Exercises for Section 6.1 2. The pH of an acid solution used to etch aluminum

varies somewhat from batch to batch. In a sample of 50 batches, the mean pH was 2.6, with a standard deviation of 0.3. Let represent the mean pH for batches of this solution. a. Find the P-value for testing versus

. b. Either the mean pH is greater than 2.5 mm, or

the sample is in the most extreme _______% of its distribution.

4. In a process that manufactures tungsten-coated

silicon wafers, the target resistance for a wafer is 85

m. In a simple random sample of 50 wafers, the

sample mean resistance was 84.8 m, and the

standard deviation was 0.5 m. Let represent the mean resistance of the wafers manufactured by this process. A quality engineer tests versus . a. Find the P-value. b. Do you believe it is plausible that the mean is

on target, or are you convinced that the mean is not on target? Explain your reasoning.

8. Lasers can provide highly accurate measurements

of small movements. To determine the accuracy of such a laser, it was used to take 100 measurements of a known quantity. The sample mean error was 25 m with a standard deviation of 60 m. The laser is properly calibrated if the mean error is . A test is made of versus . a. Find the P-value. b. Do you believe it is plausible that the laser is

properly calibrated, or are you convinced that it is out of calibration? Explain your reasoning.

Exercises for Section 6.2 6. George performed a hypothesis test. Luis checked

George’s work by redoing the calculations. Both George and Luis agree that the result was statistically significant at the 5% level, but they got different P-values. George got a P-value of 0.20, and Luis got a P-value of 0.02. a. Is it possible that George’s work is correct?

Explain. b. Is it possible that Luis’s work is correct? Explain.

12. A machine that fills cereal boxes is supposed to be

calibrated so that the mean fill weight is 12 oz. Let

denote the true mean fill weight. Assume that in a test of the hypotheses versus , the P-va1ue is 0.30. a. Should be rejected on the basis of this test?

Explain. b. Can you conclude that the machine is calibrated

to provide a mean fill weight of 12 oz.? Explain. 18. A shipment of fibers is not acceptable if the mean

breaking strength of the fibers is less than 50 N. A large sample of fibers from this shipment was tested, and a 98% lower confidence bound for the mean breaking strength was computed to be 50.1 N. Someone suggests using these data to test the hypotheses versus . a. Is it possible to determine from the confidence

bound whether P < 0.01? Explain. b. Is it possible to determine from the confidence

bound whether P < 0.05? Explain. Exercises for Section 6.3 6. A random sample of 80 bolts is sampled from a

day’s production, and 4 of them are found to have diameters below specification. It is claimed that the proportion of defective bolts among those manufactured that day is less than 0.10. Is it appropriate to use the methods of this section to determine whether we can reject this claim? If so, state the appropriate null and alternate hypotheses and compute the P-value. If not, explain why not.

8. A grinding machine will be qualified for a particular

task if it can be shown to produce less than 8% defective parts. In a random sample of 300 parts, 12 were defective. On the basis of these data, can the machine be qualified?

Exercises for Section 6.4 4. A certain manufactured product is supposed to

contain 23% potassium by weight. A sample of 10 specimens of this product had an average percentage of 23.2 with a standard deviation of 0.2. If the mean percentage is found to differ from 23, the manufacturing process will be recalibrated. a. State the appropriate null and alternate

hypotheses. b. Compute the P-value. c. Should the process be recalibrated? Explain.

6. The thicknesses of six pads designed for use in

aircraft engine mounts were measured. The results,

in mm, were 40.93, 41.11, 41.47, 40.96, 40.80, and 41.32. a. Can you conclude that the mean thickness is

greater than 41 mm? b. Can you conclude that the mean thickness is

less than 41.4 mm? c. The target thickness is 41.2 mm. Can you

conclude that the mean thickness differs from the target value?

8. The article “Solid-Phase Chemical Fractionation of

Selected Trace Metals in Some Northern Kentucky Soils” (A. Karathanasis and J. Pils, Soil and Sediment Contamination, 2005:293-308) reports that in a sample of 26 soil specimens taken in a region of northern Kentucky, the average concentration of chromium (Cr) in mg/kg was 20.75 with a standard deviation of 3.93. a. Can you conclude that the mean concentration

of Cr is greater than 20 mg/kg? b. Can you conclude that the mean concentration

of Cr is less than 25 mg/kg? Exercises for Section 6.5 6. The article referred to in Exercise 3 categorized

firms by size and percentage of full-operating-capacity labor force currently employed. The numbers of firms in each of the categories are presented in the following table.

Percent of Full Operating-Capacity Labor Force Currently Employed

Small Large

>100% 6 8 95-100% 29 45 90-94% 12 28 85-89% 20 21 80-84% 17 22 75-79% 15 21 70-74% 33 29 <70% 39 34

Can you conclude that the distribution of labor force currently employed differs between small and large firms? Compute the relevant test statistic and P-value.

