Solved: Many older homes have electrical systems that use fusesrather than circuit

For each of the following assertions, state whether it is a legitimate statistical hypothesis and why: a. b. c. d. e. f. , where l is the parameter of an exponential distribution used to model component lifetime

For the following pairs of assertions, indicate which do not comply with our rules for setting up hypotheses and why (the subscripts 1 and 2 differentiate between quantities for two different populations or samples):

To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected, and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld. Suppose the specifications state that mean strength of welds should exceed 100 lb/in2; the inspection team decides to test versus . Explain why it might be preferable to use this Ha rather than .

Let m denote the true average radioactivity level (picocuries per liter). The value 5 pCi/L is considered the dividing line between safe and unsafe water. Would you recommend testing versus or versus HaH0 : m , 5? Explain your reasoning. [Hint: Think about the consequences of a type I and type II error for each possibility.]

Before agreeing to purchase a large order of polyethylene sheaths for a particular type of high-pressure oil-filled submarine power cable, a company wants to see conclusive evidence that the true standard deviation of sheath thickness is less than .05 mm. What hypotheses should be tested, and why? In this context, what are the type I and type II errors?

Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40-amp fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the mean amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction. To verify the amperage of the fuses, a sample of fuses is to be selected and inspected. If a hypothesis test were to be performed on the resulting data, what null and alternative hypotheses would be of interest to the manufacturer? Describe type I and type II errors in the context of this problem situation.

Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most 150F, there will be no negative effects on the rivers ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above 150, 50 water samples will be taken at randomly selected times and the temperature of each sample recorded. The resulting data will be used to test the hypotheses versus . In the context of this situation, describe type I and type II errors. Which type of error would you consider more serious? Explain.

A regular type of laminate is currently being used by a manufacturer of circuit boards. A special laminate has been developed to reduce warpage. The regular laminate will be used on one sample of specimens and the special laminate on another sample, and the amount of warpage will then be determined for each specimen. The manufacturer will then switch to the special laminate only if it can be demonstrated that the true average amount of warpage for that laminate is less than for the regular laminate. State the relevant hypotheses, and describe the type I and type II errors in the context of this situation.

Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the first company over the second. Consider testing versus based on a random sample of 25 individuals. Let X denote the number in the sample who favor the first company and x represent the observed value of X. a. Which of the following rejection regions is most appropriate and why?"b. In the context of this problem situation, describe what the type I and type II errors are. c. What is the probability distribution of the test statistic X when H0 is true? Use it to compute the probability of a type I error. d. Compute the probability of a type II error for the selected region when , again when , and also for both and . e. Using the selected region, what would you conclude if 6 of the 25 queried favored company 1?"

A mixture of pulverized fuel ash and Portland cement to be used for grouting should have a compressive strength of more than 1300 KN/m2. The mixture will not be used unless experimental evidence indicates conclusively that the strength specification has been met. Suppose compressive strength for specimens of this mixture is normally distributed with . Let m denote the true average compressive strength. a. What are the appropriate null and alternative hypotheses? b. Let denote the sample average compressive strength for randomly selected specimens. Consider the test procedure with test statistic and rejection region . What is the probability distribution of the test statistic when H0 is true? What is the probability of a type I error for the test procedure? c. What is the probability distribution of the test statistic when ? Using the test procedure of part (b), what is the probability that the mixture will be judged unsatisfactory when in fact (a type II error)? d. 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)? e. Consider the standardized test statistic . What are the values of Z corresponding to the rejection region of part (b)?

The calibration of a scale is to be checked by weighing a 10-kg test specimen 25 times. Suppose that the results of different weighings are independent of one another and that the weight on each trial is normally distributed with kg. Let m denote the true average weight reading on the scale. a. What hypotheses should be tested? b. Suppose the scale is to be recalibrated if either or . What is the probability that recalibration is carried out when it is actually unnecessary? c. What is the probability that recalibration is judged unnecessary when in fact ? When ? d. Let . For what value c is the rejection region of part (b) equivalent to the two-tailed region of either or ? e. If the sample size were only 10 rather than 25, how should the procedure of part (d) be altered so that ? f. Using the test of part (e), what would you conclude from the following sample data? 9.981 10.006 9.857 10.107 9.888 9.728 10.439 10.214 10.190 9.793 g. Reexpress the test procedure of part (b) in terms of the standardized test statistic .

