christian-mann · April 1, 2012 02:25
diff --git a/hw7.tex b/hw7.tex
 \documentclass{exam}

 \usepackage{listings}
 \begin{document}
 	\title{Homework Set 7}
 	\author{}
 	\date{}
 	\maketitle
 	
 	\begin{questions}
 	\question \textbf{8-10} A mixture of pulverized fuel ash and Portland cement to be used for grouting should have a compressive strength of more than 1300 KN/m$^2$. The mixture will not be used unless experimental evidence indicated conclusively that the strength specification has been met. Suppose compressive strength for specimens of this misture is normally distributed with $\sigma = 60$. Let $\mu$ denote the true average compressive strength. \begin{parts}
 		\part What are the appropriate null and alternative hypotheses?
 		\part Let $\bar{X}$ denote the sample average compressive strength for $n = 10$ randomly selected specimens. Consider the test procedure with test statistic $\bar{X}$ and rejection region $\bar{x} \geq 1331.26$. What is the probability distribution of the test statistic when $H_0$ is true? What is the probability of a type I error for the test procedure?
 		\part What ist he probability distribution of the test statistic when $\mu = 1350?$ Using the test procedure of part (b), what is the probability that the misture will be judged unsatisfactory when in fact $\mu = 1350$ (a type II error)?
 		\part How would you change the test procedure of part (b) to obtain a test with significance level .05? What impact would this change have on the error probability of part (c)?
 		\part Consider the standardized test statistic $Z = (\bar{X} - 1300)/(\sigma/\sqrt{n}) = (\bar{X} - 1300)/13.42$. What are the values of $Z$ corresponding to the rejection region of part (b)?
 	\end{parts}
 	\question \textbf{8-14} Reconsider the situation of Exercise 11 and suppose the rejection region is $\{\bar{x}: \bar{x} \geq 10.1004 \textrm{or} \bar{x} \leq 9.98940\} = \{z: z \geq 2.51 \textrm{or} z \leq -2.65\}$. \begin{parts}
 		\part What is $\alpha$ for this procedure?
 		\part What is $\beta$ when $\mu = 10.1$? When $\mu = 9.9$? Is this desirable?
 	\end{parts}
 	\question \textbf{8-20} Lightbulbs of a certain type are advertised as having an average lifetime of 750 hours. The price of these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase arrangement unless it can be conclusively determined that the true average lifetime is smaller than what is advertised. A random sample of 50 bulbs was selected, the lifetime of each bulb determined, and the appropriate hypotheses were tested using Minitab, resulting in the accompanying output.
 		\begin{lstlisting}
 Variable   N    Mean  StDev  SEMean      Z  P-Value
 lifetime  50  738.44  38.20    5.40  -2.14    0.016
 		\end{lstlisting}
 		What conclusion would be approprate for a significance level of .05? A significance level of .01? What significance level and conclusion would you recommend?
 	\question \textbf{8-26} To obtain information on the corrosion-resistance properties of a certain type of steel conduit, 45 specimens are buried in soil for a 2-year period. The maximum penetration (in mils) for each specimen is then measured, yielding a sample average penetration of $\bar{x} = 52.7$ and a sample standard deviation of $s = 4.8$. The conduits were manufactured with the specification that true average penetration be at most 50 mils. They will be used unless it can be demonstrated conclusively that the specification has not been met. What would you conclude?
 	\question \textbf{8-38} 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. \begin{parts}
 		\part 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 signnificance level of .05. In reaching your conclusion, what type of error might you have committed?
 		\part 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.
 		\part How many plates would have to be tested to have $\beta(.15) = .10$ for the test of part (a)?
