91影视

The following observations are on stopping distance (ft) of a particular truck at \(20mph\) under specified experimental conditions (鈥淓xperimental Measurement of the Stopping Performance of a Tractor-Semitrailer from Multiple Speeds,鈥� NHTSA, DOT HS 811 488, June 2011):

\(32.1 30.6 31.4 30.4 31.0 31.9\)

The cited report states that under these conditions, the maximum allowable stopping distance is \(30\). A normal probability plot validates the assumption that stopping distance is normally distributed.

a. Does the data suggest that true average stopping distance exceeds this maximum value? Test the appropriate hypotheses using \(\alpha = .01\).

b. Determine the probability of a type II error when a 5 .01, \(\sigma = .65\), and the actual value of \(\mu \) is \(31\). Repeat this for \(\mu = 32\) (use either statistical software or Table A.17).

c. Repeat (b) using \(\sigma = .80\) and compare to the results of (b).

d. What sample size would be necessary to have \(\alpha = .01\) and \(\beta = .10\) when \(\mu = 31\) and \(\sigma = .65\)?

The article 鈥淯ncertainty Estimation in Railway Track Life-Cycle Cost鈥� (J. of Rail and Rapid Transit, 2009) presented the following data on time to repair (min) a rail break in the high rail on a curved track of a certain railway line.

\(159 120 480 149 270 547 340 43 228 202 240 218\)

A normal probability plot of the data shows a reasonably linear pattern, so it is plausible that the population distribution of repair time is at least approximately normal. The sample mean and standard deviation are \(249.7\) and \(145.1\), respectively.

a. Is there compelling evidence for concluding that true average repair time exceeds \(200\) min? Carry out a test of hypotheses using a significance level of \(.05\).

b. Using \(\sigma = 150\), what is the type II error probability of the test used in (a) when true average repair time is actually \(300\) min? That is, what is \(\beta (300)\)?

Have you ever been frustrated because you could not get a container of some sort to release the last bit of its contents? The article 鈥淪hake, Rattle, and Squeeze: How Much Is Left in That Container?鈥� (Consumer Reports, May 2009: 8) reported on an investigation of this issue for various consumer products. Suppose five \(6.0oz\) tubes of toothpaste of a particular brand are randomly selected and squeezed until no more toothpaste will come out. Then each tube is cut open and the amount remaining is weighed, resulting in the following data (consistent with what the cited article reported):

\(.53,.65,.46,.50,.37.\)

Does it appear that the true average amount left is less than \(10\% \) of the advertised net contents?

a. Check the validity of any assumptions necessary for testing the appropriate hypotheses.

b. Carry out a test of the appropriate hypotheses using a significance level of \(.05\). Would your conclusion change if a significance level of \(.01\) had been used?

c. Describe in context type I and II errors, and say which error might have been made in reaching a conclusion.

The accompanying data on cube compressive strength (MPa) of concrete specimens appeared in the article 鈥淓xperimental Study of Recycled Rubber-Filled High-Strength Concrete鈥� (Magazine of Concrete Res., 2009: 549鈥�556):

\(\begin{array}{l}112.3 97.0 92.7 86.0 102.0\\99.2 95.8 103.5 89.0 86.7\end{array}\)

a. Is it plausible that the compressive strength for this type of concrete is normally distributed?

b. Suppose the concrete will be used for a particular application unless there is strong evidence that true average strength is less than \(100MPa\). Should the concrete be used? Carry out a test of appropriate hypotheses.

A random sample of soil specimens was obtained, and the amount of organic matter (%) in the soil was determined for each specimen, resulting in the accompanying data (from 鈥淓ngineering Properties of Soil,鈥� Soil Science, 1998: 93鈥�102).