A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under specified conditions is known to be 120 ft. It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average braking distance for the new design. a. Define the parameter of interest and state the relevant hypotheses. b. Suppose braking distance for the new system is normally distributed with . Let denote the sample average braking distance for a random sample of 36 observations. Which of the following three rejection regions is appropriate: ? c. What is the significance level for the appropriate region of part (b)? How would you change the region to obtain a test with ? d. What is the probability that the new design is not implemented when its true average braking distance is actually 115 ft and the appropriate region from part (b) is used? e. Let . What is the significance level for the rejection region ? For the region ?

Let denote a random sample from a normal population distribution with a known value of s. a. For testing the hypotheses versus (where m0 is a fixed number), show that the test with test statistic and rejection region has significance level .01. b. Suppose the procedure of part (a) is used to test versus . If , and , what is the probability of committing a type I error when ? When ? In general, what can be said about the probability of a type I error when the actual value of m is less than m0? Verify your assertion.

Reconsider the situation of Exercise 11 and suppose the rejection region is . a. What is a for this procedure? b. What is b when ? When ? Is this desirable?

Let the test statistic Z have a standard normal distribution when H0 is true. Give the significance level for each of the following situations: a. , rejection region b. , rejection region c. Ha , rejection region z $ 2.88 or z # 22.88

Let the test statistic T have a t distribution when H0 is true. Give the significance level for each of the following situations:

Answer the following questions for the tire problem in Example 8.7. a. If and a level test is used, what is the decision? b. If a level .01 test is used, what is b(30,500)? c. If a level .01 test is used and it is also required that , what sample size n is necessary? d. If , what is the smallest a at which H0 can be rejected (based on )?

Reconsider the paint-drying situation of Example 8.2, in which drying time for a test specimen is normally distributed with . The hypotheses versus are to be tested using a random sample of observations. a. How many standard deviations (of ) below the null value is ? b. If , what is the conclusion using ? c. What is a for the test procedure that rejects H0 when ? d. For the test procedure of part (c), what is b(70)? e. If the test procedure of part (c) is used, what n is necessary to ensure that ? f. If a level .01 test is used with , what is the probability of a type I error when ?

The melting point of each of 16 samples of a certain brand of hydrogenated vegetable oil was determined, resulting in . Assume that the distribution of the melting point is normal with . a. Test versus using a two-tailed level .01 test. b. If a level .01 test is used, what is b(94), the probability of a type II error when ? c. What value of n is necessary to ensure that when ?

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 demonstrated 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. Variable N Mean StDev SEMean Z P-Value lifetime 50 738.44 38.20 5.40 2.14 0.016 What conclusion would be appropriate for a significance level of .05? A significance level of .01? What significance level and conclusion would you recommend?

The true average diameter of ball bearings of a certain type is supposed to be .5 in. A one-sample t test will be carried out to see whether this is the case. What conclusion is appropriate in each of the following situations? a. b. c. d.

The article The Foremans View of Quality Control (Quality Engr., 1990: 257280) described an investigation into the coating weights for large pipes resulting from a galvanized coating process. Production standards call for a true average weight of 200 lb per pipe. The accompanying descriptive summary and boxplot are from Minitab. Variable N Mean Median TrMean StDev SEMean ctg wt 30 206.73 206.00 206.81 6.35 1.16 Variable Min Max Q1 Q3 ctg wt 193.00 218.00 202.75 212.00 n 5 25, t 5 23.9 n 5 25, t 5 22.6, a 5 .01a. What does the boxplot suggest about the status of the specification for true average coating weight? b. A normal probability plot of the data was quite straight. Use the descriptive output to test the appropriate hypotheses.

Exercise 36 in Chapter 1 gave observations on escape time (sec) for oil workers in a simulated exercise, from which the sample mean and sample standard deviation are 370.69 and 24.36, respectively. Suppose the investigators had believed a priori that true average escape time would be at most 6 min. Does the data contradict this prior belief? Assuming normality, test the appropriate hypotheses using a significance level of .05.

Reconsider the sample observations on stabilized viscosity of asphalt specimens introduced in Exercise 46 in Chapter 1 (2781, 2900, 3013, 2856, and 2888). Suppose that for a particular application it is required that true average viscosity be 3000. Does this requirement appear to have been satisfied? State and test the appropriate hypotheses.

The desired percentage of SiO2 in a certain type of aluminous cement is 5.5. To test whether the true average percentage is 5.5 for a particular production facility, 16 independently obtained samples are analyzed. Suppose that the percentage of SiO2 in a sample is normally distributed with and that . a. Does this indicate conclusively that the true average percentage differs from 5.5? Carry out the analysis using the sequence of steps suggested in the text. b. If the true average percentage is and a level test based on is used, what is the probability of detecting this departure from H0? c. What value of n is required to satisfy and b(5.6) 5 .01?

