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HW#10 - Provide a practical interpretation of this...

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Unformatted text preview: Provide a practical interpretation of this interval. (0) Calculate a 99% lower confidence bound on the mean Compare this bound with the Inner bound of the two-side confidence interval and discuss why they are difierent. 8—37. The compressive strength of concrete is being teste by a civil engineer. He tests 12 specimens and obtains th followmg data. 2216 2237 2249 2204 2225 2301 228] 2263 - 2318 2255 2275 2295 (a) Check the assumption that compressive strength is normally I distributed. Include a graphical display in your answer. (b) Construct a 95% two-sided confidence interval on the mean strength. - (0) Construct a 95% IoWer confidence bound on the mean strength. Compare this'bound with the lower bound of the two-sided confidence interval and discuss why they are different. 8-38. A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected, and the diameter is measured. The resulting data (in millime- ters) are as follOWS: 8.24 8.25 8.20 8.23 8.24 5 8.21 8.26 8.26 8.20 8.25 :3 8.23 8.23 8.19 8.28 8.24 I: (a) Check the assumption ofnormality for rod diameter. (b) Calculate a 95% two-sided confidence interval on mean rod diameter. (c) Calculate a 95% upper confidence bound on the mean. Compare this bound with the upper bound of the two-sided confidence interval and discuss why they are different. fidence intervals. 266 CHAPTER 8 STATISTICAL mrERVALs FOR A SINGLE s (b) Calculate a 99% confidence interval on the mean it. 8—3 CONFIDENCE INTERVAL ON THE VARIANCE AND A 7 , STANDARD DEVIATION OF A NORMAL DISTRIBUTION Sometimes confidence intervals on the population varianceor standard deviation are needed. When the population is modeled by a normal distribution, the tests and intervals described in this section are applicable. The following result provides the basis of constructing these con- F‘wm Ml“ have. 8-39. An article in Computers a Electrical Engineering [“Parailel Simulation of Cellular Neural Networks” (1996, Vol. 22, pp. 61—84)] considered the speed-up of cellular neural new/0mg (CNN) for a parallel general-purpose computing architecture based on six transputers in different areas. The data follow: 3.775302 3.350679 4.217981 4.030324 4.639692 4.139665 4.395575 4.824257 4.268119 4.584193 4.930027 4.315973 4.600101 (a) Is there evidence to support the assumption that speed-up of CNN is normally distributed? Include a graphical dis— play in your answer. (b) Construct a 95% two-sided confidence interval on the mean speed-up. (c) Construct a 95% lower confidence bound on the mean seed—up. The wall thickness of 25 glass 2-liter bottles was mea- sured by a quality-control engineer. The sample mean was 3 = 4.05 millimeters, and the sample standard deviation was s = 0.08 millimeter. Find a 95% lower confidence bound for mean wall thickness. Interpret the interval you have obtained. 841. An article in Nuclear Engineering International (February 1988, p. 33) describes several characteristics of fuel rods used in a reactor owned by an electric utility in Norway. Measurements on the percentage of enrichment of 12 rods were reported as follows: 2.94 3.00 2.90 2.75 3.00 2.95 2.90 2.75 2.95 2.82 2.81 3.05 (21) Use a normal probability plot to check the normality as- sumption. (b) Find a 99% two-sided confidence interval on the mean per- centage of enrichment. Are you comfortable with the state- ment that the mean percentage of enrichment is 2.95%? Why? Xz Distribution Let X1, X2, . .. , X" be a random sample from a normal distribution with mean u and variance oz, and let S2 be the sample variance. Then the random variable has a chi-square (38) distribution with ii — 1 degrees of freedom. _ {n — l).512 ‘ _ —_._02 _ _ . (8—17) X2 298 CHAPTER 9 TESTS OF HYPOTI-IESES FOR A SINGLE SAMPLE 9—2. A semiconductor manufacturer collects data from a new tool and conducts a hypothesis test with the null hypothe- sis that a critical dimension mean width equals 100 nm. The conclusion is to not reject the null hypothesis. Does this result provide strong evidence that the critical dimension mean equals 100 um? Explain. 