8. For the given table of observed values:

a. Construct the corresponding table of expected values.

b. If appropriate, perform the chi-square test for the null hypothesis that the row and column

outcomes are independent. If not appropriate, explain why.

Observed Values

1 2 3

A 25 4 11 B 3 3 4 C 42 3 5

12 The article “Determination’ of Carboxyhemoglobin

Levels and Health Effects on Officers Working at the Istanbul Bosphorus Bridge” (G. Kocasoy and H. Yalin, Journal of Environmental Science and Health, 2004:1129-1139) presents assessments of health outcomes of people working in an environment with high levels of carbon monoxide (CO). Following are the numbers of workers reporting various symptoms, categorized by work shift. The numbers were read from a graph.

Shift

Morning Evening Night

Influenza 16 13 18 Headache 24 33 6 Weakness 11 16 5 Shortness of Breath

7 9 9

Can you conclude that the proportions of workers with the various symptoms differ among the shifts?

Exercises for Section 6.6 4. A hypothesis test is to be performed, and the null

hypothesis will be rejected if P 5 0.05. If is in fact true, what is the maximum probability that it will be rejected?

6. A wastewater treatment program is designed to

produce treated water with a pH of 7. Let represent the mean pH of water treated by this process. The pH of 60 water specimens will be measured, and a test of the hypotheses versus will be made. Assume it is known from previous experiments that the standard deviation of the pH of water specimens is approximately 0.5. a. If the test is made at the 5% level, what is the

rejection region? b. If the sample mean pH is 6.87, will be

rejected at the 10% level? c. lf the sample mean pH is 6.87, will be

rejected at the 1% level? d. If the value 7.2 is a critical point, what is the

level of the test?

Exercises for Section 6.7 6. A process that manufactures glass sheets is

supposed to be calibrated so that the mean thickness of the sheets is more than 4 mm. The standard deviation of the sheet thicknesses is known to be well approximated by = 0.20 mm. Thicknesses of each sheet in a sample of sheets will be measured, and a test of the hypothesis versus will be performed. Assume that, in fact, the true mean thickness is 4.04 mm. a. If 100 sheets are sampled, what is the power of

a test made at the 5% level? b. How many sheets must be sampled so that a 5%

level test has power 0.95? c. If 100 sheets are sampled, at what level must

the test be made so that the power is 0.90? d. If 100 sheets are sampled, and the rejection

region is ̅ , what is the power ofthe test?

8. Water quality in a large estuary is being monitored

in order to measure the PCB concentration (in parts per billion). a. If the population mean is 1.6 ppb and the

population standard deviation is 0.33 ppb, what is the probability that the null hypothesis is rejected at the 5% level, if the sample size is 80?

b. If the population mean is 1.6 ppb and the population standard deviation is 0.33 ppb, what sample size is needed so that the probability is 0.99 that is rejected at the 5% level?

Exercises for Section 6.8 2. Five different variations of a bolt-making process

are run to see if any of them can increase the mean breaking strength of the bolts over that of the current process. The P-values are 0.13, 0.34, 0.03, 0.28, and 0.38. Of the following choices, which is the best thing to do next? i. Implement the process whose P-value was 0.03,

since it performed the best. ii. Since none of the processes had Bonferroni-

adjusted P-values less than 0.05, we should stick with the current process.

iii. Rerun the process whose P-value was 0.03 to see if it remains small in the absence of multiple testing.

iv. Rerun all the five variations again, to see if any of them produce a small P-value the second time around.

4. Five new paint additives have been tested t0 see if

any of them can reduce the mean drying time from the current value of 12 minutes. Ten specimens have been painted with each 0f the new types of paint, and the drying times (in minutes) have been measured. The results are as follows:

Additive

A B C D E

1 14.573 10.393 15.497 10.350 11.263 2 12.012 10.435 9.162 7.324 10.848 3 13.449 11.440 11.394 10.338 11.499 4 13.928 9.719 10.766 11.600 10.493 5 13.123 11.045 11.025 10.725 13.409 6 13.254 11.707 10.636 12.240 10.219 7 12.772 11.141 15.066 10.249 10.997 8 10.948 9.852 11.991 9.326 13.196 9 13.702 13.694 13.395 10.774 12.259 10 11.616 9.474 8.276 11.803 11.056

For each additive, perform a hypothesis test of the null hypothesis against the alternate You may assume that each population is approximately normal. a. What are the P-values for the five tests? b. On the basis of the results, which of the three

following conclusions seems most appropriate? Explain your answer. i. At least one of the new additives results in

an improvement. ii. None of the new additives result in an

improvement. iii. Some of the new additives may result in

improvement, but the evidence is inconclusive.