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 and a sample standard deviation of . 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?

Automatic identification of the boundaries of significant structures within a medical image is an area of ongoing research. The paper Automatic Segmentation of Medical Images Using Image Registration: Diagnostic and Simulation Applications (J. of Medical Engr. and Tech., 2005: 5363) discussed a new technique for such identification. A measure of the accuracy of the automatic region is the average linear displacement (ALD). The paper gave the following ALD observations for a sample of 49 kidneys (units of pixel dimensions).1.38 0.44 1.09 0.75 0.66 1.28 0.51 0.39 0.70 0.46 0.54 0.83 0.58 0.64 1.30 0.57 0.43 0.62 1.00 1.05 0.82 1.10 0.65 0.99 0.56 0.56 0.64 0.45 0.82 1.06 0.41 0.58 0.66 0.54 0.83 0.59 0.51 1.04 0.85 0.45 0.52 0.58 1.11 0.34 1.25 0.38 1.44 1.28 0.51 a. Summarize/describe the data. b. Is it plausible that ALD is at least approximately normally distributed? Must normality be assumed prior to calculating a CI for true average ALD or testing hypotheses about true average ALD? Explain. c. The authors commented that in most cases the ALD is better than or of the order of 1.0. Does the data in fact provide strong evidence for concluding that true average ALD under these circumstances is less than 1.0? Carry out an appropriate test of hypotheses. d. Calculate an upper confidence bound for true average ALD using a confidence level of 95%, and interpret this bound.

Minor surgery on horses under field conditions requires a reliable short-term anesthetic producing good muscle relaxation, minimal cardiovascular and respiratory changes, and a quick, smooth recovery with minimal aftereffects so that horses can be left unattended. The article A Field Trial of Ketamine Anesthesia in the Horse (Equine Vet. J., 1984: 176179) reports that for a sample of horses to which ketamine was administered under certain conditions, the sample average lateral recumbency (lying-down) time was 18.86 min and the standard deviation was 8.6 min. Does this data suggest that true average lateral recumbency time under these conditions is less than 20 min? Test the appropriate hypotheses at level of significance .10.

The article Uncertainty Estimation in Railway Track Life- Cycle Cost (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 21 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 , 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 b(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 Shake, Rattle, and Squeeze: How Much Is Left in That Container? (Consumer Reports, May 2009: 8) reported on an investigation of this issue for various consumer products. Suppose five 6.0 oz 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.

A well-designed and safe workplace can contribute greatly to increased productivity. It is especially important that workers not be asked to perform tasks, such as lifting, that exceed their capabilities. The accompanying data on maximum weight of lift (MAWL, in kg) for a frequency of four lifts/min was reported in the article The Effects of Speed, Frequency, and Load on Measured Hand Forces for a Floor-to-Knuckle Lifting Task (Ergonomics, 1992: 833843); subjects were randomly selected from the population of healthy males ages 1830. Assuming that MAWL is normally distributed, does the data suggest that the population mean MAWL exceeds 25? Carry out a test using a significance level of .05. 25.8 36.6 26.3 21.8 27.2

The recommended daily dietary allowance for zinc among males older than age 50 years is 15 mg/day. The article Nutrient Intakes and Dietary Patterns of Older Americans: A National Study (J. of Gerontology, 1992: M145150) reports the following summary data on intake for a sample of males age 6574 years: , , and . Does this data indicate that average daily zinc intake in the population of all males ages 6574 falls below the recommended allowance?

Reconsider the accompanying sample data on expense ratio (%) for large-cap growth mutual funds first introduced in Exercise 1.53. 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 for the population of all 762 such funds. So did you actually commit an error in reaching your conclusion? c. Supposing that , determine and interpret the power of the test in (a) for the actual value of m stated in (b).

A sample of 12 radon detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6 90.9 91.2 96.9 96.5 91.3 100.1 105.0 99.6 107.7 103.3 92.4 a. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypotheses using . b. Suppose that prior to the experiment a value of had been assumed. How many determinations would then have been appropriate to obtain for the alternative ?

Show that for any , when the population distribution is normal and s is known, the two-tailed test satisfies , so that is symmetric about m0.

For a fixed alternative value m, show that as for either a one-tailed or a two-tailed z test in the case of a normal population distribution with known s.