9—3. The standard deviation of critical dimension thickness in semiconductor manufacturing is cr = 20 nm. (a) State the null and alternative hypotheses used to demon— strate that the standard deviation is reduced. (b) Assume that thebprevious test does not reject the null hypothesis. Does this result provide strong evidence that the standard deviation has not been reduced? Explain. 9—4. The mean pull-off force of a connector depends on cure e. (a) State the null and alternative hypotheses used to demon- strate that the pull-off force is below 25 newtons. (h) Assume that the previous test does not reject the null hypoth- esis. Does this result provide strong evidence that the pull-off force is greater than or equal to 25 newtons? Explain. 9—5. A textile fiber manufacturer is hivestigating a new drap- ery yam, which the company claims has a mean thread elonga— tion of 12 kilograms with a standard deviation of 0.5 kilo- grams. The company wishes to test the hypothesis H0: p. = 12 against H1: p._< 12, using a random sample of four specimens. (a) What is the type I error probability if the critical region is defined as i < 11.5 kilograms? (b) Find B for the case where the true mean elongation is 11.25 kilograms. (c) Find B for the case where the true mean is 11.5 kilograms. 9—6. Repeat Exercise 9-5 using a sample size of n = 16 and the same critical region. 9—7. In Exercise 9—5, find the boundary of the critical region it" the type I error probability is (a) 0L=0.01andn=4 (c) ot=0.01andn=16 (b) or = 0.05 andn = 4 ((1)0 = 0.05 andn :16 9—8. In Exercise 9-5, calculate the probability of a type II error if the true mean elongation is l 1.5 kilograms and (5001 = 0.05 andn : 4 (b) or = 0.05 andn =16 (c) Compare the values of B calculated in the previous parts. What conclusion can you draw? 9—9. In Exercise 9-5, calculate the P—value if the observed statistic is (a) i=11.25 (h)i=11.0 (c) 5:11.75 9—10. The heat evolved in calories per gram of a cement - mixture is approximately normally distributed. The mean is thought to be 100 and the standard deviation is 2. We wish to test H0: p. = 100 versus H1: p. at 100 with a sample ofri! = 9 specimens. (21) If the acceptance region is defined as 98.5 S 2? 5 101.5, find the type I error probability 0:. (b) Find B for the caso where the true mean heat evolved is 103. HWtHO (c) Find B for the case where the true mean heat evolved is 105. This value of B is smaller than the one found in part (b) above. Why? 9—11. Repeat Exercise 9—10 using a sample size of n = 5 and the same acceptance region. 9-12.. In Exercise 9-10, find the boundary of the critical region if the type I error probability is (a)a=0.01andn=9 (c)or=0.01andn=5 (b)a=0.05andn=9 (d)o.=0.05andn=5 9—13. In Exercise 9-10, calculate the probability of a type 11 error if the true mean heat evolved is 103 and (a) ct = 0.05 andn = 9 (b) ot=0.05andn=5 (c) Compare the values of B calculated in the previous parts. What conclusion can you draw? 9-14. In Exercise 9-10, calculate the P—value if the observed statistic is (a) f=98 (b) i=101 (c) f=102 9—15. A consumer products company is formulating a new shampoo and is interested in foam height (in millimeters). Foam height is approximately normally distributed and has a standard deviation of 20 millimeters. The company wishes to test H0: p. = 175 millimeters versus H1: p. > 175 millime- ters, using the results of n : 10 samples. (a) Find the type 1 error probabilityo if the critical region is f > 185. I i (b) What is the probability of type 11 error if the true mean foam height is 185 millimeters? (c) Find B for the true mean of 195 millimeters. 