 	\end{parts}
 	\question \textbf{8-50} Newly purchased tires of a certain type are supposed to be filled to a pressure of 30 lb/in$^2$. Let $\mu$ denote the true average pressure. Find the P-value associated with each given $z$ statistic value for testing $H_0: \mu = 30$ versus $H_a: \mu \not = 30$. \begin{parts}
 		\part 2.10
 		\part -1.75
 		\part -.55
 		\part 1.41
 		\part -5.3
 	\end{parts}
 	\question \textbf{8-52} The paint used to make lines on roads must reflect enough light to be clearly visible at night. Let $\mu$ denote the true average reflectometer reading for a new type of paint under consideration. A test of $H_0: \mu = 20$ versus $H_a: \mu > 20$ will be based on a random sample of size $n$ from a normal population distribution. What conclusion is appropriate in each of the following situations? \begin{parts}
 		\part $n = 15, t = 3.2, \alpha = .05$
 		\part $n = 9, t = 1.8, \alpha = .01$
 		\part $n = 24, t = -.2$
 	\end{parts}
 	\question \textbf{8-58} 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 "Engineering Properties of Soil," \it Soil Science, \rm 1998: 93-102). \begin{center}
 		1.10 5.09 0.97 1.59 4.60 0.32 0.55 1.34 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{center}
 		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 hypothesess at significance level .10 by first determining the P-value. Would your conclusion be different if $\alpha =.05$ had been used? [\it Note: \rm A normal probability plot of the data shows an acceptable pattern in light of the reasonably large sample size]
 	\question \textbf{8-68} One method for straightening wire before coiling it to make a spring is called "roller straightening." The article "The Effect of Roller and Spinner Wire Straightening on Coiling Performance and Wire Properties" (\it Springs, \rm 1987: 27-28) reports on the tensile properties of wire. Suppose a sample of 16 wires is selected and each is tested to determine tensile strength (N/mm$^2$). The resulting sample mean and standard deviation are 2160 and 30, respectively. \begin{parts}
 		\part The mean tensile strength for springs made using spinner straightening is 2150 N/mm$^2$. What hypotheses should be tested to determine whether the mean tensile strength for the roller method exceeds 2150?
 		\part Assuming that the tensile strength distribution is approximately normal, what test statistic woul dyou use to test the hypotheses in part (a)?
 		\part What is the value of the test statistic for this data?
 		\part What is the P-value for the value of the test statistic computed in part (c)?
 		\part For a level .05 test, what conclusion would you reach?
 	\end{parts}
 	\question \textbf{8-80} An article in the Nov. 11, 2005, issue of the San Luis Obispo \it Tribune \rm reported that researchers making random purchases at California Wal-Mart stores found scanner coming up with the wrong price 8.3\% of the time. Suppose this was based on 200 purchases. The National Institute for Standards and Technology says that in the long run at most two out of every 100 items should have incorrectly scanned prices. \begin{parts}
 		\part Develop a test procedure with a significance level of (approximately) .05, and then carry out the test to decide whether the NIST benchmark is not satisfied.
 		\part For the test procedure you employed in (a), what is the probability of deciding that the NIST benchmark has been satisifed when in fact the mistake rate is 5\%?
 	\end{parts}
 	\end{questions}
 \end{document}
	\documentclass{exam}

	\usepackage{listings}
	\begin{document}
	\title{Homework Set 7}
	\author{}
	\date{}
	\maketitle

	\begin{questions}
	\question \textbf{8-10} A mixture of pulverized fuel ash and Portland cement to be used for grouting should have a compressive strength of more than 1300 KN/m$^2$. The mixture will not be used unless experimental evidence indicated conclusively that the strength specification has been met. Suppose compressive strength for specimens of this misture is normally distributed with $\sigma = 60$. Let $\mu$ denote the true average compressive strength. \begin{parts}
	\part What are the appropriate null and alternative hypotheses?
	\part Let $\bar{X}$ denote the sample average compressive strength for $n = 10$ randomly selected specimens. Consider the test procedure with test statistic $\bar{X}$ and rejection region $\bar{x} \geq 1331.26$. What is the probability distribution of the test statistic when $H_0$ is true? What is the probability of a type I error for the test procedure?
	\part What ist he probability distribution of the test statistic when $\mu = 1350?$ Using the test procedure of part (b), what is the probability that the misture will be judged unsatisfactory when in fact $\mu = 1350$ (a type II error)?
	\part How would you change the test procedure of part (b) to obtain a test with significance level .05? What impact would this change have on the error probability of part (c)?
	\part Consider the standardized test statistic $Z = (\bar{X} - 1300)/(\sigma/\sqrt{n}) = (\bar{X} - 1300)/13.42$. What are the values of $Z$ corresponding to the rejection region of part (b)?
	\end{parts}
	\question \textbf{8-14} Reconsider the situation of Exercise 11 and suppose the rejection region is $\{\bar{x}: \bar{x} \geq 10.1004 \textrm{or} \bar{x} \leq 9.98940\} = \{z: z \geq 2.51 \textrm{or} z \leq -2.65\}$. \begin{parts}
	\part What is $\alpha$ for this procedure?
	\part What is $\beta$ when $\mu = 10.1$? When $\mu = 9.9$? Is this desirable?