A common characterization of obese individuals is that their body mass index is at least 30 [ , where height is in meters and weight is in kilograms]. The article The Impact of Obesity on Illness Absence and Productivity in an Industrial Population of Petrochemical Workers (Annals of Epidemiology, 2008: 814) 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 using the rejection region approach 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 for the test of part (a)?

A random sample of 150 recent donations at a certain blood bank reveals that 82 were type A blood. Does this suggest that the actual percentage of type A donations differs from 40%, the percentage of the population having type A blood? Carry out a test of the appropriate hypotheses using a significance level of .01. Would your conclusion have been different if a significance level of .05 had been used?

It is known that roughly 2/3 of all human beings have a dominant right foot or eye. Is there also right-sided dominance in kissing behavior? The article Human Behavior: Adult Persistence of Head-Turning Asymmetry (Nature, 2003: 771) reported that in a random sample of 124 kissing couples, both people in 80 of the couples tended to lean more to the right than to the left. a. If 2/3 of all kissing couples exhibit this right-leaning behavior, what is the probability that the number in a sample of 124 who do so differs from the expected value by at least as much as what was actually observed? b. Does the result of the experiment suggest that the 2/3 figure is implausible for kissing behavior? State and test the appropriate hypotheses.

The article referenced in Example 8.11 also reported that in a sample of 106 wine consumers, 22 (20.8%) thought that screw tops were an acceptable substitute for natural corks. Suppose a particular winery decided to use screw tops for one of its wines unless there was strong evidence to suggest that fewer than 25% of wine consumers found this acceptable. a. Using a significance level of .10, what would you recommend to the winery? b. For the hypotheses tested in (a), describe in context what the type I and II errors would be, and say which type of error might have been committed.

With domestic sources of building supplies running low several years ago, roughly 60,000 homes were built with imported Chinese drywall. According to the article Report Links Chinese Drywall to Home Problems (New York Times, Nov. 24, 2009), federal investigators identified a strong association between chemicals in the drywall and electrical problems, and there is also strong evidence of respiratory difficulties due to the emission of hydrogen sulfide gas. An extensive examination of 51 homes found that 41 had such problems. Suppose these 51 were randomly sampled from the population of all homes having Chinese drywall. a. Does the data provide strong evidence for concluding that more than 50% of all homes with Chinese drywall have electrical/environmental problems? Carry out a test of hypotheses using . b. Calculate a lower confidence bound using a confidence level of 99% for the percentage of all such homes that have electrical/environmental problems. c. If it is actually the case that 80% of all such homes have problems, how likely is it that the test of (a) would not conclude that more than 50% do?

A plan for an executive travelers club has been developed by an airline on the premise that 5% of its current customers would qualify for membership. A random sample of 500 customers yielded 40 who would qualify. a. Using this data, test at level .01 the null hypothesis that the companys premise is correct against the alternative that it is not correct. b. What is the probability that when the test of part (a) is used, the companys premise will be judged correct when in fact 10% of all current customers qualify?

Each of a group of 20 intermediate tennis players is given two rackets, one having nylon strings and the other synthetic gut strings. After several weeks of playing with the two rackets, each player will be asked to state a preference for one of the two types of strings. Let p denote the proportion of all such players who would prefer gut to nylon, and let X be the number of players in the sample who prefer gut. Because gut strings are more expensive, consider the null hypothesis that at most 50% of all such players prefer gut. We simplify this to , planning to reject H0 only if sample evidence strongly favors gut strings. a. Which of the rejection regions {15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5}, or {0, 1, 2, 3, 17, 18, 19, 20} is most appropriate, and why are the other two not appropriate? b. What is the probability of a type I error for the chosen region of part (a)? Does the region specify a level .05 test? Is it the best level .05 test? c. If 60% of all enthusiasts prefer gut, calculate the probability of a type II error using the appropriate region from part (a). Repeat if 80% of all enthusiasts prefer gut. d. If 13 out of the 20 players prefer gut, should H0 be rejected using a significance level of .10?

A manufacturer of plumbing fixtures has developed a new type of washerless faucet. Let (a randomly selected faucet of this type will develop a leak within 2 years under normal use). The manufacturer has decided to proceed with production unless it can be determined that p is too large; the borderline acceptable value of p is specified as .10. The manufacturer decides to subject n of these faucets to accelerated testing (approximating 2 years of normal use). With the number among the n faucets that leak before the test concludes, production will commence unless the observed X is too large. It is decided that if , the probability of not proceeding should be at most .10, whereas if the probability of proceeding should be at most .10. Can be used? ? ? What is the appropriate rejection region for the chosen n, and what are the actual error probabilities when this region is used?