9-16. Repeat Exercise 9-15 assuming that the sample size is r n = 16 and the boundary of the critical region is the same. 9—17. In Exercise 9-15, find the boundary of the critical region if the type I error probability is (a) (1:001 andn= 10 (c) ot=0.01andn= 16 (b or=0.05andn=10 (d)u=0.05andn=16 @ In Exercise 9-15, calculate the probability of a type II eor if the true mean foam height is 185 millimeters and (a) 01': 0.05 andn =10 (b) (I: (1.05de =16 (0) Compare the values of [3 calculated in the previous parts. What conclusion can you draw? 9—19. In Exercise 9-15, calculate the P-value if the observed statistic is . (a) 3-: =180 (b) i = 190 (c) i =‘ll70 9—20. A manufacturer is interested in the output voltage of a‘ power supply used in a PC. Output voltage is assumed to be normally distributed, With standard deviation 0.25 volt, and the manufacturer wishes to test H0: p. = 5 volts against H12“. #3 5 volts= using 7: = 8 units. (a) The acceptance region is 4.85 5 3: S 5.15. Find the value of or. (b) Find the power of the test for detecting a true mean output voltage of 5.1 volts. ' 9-2 rssrs ON THE MEAN or ANoaMAL DISTRIBUTION. VARIANCE KNOWN 299 9.21. Rework Exercise 9-20 when the sample size is 16 and the boundaries of the acceptance region do not change. What impact does the change in sample size have on the results of parts (a) and (b)? @ In Exercise 9-20, find the boundary of the critical re- gion if the type I error probability is (a) 0L=0.01811dn=8 (c) or=0.01andn= 16 (b)or=0.05andn=8 (d) o=0.05andn=16 9.23. In Exercise 9-20, calculate the P-value if the observed statistic is (a) i=5.2 Ga) 2:43 (C) i=5.l 9-24. In Exercise 9—20, calculate the probability of a type 11 error if the true mean output is 5.05 volts and (a) at = 0.05 andn =10 (b) or: 0.05 andn =16 (c) Compare the values of B calculated in the previous parts. What conclusion can you draw? 9—25. The proportion of adults living in Tempe, Arizona, who are college graduates is estimated to hep = 0.4. To test this hypothesis, a random sample of 15 Tempe adults is selected. If the number of college graduates is between 4 and 8, the hypothesis will be accepted; otherwise, we will conclude that p a5 0.4 . (a) Find the type I error probability for this procedure, assum- ing thatp = 0.4. ' (b) Find the probability of committing a type II error if the true proportion is really p = 0.2. 9—26. The proportion of residents in Phoenix favoring the building of toll roads to complete the freeway system is believed to be p = 0.3. If a random sample of 10 residents shows that l or fewer favor this proposal, we will conclude that p < 0.3. (2.) Find the probability of type I error if the true proportion is p = 0.3. . (b) Find the probability of committing a type H error with this procedure ifp = 0.2. (c) What is the power of this procedure if the true proportion is p = 0.2? 9—27. A random sample of 500 registered voters in Phoenix is asked if they favor the use of oxygenated fuels year-round to reduce air pollution. If more than 400 voters respond posi- tively, we will conclude that more than 60% of the voters favor the use of these fuels. (a) Find the probability of type I error if exactly 60% of the voters favor the use of these fuels. (b) What is the type II error probability B if 75% of the voters favor this action? ' Hint: use the normal approximation to the binomial. 9'28° If we plot the probability of accepting Hozu = no versus various values of p. and connect the points with a smooth curve, we obtain the operating characteristic curve (or the 0C curve) of the test procedure. These curves are used extensively in industrial applications of hypothesis testing to display the sensitivity and relative performance of the test. When the true mean is really equal to no, the probability of accepting H0 is 1 i or. (a) Construct an 0C curve for Exercise 9~15, using values ofthe true mean u of 178, 181, 184, 187, 190, 193, 196, and 199. (b) Convert the 0C curve into a plot of the power function of the test. 9-2 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN In this section, we consider hypothesis testing about the mean u. of a single normal population where the variance of the population 0*2 is known. We will assume that a random sample X1, X2, ... , X, has been taken fiom the population. Based on our previous discussion, the sample mean I? is an unbiased point estimator of u with variance 02/11. 