses (these numbers are very consistent with information that appeared in a Jan. 15. 2003, report by the Associated Press). Does this data strongly suggest that a resume with a "black" name is less likely to result in a response than is a resume with a "white" name?

Consider the accompanying data on breaking load (\(kg/25\;mm\)width) for various fabrics in both an unabraded condition and an abraded condition (\(^6\)The Effect of Wet Abrasive Wear on the Tensile Properties of Cotton and Polyester-Cotton Fabrics," J. Testing and Evaluation, \(\;1993:84 - 93\)). Use the paired\(t\)test, as did the authors of the cited article, to test\({H_0}:{\mu _D} = 0\)versus\({H_a}:{\mu _D} > 0\)at significance level . \(01\)

Lactation promotes a temporary loss of bone mass to provide adequate amounts of calcium for milk production. The paper "Bone Mass Is Recovered from Lactation to Postweaning in Adolescent Mothers with Low Calcium Intakes" (Amer. J. of Clinical Nutr., 2004: 1322-1326) gave the following data on total body bone mineral content (TBBMC) (g) for a sample both during lactation (L) and in the postweaning period (P).

\(\begin{array}{*{20}{c}}{Subject}&{}&{}&{}&{}&{}&{}&{}&{}&{}\\1&2&3&4&5&6&7&8&9&{10}\\{1928}&{2549}&{2825}&{1924}&{1628}&{2175}&{2114}&{2621}&{1843}&{2541}\\{2126}&{2885}&{2895}&{1942}&{1750}&{2184}&{2164}&{2626}&{2006}&{2627}\end{array}\)

a. Does the data suggest that true average total body bone mineral content during postweaning exceeds that during lactation by more than\(25\;g\)? State and test the appropriate hypotheses using a significance level of .05. (Note: The appropriate normal probability plot shows some curvature but not enough to cast substantial doubt on a normality assumption.)

b. Calculate an upper confidence bound using a\(95\% \)confidence level for the true average difference between TBBMC during postweaning and during lactation.

c. Does the (incorrect) use of the two-sample\(t\)test to test the hypotheses suggested in (a) lead to the same
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