Scientists think that robots will play a crucial role in factories in the next several decades. Suppose that in an experiment to determine whether the use of robots to weave computer cables is feasible, a robot was used to assemble 500 cables. The cables were examined and there were 15 defectives. If human assemblers have a defect rate of .035 (3.5%), does this data support the hypothesis that the proportion of defectives is lower for robots than for humans? Use a .01 significance level.

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

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

Let m denote the mean reaction time to a certain stimulus. For a large-sample z test of versus , find the P-value associated with each of the given values of the z test statistic. a. 1.42 b. .90 c. 1.96 d. 2.48 e. 2.11

Newly purchased tires of a certain type are supposed to be filled to a pressure of 30 lb/in2. Let m denote the true average pressure. Find the P-value associated with each given z statistic value for testing versus

Give as much information as you can about the P-value of a t test in each of the following situations: a. Upper-tailed test, b. Lower-tailed test, c. Two-tailed test, d. Upper-tailed test, e. Upper-tailed test, f. Two-tailed test

The paint used to make lines on roads must reflect enough light to be clearly visible at night. Let m denote the true average reflectometer reading for a new type of paint under consideration. A test of versus H wil lbe based on a random sample of size n from a normal population distribution. What conclusion is appropriate in each of the following situations?

Let m denote true average serum receptor concentration for all pregnant women. The average for all women is known to be 5.63. The article Serum Transferrin Receptor for the Detection of Iron Deficiency in Pregnancy (Amer. J. of Clinical Nutr., 1991: 10771081) reports that for a test of versus based on pregnant women. Using a significance level of .01, what would you conclude?

The article Analysis of Reserve and Regular Bottlings: Why Pay for a Difference Only the Critics Claim to Notice? (Chance, Summer 2005, pp. 915) reported on an experiment to investigate whether wine tasters could distinguish between more expensive reserve wines and their regular counterparts. Wine was presented to tasters in four containers labeled A, B, C, and D, with two of these containing the reserve wine and the other two the regular wine. Each taster randomly selected three of the containers, tasted the selected wines, and indicated which of the three he/she believed was different from the other two. Of the tasting trials, 346 resulted in correct distinctions (either the one reserve that differed from the two regular wines or the one regular wine that differed from the two reserves). Does this provide compelling evidence for concluding that tasters of this type have some ability to distinguish between reserve and regular wines? State and test the relevant hypotheses using the P-value approach. Are you particularly impressed with the ability of tasters to distinguish between the two types of wine?

An aspirin manufacturer fills bottles by weight rather than by count. Since each bottle should contain 100 tablets, the average weight per tablet should be 5 grains. Each of 100 tablets taken from a very large lot is weighed, resulting in a sample average weight per tablet of 4.87 grains and a sample standard deviation of .35 grain. Does this information provide strong evidence for concluding that the company is not filling its bottles as advertised? Test the appropriate hypotheses using by first computing the P-value and then comparing it to the specified significance level.

Because of variability in the manufacturing process, the actual yielding point of a sample of mild steel subjected to increasing stress will usually differ from the theoretical yielding point. Let p denote the true proportion of samples that yield before their theoretical yielding point. If on the basis of a sample it can be concluded that more than 20% of all specimens yield before the theoretical point, the production process will have to be modified. a. If 15 of 60 specimens yield before the theoretical point, what is the P-value when the appropriate test is used, and what would you advise the company to do?b. If the true percentage of early yields is actually 50% (so that the theoretical point is the median of the yield distribution) and a level .01 test is used, what is the probability that the company concludes a modification of the process is necessary?

The article Heavy Drinking and Polydrug Use Among College Students (J. of Drug Issues, 2008: 445466) stated that 51 of the 462 college students in a sample had a lifetime abstinence from alcohol. Does this provide strong evidence for concluding that more than 10% of the population sampled had completely abstained from alcohol use? Test the appropriate hypotheses using the P-value method. [Note: The article used more advanced statistical methods to study the use of various drugs among students characterized as light, moderate, and heavy drinkers.]

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, Soil Science, 1998: 93102). 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 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 by first determining the P-value. Would your conclusion be different if had been used? [Note: A normal probability plot of the data shows an acceptable pattern in light of the reasonably large sample size.]

The accompanying data on cube compressive strength (MPa) of concrete specimens appeared in the article Experimental Study of Recycled Rubber-Filled High- Strength Concrete (Magazine of Concrete Res., 2009: 549556): 112.3 97.0 92.7 86.0 102.0 99.2 95.8 103.5 89.0 86.7 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 100 MPa. Should the concrete be used? Carry out a test of appropriate hypotheses using the P-value method.

A certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of a writing machine) is at least 10 hours. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting data supports the use of a one-sample t test. a. What hypotheses should be tested if the investigators believe a priori that the design specification has been satisfied? b. What conclusion is appropriate if the hypotheses of part (a) are tested, , and ? c. What conclusion is appropriate if the hypotheses of part (a) are tested, , and ? d. What should be concluded if the hypotheses of part (a) are tested and ?

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 70 ppm. 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 readings85, 77, 82, 68, 72, and 69is recalibration necessary? Carry out a test of the relevant hypotheses using the P-value approach with .

The relative conductivity of a semiconductor device is determined by the amount of impurity doped into the device during its manufacture. A silicon diode to be used for a specific purpose requires an average cut-on voltage of .60 V, and if this is not achieved, the amount of impurity must be adjusted. A sample of diodes was selected and the cut-on voltage was determined. The accompanying SAS output resulted from a request to test the appropriate hypotheses. N Mean Std Dev T Prob. T 15 0.0453333 0.0899100 1.9527887 0.0711 [Note: SAS explicitly tests , so to test the null value .60 must be subtracted from each xi; the reported mean is then the average of the values. Also, SASs P-value is always for a two-tailed test.] What would be concluded for a significance level of .01? .05? .10?

A sample of 50 lenses used in eyeglasses yields a sample mean thickness of 3.05 mm and a sample standard deviation of .34 mm. The desired true average thickness of such lenses is 3.20 mm. Does the data strongly suggest that the true average thickness of such lenses is something other than what is desired? Test using .

In Exercise 65, suppose the experimenter had believed before collecting the data that the value of s was approximately .30. If the experimenter wished the probability of a type II error to be .05 when , was a sample size 50 unnecessarily large?

A sample of 50 lenses used in eyeglasses yields a sample mean thickness of 3.05 mm and a sample standard deviation of .34 mm. The desired true average thickness of such lenses is 3.20 mm. Does the data strongly suggest that the true average thickness of such lenses is something other than what is desired? Test using .

In Exercise 65, suppose the experimenter had believed before collecting the data that the value of s was approximately .30. If the experimenter wished the probability of a type II error to be .05 when , was a sample size 50 unnecessarily large?

It is specified that a certain type of iron should contain .85 g of silicon per 100 g of iron (.85%). The silicon content of each of 25 randomly selected iron specimens was determined, and the accompanying Minitab output resulted from a test of the appropriate hypotheses. Variable N Mean StDev SE Mean T P sil cont 25 0.8880 0.1807 0.0361 1.05 0.30 a. What hypotheses were tested? b. What conclusion would be reached for a significance level of .05, and why? Answer the same question for a significance level of .10.

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 (Springs, 1987: 2728) reports on the tensile properties of wire. Suppose a sample of 16 wires is selected and each is tested to determine tensile strength (N/mm2). The resulting sample mean and standard deviation are 2160 and 30, respectively. a. The mean tensile strength for springs made using spinner straightening is 2150 N/mm2. What hypotheses should be tested to determine whether the mean tensile strength for the roller method exceeds 2150? b. Assuming that the tensile strength distribution is approximately normal, what test statistic would you use to test the hypotheses in part (a)? c. What is the value of the test statistic for this data? d. What is the P-value for the value of the test statistic computed in part (c)? e. For a level .05 test, what conclusion would you reach?

Contamination of mine soils in China is a serious environmental problem. The article Heavy Metal Contamination in Soils and Phytoaccumulation in a Manganese Mine Wasteland, South China (Air, Soil, and Water Res., 2008: 3141) reported that, for a sample of 3 soil specimens from a certain restored mining area, the sample mean concentration of Total Cu was 45.31 mg/kg with a corresponding (estimated) standard error of the mean of 5.26. It was also stated that the China background value for this concentration was 20. The results of various statistical tests described in the article were predicated on assuming normality. a. Does the data provide strong evidence for concluding that the true average concentration in the sampled region exceeds the stated background value? Carry out a test at significance level .01 using the P-value method. Does the result surprise you? Explain. b. Referring back to the test of (a), how likely is it that the P-value would be at least .01 when the true average concentration is 50 and the true standard deviation of concentration is 10?

