91影视

An experiment was performed to compare the fracture toughness of high-purity \(18Ni\) maraging steel with commercial-purity steel of the same type (Corrosion Science, 1971: 723鈥�736). For \(m = 32\)specimens, the sample average toughness was \(\overline x = 65.6\) for the high purity steel, whereas for \(n = 38\)specimens of commercial steel \(\overline y = 59.8\). Because the high-purity steel is more expensive, its use for a certain application can be justified only if its fracture toughness exceeds that of commercial purity steel by more than 5. Suppose that both toughness distributions are normal.

a. Assuming that \({\sigma _1} = 1.2\) and \({\sigma _2} = 1.1\), test the relevant hypotheses using \(\alpha = .001\).

b. Compute \(\beta \) for the test conducted in part (a) when \({\mu _1} - {\mu _2} = 6.\)

The level of lead in the blood was determined for a sample of \(152\) male hazardous-waste workers ages \(20 - 30\) and also for a sample of \(86\) female workers, resulting in a mean \(6\) standard error of \(5.5 \pm 0.3\)for the men and \(3.8 \pm 0.2\) for the women (鈥淭emporal Changes in Blood Lead Levels of Hazardous Waste Workers in New Jersey, 1984鈥�1987,鈥� Environ. Monitoring and Assessment, 1993: 99鈥�107). Calculate an estimate of the difference between true average blood lead levels for male and female workers in a way that provides information about reliability and precision.

The accompanying summary data on total cholesterol level (mmol/l) was obtained from a sample of Asian postmenopausal women who were vegans and another sample of such women who were omnivores (鈥淰egetarianism, Bone Loss, and Vitamin D: A Longitudinal Study in Asian Vegans and Non-Vegans,鈥� European J. of Clinical Nutr., 2012: 75鈥�82)

Diet sample sample sample

Size mean SD

\(\overline {\underline {\begin{array}{*{20}{l}}{ Vegan }&{88}&{5.10}&{1.07}\\{ Omnivore }&{93}&{5.55}&{1.10}\\{}&{}&{}&{}\end{array}} } \)

Calculate and interpret a \(99\% \) \(CI\) for the difference between population mean total cholesterol level for vegans and population mean total cholesterol level for omnivores (the cited article included a \(95\% \)\(CI\)). (Note: The article described a more sophisticated statistical analysis for investigating bone density loss taking into account other characteristics (鈥渃ovariates鈥�) such as age, body weight, and various nutritional factors; the resulting CI included 0, suggesting no diet effect.

The level of monoamine oxidase (MAO) activity in blood platelets (nm/mg protein/h) was determined for each individual in a sample of \(43\) chronic schizophrenics, resulting in \(\bar x = 2.69\) and \({s_1} = 2.30,\), as well as for \(45\) normal subjects, resulting in \(\bar y = 6.35\) and \({s_2} = 4.03.\). Does this data strongly suggest that true average MAO activity for normal subjects is more than twice the activity level for schizophrenics? Derive a test procedure and carry out the test using \(\alpha = .01\)

. (Hint: \({H_0}\) and \({H_a}\) here have a different form from the three standard cases. Let \({\mu _1}\) and \({\mu _2}\) refer to true average MAO activity for schizophrenics and normal subjects, respectively, and consider the parameter \(\theta = 2{\mu _1} - {\mu _2}\). Write \({H_0}\) and \({H_a}\) in terms of \(\theta \), estimate \(\theta \), and derive \({\hat \sigma _{\tilde \theta }}\) (鈥淩educed Monoamine Oxidase Activity in Blood Platelets from Schizophrenic Patients,鈥� Nature, July 28, 1972: 225鈥�226).) \(\alpha = .01\)

a. Show for the upper-tailed test with \({\sigma _1}\) and \({\sigma _2}\)known that as either\(m\) or\(n\) increases, \(\beta \)decreases when \({\mu _1} - {\mu _2} > {\Delta _0}\).

b. For the case of equal sample sizes \(\left( {m = n} \right)\)and fixed \(\alpha \),what happens to the necessary sample size \(n\) as \(\beta \) is decreased, where \(\beta \) is the desired type II error probability at a fixed alternative?