	\end{parts}
	\question \textbf{8-20} Lightbulbs of a certain type are advertised as having an average lifetime of 750 hours. The price of these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase arrangement unless it can be conclusively determined that the true average lifetime is smaller than what is advertised. A random sample of 50 bulbs was selected, the lifetime of each bulb determined, and the appropriate hypotheses were tested using Minitab, resulting in the accompanying output.
	\begin{lstlisting}
	Variable N Mean StDev SEMean Z P-Value
	lifetime 50 738.44 38.20 5.40 -2.14 0.016
	\end{lstlisting}
	What conclusion would be approprate for a significance level of .05? A significance level of .01? What significance level and conclusion would you recommend?
	\question \textbf{8-26} To obtain information on the corrosion-resistance properties of a certain type of steel conduit, 45 specimens are buried in soil for a 2-year period. The maximum penetration (in mils) for each specimen is then measured, yielding a sample average penetration of $\bar{x} = 52.7$ and a sample standard deviation of $s = 4.8$. The conduits were manufactured with the specification that true average penetration be at most 50 mils. They will be used unless it can be demonstrated conclusively that the specification has not been met. What would you conclude?
	\question \textbf{8-38} 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. \begin{parts}
	\part 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 signnificance level of .05. In reaching your conclusion, what type of error might you have committed?
	\part 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.
	\part How many plates would have to be tested to have $\beta(.15) = .10$ for the test of part (a)?
	\end{parts}
	\question \textbf{8-50} Newly purchased tires of a certain type are supposed to be filled to a pressure of 30 lb/in$^2$. Let $\mu$ denote the true average pressure. Find the P-value associated with each given $z$ statistic value for testing $H_0: \mu = 30$ versus $H_a: \mu \not = 30$. \begin{parts}
	\part 2.10
	\part -1.75
	\part -.55
	\part 1.41
	\part -5.3
	\end{parts}
	\question \textbf{8-52} The paint used to make lines on roads must reflect enough light to be clearly visible at night. Let $\mu$ denote the true average reflectometer reading for a new type of paint under consideration. A test of $H_0: \mu = 20$ versus $H_a: \mu > 20$ will be based on a random sample of size $n$ from a normal population distribution. What conclusion is appropriate in each of the following situations? \begin{parts}
	\part $n = 15, t = 3.2, \alpha = .05$
	\part $n = 9, t = 1.8, \alpha = .01$
	\part $n = 24, t = -.2$
	\end{parts}
	\question \textbf{8-58} 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 "Engineering Properties of Soil," \it Soil Science, \rm 1998: 93-102). \begin{center}
	1.10 5.09 0.97 1.59 4.60 0.32 0.55 1.34 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{center}
	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 hypothesess at significance level .10 by first determining the P-value. Would your conclusion be different if $\alpha =.05$ had been used? [\it Note: \rm A normal probability plot of the data shows an acceptable pattern in light of the reasonably large sample size]
	\question \textbf{8-68} One method for straightening wire before coiling it to make a spring is called "roller straightening." The article "The Effect of Roller and Spinner Wire Straightening on Coiling Performance and Wire Properties" (\it Springs, \rm 1987: 27-28) reports on the tensile properties of wire. Suppose a sample of 16 wires is selected and each is tested to determine tensile strength (N/mm$^2$). The resulting sample mean and standard deviation are 2160 and 30, respectively. \begin{parts}
	\part The mean tensile strength for springs made using spinner straightening is 2150 N/mm$^2$. What hypotheses should be tested to determine whether the mean tensile strength for the roller method exceeds 2150?
	\part Assuming that the tensile strength distribution is approximately normal, what test statistic woul dyou use to test the hypotheses in part (a)?
	\part What is the value of the test statistic for this data?
	\part What is the P-value for the value of the test statistic computed in part (c)?
	\part For a level .05 test, what conclusion would you reach?
	\end{parts}
	\question \textbf{8-80} An article in the Nov. 11, 2005, issue of the San Luis Obispo \it Tribune \rm reported that researchers making random purchases at California Wal-Mart stores found scanner coming up with the wrong price 8.3\% of the time. Suppose this was based on 200 purchases. The National Institute for Standards and Technology says that in the long run at most two out of every 100 items should have incorrectly scanned prices. \begin{parts}
	\part Develop a test procedure with a significance level of (approximately) .05, and then carry out the test to decide whether the NIST benchmark is not satisfied.
	\part For the test procedure you employed in (a), what is the probability of deciding that the NIST benchmark has been satisifed when in fact the mistake rate is 5\%?
	\end{parts}
	\end{questions}
	\end{document}