\(\begin{array}{l}1.10 5.09 0.97 1.59 4.60 0.32 0.55 1.45\\0.14 4.47 1.20 3.50 5.02 4.67 5.22 2.69\\3.98 3.17 3.03 2.21 0.69 4.47 3.31 1.17\\0.76 1.17 1.57 2.62 1.66 2.05\end{array}\)

The values of the sample mean, sample standard deviation, and (estimated) standard error of the mean are \(2.481,1.616,\) and \(.295,\) respectively. Does this data suggest that the true average percentage of organic matter in such soil is something other than \(3\% \)? Carry out a test of the appropriate hypotheses at significance level \(.10\). Would your conclusion be different if a \(\alpha = .05\) had been used? (Note: A normal probability plot of the data shows an acceptable pattern in light of the reasonably large sample size.)

Reconsider the accompanying sample data on expense ratio (%) for large-cap growth mutual funds first introduced in Exercise 1.53.

\(\begin{array}{l}0.52 1.06 1.26 2.17 1.55 0.99 1.10 1.07 1.81 2.05\\0.91 0.79 1.39 0.62 1.52 1.02 1.10 1.78 1.01 1.15\end{array}\)

A normal probability plot shows a reasonably linear pattern.

a. Is there compelling evidence for concluding that the population mean expense ratio exceeds \(1\% \)? Carry out a test of the relevant hypotheses using a significance level of \(.01\).

b. Referring back to (a), describe in context type I and II errors and say which error you might have made in reaching your conclusion. The source from which the data was obtained reported that \(\mu = 1.33\) for the population of all \(762\) such funds. So, did you actually commit an error in reaching your conclusion?

c. Supposing that \(\sigma = .5\), determine and interpret the power of the test in (a) for the actual value of m stated in (b).

For which of the given P-values would the null hypothesis be rejected when performing a level .05 test?

a. .001 b. .021 c. .078

d..047 e. .148

A spectrophotometer used for measuring CO concentration (ppm (parts per million) by volume) is checked for accuracy by taking readings on a manufactured gas (called span gas) in which the CO concentration is very precisely controlled at \(70ppm\). If the readings suggest that the spectrophotometer is not working properly, it will have to be recalibrated. Assume that if it is properly calibrated, measured concentration for span gas samples is normally distributed. On the basis of the six readings \(85,{\rm{ }}77,{\rm{ }}82,{\rm{ }}68,{\rm{ }}72,\) and \(69\) is recalibration necessary? Carry out a test of the relevant hypotheses using a \(\alpha = .05\).

A common characterization of obese individuals is that their body mass index is at least \(30\) (BMI 5 weighty(height)², where height is in meters and weight is in kilograms). The article 鈥淭he Impact of Obesity on Illness Absence and Productivity in an Industrial Population of Petrochemical Workers鈥� (Annals of Epidemiology, 2008: 8鈥�14) reported that in a sample of female workers, \(262\) had BMIs of less than \(25,159\) had BMIs that were at least \(25\) but less than \(30\), and \(120\) had BMIs exceeding \(30\). Is there compelling evidence for concluding that more than \(20\% \) of the individuals in the sampled population are obese? a. State and test appropriate hypotheses with a significance level of \(.05\). b. Explain in the context of this scenario what constitutes type I and II errors. c. What is the probability of not concluding that more than \(20\% \) of the population is obese when the actual percentage of obese individuals is \(25\% \)?

A manufacturer of nickel-hydrogen batteries randomly selects \(100\) nickel plates for test cells, cycles them a specified number of times, and determines that \(14\) of the plates have blistered.

a. Does this provide compelling evidence for concluding that more than \(10\% \) of all plates blister under such circumstances? State and test the appropriate hypotheses using a significance level of \(.05\). In reaching your conclusion, what type of error might you have committed?

b. If it is really the case that \(15\% \) of all plates blister under these circumstances and a sample size of \(100\) is used, how likely is it that the null hypothesis of part (a) will not be rejected by the level \(.05\) test? Answer this question for a sample size of 200.

c. How many plates would have to be tested to have \(\beta (.15) = 10\) for the test of part (a)?