To decide whether two different types of steel have the same true average fracture toughness values, n specimens of each type are tested, yielding the following results:

\(\begin{array}{l}\underline {\begin{array}{*{20}{c}}{ Type }&{ Sample Average }&{ Sample SD }\\1&{60.1}&{1.0}\\2&{59.9}&{1.0}\\{}&{}&{}\end{array}} \\\end{array}\)

Calculate the P-value for the appropriate two-sample \(z\) test, assuming that the data was based on \(n = 100\). Then repeat the calculation for \(n = 400\). Is the small P-value for \(n = 400\) indicative of a difference that has practical significance? Would you have been satisfied with just a report of the P-value? Comment briefly.a

Which way of dispensing champagne, the traditional vertical method or a tilted beer-like pour,preserves more of the tiny gas bubbles that improve flavor and aroma? The following data was reported in the article 鈥淥n the Losses of Dissolved \(C{O_2}\) during Champagne Serving鈥� (J. Agr. Food Chem., 2010: 8768鈥�8775)

\(\begin{array}{*{20}{c}}{ Temp \left( {^^\circ C} \right)}&{ Type of Pour }&n&{ Mean (g/L)}&{ SD }\\{18}&{ Traditional }&4&{4.0}&{.5}\\{18}&{ Slanted }&4&{3.7}&{.3}\\{12}&{ Traditional }&4&{3.3}&{.2}\\{12}&{ Slanted }&4&{2.0}&{.3}\\{}&{}&{}&{}&{}\end{array}\)

Assume that the sampled distributions are normal.

a. Carry out a test at significance level \(.01\) to decide whether true average\(C{O_2}\)loss at \(1{8^o}C\) for the traditional pour differs from that for the slanted pour.

b. Repeat the test of hypotheses suggested in (a) for the \(1{2^o}\) temperature. Is the conclusion different from that for the \(1{8^o}\) temperature? Note: The \(1{2^o}\) result was reported in the popular media

Suppose \({\mu _1}\) and \({\mu _2}\) are true mean stopping distances at \(50mph\) for cars of a certain type equipped with two different types of braking systems. Use the two-sample t test at significance level

t test at significance level \(.01\) to test \({H_0}:{\mu _1} - {\mu _2} = - 10\) versus \({H_a}:{\mu _1} - {\mu _2} < - 10\) for the following data: \(m = 6,\;\;\;\bar x = 115.7,{s_1} = 5.03,n = 6,\bar y = 129.3,\;\)and \({s_2} = 5.38.\)

1. An article in the November \(1983\) Consumer Reports compared various types of batteries. The average lifetimes of Duracell Alkaline \(AA\)batteries and Eveready Energizer Alkaline \(AA\) batteries were given as \(4.1\) hours and \(4.5\) hours, respectively. Suppose these are the population average lifetimes.

a. Let \(\bar X\) be the sample average lifetime of \(100\) Duracell batteries and \(\bar Y\) be the sample average lifetime of \(100\) Eveready batteries. What is the mean value of \(\bar X - \bar Y\) (i.e., where is the distribution of \({\bf{\bar X - \bar Y}}\) centered)? How does your answer depend on the specified sample sizes?

b. Suppose the population standard deviations of lifetime are \(1.8\) hours for Duracell batteries and \(2.0\) hours for Eveready batteries. With the sample sizes given in part (a), what is the variance of the statistic \(\bar X - \bar Y\), and what is its standard deviation?

c. For the sample sizes given in part (a), draw a picture of the approximate distribution curve of \(\bar X - \bar Y\) (include a measurement scale on the horizontal axis). Would the shape of the curve necessarily be the same for sample sizes of \(10\) batteries of each type? Explain

Quantitative noninvasive techniques are needed for routinely assessing symptoms of peripheral neuropathies, such as carpal tunnel syndrome (CTS). The article "A Gap Detection Tactility Test for Sensory Deficits Associated with Carpal Tunnel Syndrome" (Ergonomics, \(1995: 2588 - 2601\)) reported on a test that involved sensing a tiny gap in an otherwise smooth surface by probing with a finger; this functionally resembles many work-related tactile activities, such as detecting scratches or surface defects. When finger probing was not allowed, the sample average gap detection threshold for\(m = 8\)normal subjects was\(1.71\;mm\), and the sample standard deviation was\(.53\); for\(n = 10\)CTS subjects, the sample mean and sample standard deviation were\(2.53\)and\(.87\), respectively. Does this data suggest that the true average gap detection threshold for CTS subjects exceeds that for normal subjects? State and test the relevant hypotheses using a significance level of\(.01\).