The article Orchard Floor Management Utilizing Soil- Applied Coal Dust for Frost Protection (Agri. and Forest Meteorology, 1988: 7182) reports the following values for soil heat flux of eight plots covered with coal dust. 34.7 35.4 34.7 37.7 32.5 28.0 18.4 24.9 The mean soil heat flux for plots covered only with grass is 29.0. Assuming that the heat-flux distribution is approximately normal, does the data suggest that the coal dust is effective in increasing the mean heat flux over that for grass? Test the appropriate hypotheses using .

The article Caffeine Knowledge, Attitudes, and Consumption in Adult Women (J. of Nutrition Educ., 1992: 179184) reports the following summary data on daily caffeine consumption for a sample of adult women: , and a. Does it appear plausible that the population distribution of daily caffeine consumption is normal? Is it necessary to assume a normal population distribution to test hypotheses about the value of the population mean consumption? Explain your reasoning. b. Suppose it had previously been believed that mean consumption was at most 200 mg. Does the given data contradict this prior belief? Test the appropriate hypotheses at significance level .10 and include a P-value in your analysis.

Annual holdings turnover for a mutual fund is the percentage of a funds assets that are sold during a particular year. Generally speaking, a fund with a low value of turnover is more stable and risk averse, whereas a high value of turnover indicates a substantial amount of buying and selling in an attempt to take advantage of short-term market fluctuations. Here are values of turnover for a sample of 20 large-cap blended funds (refer to Exercise 1.53 for a bit more information) extracted from Morningstar.com: 1.03 1.23 1.10 1.64 1.30 1.27 1.25 0.78 1.05 0.64 0.94 2.86 1.05 0.75 0.09 0.79 1.61 1.26 0.93 0.84 a. Would you use the one-sample t test to decide whether there is compelling evidence for concluding that the population mean turnover is less than 100%? Explain. b. A normal probability plot of the 20 ln(turnover) values shows a very pronounced linear pattern, suggesting it is reasonable to assume that the turnover distribution is lognormal. Recall that X has a lognormal distribution if ln(X) is normally distributed with mean value m and variance s2. Because m is also the median of the ln(X) distribution, em is the median of the X distribution. Use this information to decide whether there is compelling evidence for concluding that the median of the turnover population distribution is less than 100%.

The true average breaking strength of ceramic insulators of a certain type is supposed to be at least 10 psi. They will be used for a particular application unless sample data indicates conclusively that this specification has not been met. A test of hypotheses using is to be based on a random sample of ten insulators. Assume that the breaking-strength distribution is normal with unknown standard deviation. a. If the true standard deviation is .80, how likely is it that insulators will be judged satisfactory when true average breaking strength is actually only 9.5? Only 9.0? b. What sample size would be necessary to have a 75% chance of detecting that the true average breaking strength is 9.5 when the true standard deviation is .80?

The accompanying observations on residual flame time (sec) for strips of treated childrens nightwear were given in the article An Introduction to Some Precision and Accuracy of Measurement Problems (J. of Testing and Eval., 1982: 132140). Suppose a true average flame time of at most 9.75 had been mandated. Does the data suggest that this condition has not been met? Carry out an appropriate test after first investigating the plausibility of assumptions that underlie your method of inference. 9.85 9.93 9.75 9.77 9.67 9.87 9.67 9.94 9.85 9.75 9.83 9.92 9.74 9.99 9.88 9.95 9.95 9.93 9.92 9.89

The incidence of a certain type of chromosome defect in the U.S. adult male population is believed to be 1 in 75. A random sample of 800 individuals in U.S. penal institutions reveals 16 who have such defects. Can it be concluded that the incidence rate of this defect among prisoners differs from the presumed rate for the entire adult male population? a. State and test the relevant hypotheses using . What type of error might you have made in reaching a conclusion? b. What P-value is associated with this test? Based on this P-value, could H0 be rejected at significance level .20?

In an investigation of the toxin produced by a certain poisonous snake, a researcher prepared 26 different vials, each containing 1 g of the toxin, and then determined the amount of antitoxin needed to neutralize the toxin. The sample average amount of antitoxin necessary was found to be 1.89 mg, and the sample standard deviation was .42. Previous research had indicated that the true average neutralizing amount was 1.75 mg/g of toxin. Does the new data contradict the value suggested by prior research? Test the relevant hypotheses using the P-value approach. Does the validity of your analysis depend on any assumptions about the population distribution of neutralizing amount? Explain.

The sample average unrestrained compressive strength for 45 specimens of a particular type of brick was computed to be 3107 psi, and the sample standard deviation was 188. The distribution of unrestrained compressive strength may be somewhat skewed. Does the data strongly indicate that the true average unrestrained compressive strength is less than the design value of 3200? Test using .