9—2.,1 Hypothesis Tests on the Mean Suppose that we wish to test the hypotheses where no is a specified constant. We have a random sample X 1 ,X2, . ,Xn from a normal pop- . (9-7) ulation. Since f has a normal distribution (i.e., the sampling distribution of f is normal) r’r‘-r-'—- —-—-. . L. 1: «Sis-"=- .Qzl!#m-‘—.Aba%'r—éwwm-Hs—zhaswsemnwa;+eeaari‘newinie—-.w.n' 308 converted into a large—sample test procedure for unknown 0'2 that is valid regardless ofth-e form of the distribution of the population. This large-sample test relies on the central limit the. orem just as the large—sample confidence interval on p. that was presented in the previous chapter did. Exact treatment of the case where the population is normal, 0'2 is unknovm, anfi n is small involves use of the 1‘ distribution and will be deferred until Section 9-3. ' EXERCISES FOR SECTION 9-2 9-29. State the null and alternative hypothesis in each case. (a) A hypothesis test will he used to potentially provide evi- dence that the population-mean is greater than 10. (b) A hypothesis test will be used to potentially provide evi— dence that the population mean is not equal to 7. (c) A hypothesis test will be used to potentially provide evi- dence that the population mean is less than 5. 9-30. A hypothesis will be used to test that a population mean equals 7 against the alternative that the population mean does not equal 7 with known variance 0'. What are the critical Values for the test statistic 20 for the following significance levels? (a) 0.01 (b) 0.05 (c) 0.10 9-31. A hypothesis will be used to test that a population mean equals 10 against the alternative that the population mean is greater than 10 with known variance 0. What is the critical value for the test statistic 20' for the following sigmficance levels? (a) 0.01 (b) 0.05 (c) 0.10 9-32. A hypothesis will be used to test that a population mean equals 5 against the alternative that the population mean is less than 5 with known variance 0'. What is the critical value for the test statistic Zn for the following significance levels? a 0.01 (b) 0.05 (c) 0.10 a For the hypothesis test H0: p. = 7 against H]: n. =15 7 and variance known, calculate the P-value for each of the following test statistics. ' (a) 20 = 2.05 V (b) 20 2 .—1.84 (c) 20 = 0.4 9—34. Forthe hypothesis testHg: u = 10 against H1: is > 10 and variance known, calculate the P—value for each of the following test statistics. ' . (a) 20 = 2.05 (b) 20 = —1.84 (c) 20 = 0.4 9—35. For the hypothesis test H0: p. = 5 against H1: pt < 5 and variance known, calculate the P-value for each of the following test statistics. (3 Z : 20 = (C) In : @ Output from a sofiware package is given below: One-Sample Z: Test ofmu = 35 = vs not = 35 The assumed standard deflation = 1.8 Variable. N StDev SE Mean Z P x _ 25 1.475 ? ? ? Mean 35.710 (a) Fill in the missing items. What conclusions would you draw? (b) Is this a one-sided or a two-sided test? CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE (0) Use the normal table and the above data to construct a r 95% two-sided CI on the mean. (d) What would the P-value be if the alternative hypothesis is H]: u, > 35? 9-37. Output from a software package is given below: One-Sample Z: Test of mu = 20 vs > 20 The assumed standard deviation = 0.75 Variable N Mean StDev -SE Mean Z P x 10 19.889 ? (a) Fill in the missing items. What conclgsions would you draw? (b) Is this a one—sided or a two-sided test? (c) Use the normal table andrthe above data to construct a 95% two-sided CI 011 the mean. (d) What would the P-value be‘if the alternative hypothesis is H}: p. as 20? 