The Dec. 30, 2009, the NewYork Times reported that in a survey of 948 American adults who said they were at least somewhat interested in college football, 597 said the current Bowl Championship System should be replace by a playoff similar to that used in college basketball. Does this provide compelling evidence for concluding that a majority of all such individuals favor replacing the B.C.S. with a playoff? Test the appropriate hypotheses using the P-value method.

When are independent Poisson variables, each with parameter m, and n is large, the sample mean has approximately a normal distribution with and . This implies that has approximately a standard normal distribution. For testing , we can replace m by m0 in the equation for Z to obtain a test statistic. This statistic is actually preferred to the large-sample statistic with denominator (when the Xis are Poisson) because it is tailored explicitly to the Poisson assumption. If the number of requests for consulting received by a certain statistician during a 5-day work week has a Poisson distribution and the total number of consulting requests during a 36-week period is 160, does this suggest that the true average number of weekly requests exceeds 4.0? Test using a 5 .02.

An article in the Nov. 11, 2005, issue of the San Luis Obispo Tribune reported that researchers making random purchases at California Wal-Mart stores found scanners 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. a. 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. b. For the test procedure you employed in (a), what is the probability of deciding that the NIST benchmark has been satisfied when in fact the mistake rate is 5%?

A hot-tub manufacturer advertises that with its heating equipment, a temperature of 100F can be achieved in at most 15 min. A random sample of 42 tubs is selected, and the time necessary to achieve a 100F temperature is determined for each tub. The sample average time and sample standard deviation are 16.5 min and 2.2 min, respectively. Does this data cast doubt on the companys claim? Compute the P-value and use it to reach a conclusion at level .05.

Chapter 7 presented a CI for the variance s2 of a normal population distribution. The key result there was that the rv has a chi-squared distribution with df. Consider the null hypothesis (equivalently, ). Then when H0 is true, the test statistic has a chi-squared distribution with df. If the relevant alternative is , rejecting H0 if gives a test with significance level a. To ensure reasonably uniform characteristics for a particular application, it is desired that the true standard deviation of the softening point of a certain type of petroleum pitch be at most .50C. The softening points of ten different specimens were determined, yielding a sample standard deviation of .58C. Does this strongly contradict the uniformity specification? Test the appropriate hypotheses using .

Referring to Exercise 82, suppose an investigator wishes to test versus based on a sample of 21 observations. The computed value of 20s2/.04 is 8.58. Place bounds on the P-value and then reach a conclusion at level .01.

When the population distribution is normal and n is large, the sample standard deviation S has approximately a normal distribution with and . We already know that in this case, for any n, is normal with and . a. Assuming that the underlying distribution is normal, what is an approximately unbiased estimator of the 99th percentile ? b. When the Xis are normal, it can be shown that and S are independent rvs (one measures location whereas the other measures spread). Use this to compute and for the estimator of part (a). What is the estimated standard error ? c. Write a test statistic for testing that has approximately a standard normal distribution when H0 is true. If soil pH is normally distributed in a certain region and 64 soil samples yield , does this provide strong evidence for concluding that at most 99% of all possible samples would have a pH of less than 6.75? Test using .

Let be a random sample from an exponential distribution with parameter l. Then it can be shown that has a chi-squared distribution with (by first showing that has a chi-squared distribution with ). a. Use this fact to obtain a test statistic and rejection region that together specify a level a test for versus each of the three commonly encountered alternatives. [Hint: , so is equivalent to .] b. Suppose that ten identical components, each having exponentially distributed time until failure, are tested. The resulting failure times are 95 16 11 3 42 71 225 64 87 123 Use the test procedure of part (a) to decide whether the data strongly suggests that the true average lifetime is less than the previously claimed value of 75.

Suppose the population distribution is normal with known s. Let g be such that . For testing versus , consider the test that rejects H0 if either or , where the test statistic is . a. Show that . b. Derive an expression for . [Hint: Express the test in the form reject H0 if either .] c. Let . For what values of g (relative to a) will ?

After a period of apprenticeship, an organization gives an exam that must be passed to be eligible for membership. Let (randomly chosen apprentice passes). The organization wishes an exam that most but not all should be able to pass, so it decides that is desirable. For a particular exam, the relevant hypotheses are versus the alternative . Suppose ten people take the exam, and let the number who pass. a. Does the lower-tailed region specify a level .01 test? b. Show that even though Ha is two-sided, no two-tailed test is a level .01 test. c. Sketch a graph of as a function of p for this test. Is this desirable?