9'38. Output from a software package is given below: _ One-Sample Z: Test ofmu = 14.5 vs > 14.5 The assumed standard deviation = 1.1 Variable N Mean StDev SE Mean Z P x 16 15.016 1.015 '1 ‘ :2 ‘? (a) Fill in the missing items. What conclusions would you draw? (b) Is this a one—sided or a two-sided test? (c) Use the normal table and the above data to construct a 95% lower bound on the mean. (d) What would the P—valuc be if the alternative hypothesis is H]: a we 14.5? ' 9'39. Output from a software package is given below: One-Sample Z: Test of mu = 99 vs > 99 The assumed standard deviation = 2.5 Variable N Mean StDev SE Mean Z P x 12 '100.039- 2.365 ? 1.44 0.075 _____‘—_—_4—___——- (a) Fill in the missing items. What conclusions would you draw? (b) Is this a one-sided or a two-sided test? 0.237 ? '2. mm. _-;4al‘.%_.uu.amammmmmm. ‘ ' -' ‘39 31"?" 1. '- 9c2 TESTS ON THE MEAN OF. A NORMAL DISTRlBUTION. VARIANCE KNOWN 309 (9) If the hypothesis had been H0: u. = 98 versus H .1 u. > 98, would yOu reject the null hypothesis at the 0.05 level of significauCe? Can you answer this without referring to the normal table? (d) Use the normal table and the above data to construct a 95% lower bound on the mean. (3) What w0uld the P-value be if the alternative hypothesis is H1: u. at 99? . 9,40. The mean water temperature downstream from a power plant cooling tower discharge pipe should be no more than 100°F. Past experience has indicated that the standard deviation of temperature is 2°F. The water temperature is measured on nine randomly chosen days, and the average temperature is found to be 98°F. - ' (a) Is there evidence that the water temperature is acceptable 'at or = 0.05? (b) What is the P—value for this test? (c) What is the probability of accepting the null hypothesis ' at or _= 0.05 if the water has a true mean temperature of 104°F? a 9-41. A manufacturer produces crankshafts for an automo- bile engine. The wear of the crankshaft after 100,000 miles (0.0001 inch) is of interest because it is likely to have an impact on warranqr claims. A random sample of n = 15 shafis is tested and i = 2.78. It is known that 0' : 0.9 and that wear is normally distributed. (a) Test H0: p. = 3 versus H,: p. at 3 using or = 0.05. i (b) What is the power of this test if p. = 3.25? (c) What sample size would be required to detect a true mean of 3.75 if we wanted the power to be at least 0.9? 9-42. A melting point test of a = 10 samples of a binder used in manufacturing ‘a rocket propellant resulted in i = 154.2”F. Assume that the melting point is normally dis- tributed with 0' = 15°F. (a) Test H0: p. = 155 versus H1: p. at 155 usingoi = 0.01. (b) What is the P-value for this test? (0) What is the B-error if the true mean is p. = 150? (d) :What value of u would be required if we want B < 0.1 when p. = 150? Assume that or = 0.01. The life in hours of a battery is known to be approxi- ately normally distributed with standard deviation 0' = 1.25 hours. A random sample of 10 batteries has a mean life of J? = 40.5 hours. ' (a) Is there evidence to support the claim that battery life exceeds 40 hours? Use a = 0.05. (b) What is the P-value for the test in part (a)? (c) What is the B—error for the test in part (a) if the true mean life is 42 hours? (d) What sample size would be required to ensure that B does not exceed 0.10 if the true meanllife is 44 hours? (8) Explain how you could answer the question in part (a) by calculating an appropriate confidence bound on life. 9-44. An engineer who is studying the tensile strength of a steel alloy intended for use in golf club shafts knows that tensile strength is approximately normally distributed with o = 60 psi. A random sample of 12 specimens has a mean tensile strength of it = 3450 psi. (a) Test the hypothesis that mean strength is 3500 psi. Use or = 0.01. . (b) What is the smallest level of significance at which you would be willing to reject the null hypothesis? (c) What is the B-error for the test in part (a) if the true mean is 3470? (d) Suppose that we wanted to reject the null hypothesis with probability at least 0.8 if mean strength u. = 3500. What sample size should be used? ' ...
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