HW#11 - 9,75 Consider the test of H0 0'2 5 against H1...

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Unformatted text preview: 9,75. Consider the test of H0: 0'2 : 5 against H1: 0'2 < 5. Approximate the P—value for each of the following test statistics. (3) xfi = 25.2 andn = 20 (b) 1%, =15.2andn =12 = 4.2 and r1 = 15 Consider the hypothesis test of H0: CI"2 = 10 against 1: 02 > 10. Approximate the P-value for each of the follow— ing test statistics. (a) xfi = 25.2 andn = 20 ('3) x3 =15.2andn =12 (c) 1% = 4.23am =15 9-77. The data from Medicine and Science in Sports and Exercise described in Exercise 8-48 considered ice hockey player performance afier electrostirnulation training. In summary, there were 17 players and the sample standard devi— ation of performance was 0.09 seconds. (a) Is there strong evidence to conclude that the standard devi- ation of performance time exceeds the historical value of 0.75 seconds? Use or = 0.05. Find the P—value for this test. (b) Discuss how part (a) could be answered by constructing a 95% one-sided confidence interval for 0. 9-78 The data from Technometrics described in Exercise 8- 1 considered the variability in repeated measurements of the weight of a sheet of paper. In summary, the sample stan- C ' (lard deviation from 15 measurements was 0.0083 grams. (a) Does the measurement standard deviation differ from 0.01 grams at ct = 0.05? Find the P—value for this test. (b) Discuss how part (a) could be answered by constructing a confidence interval for 0'. 9-79. Reconsider the percentage of titanium in an alloy used in aerospace castings from Exercise 8-47. Recall thats = 0.37 andn = 51. . (a) Test the hypothesis Ho: 0 = 0.25 versus H1: 0‘ sh 0.25 using or = 0.05. State any necessary assumptions about the underlying distribution of the data. Find the P—value. (b) Explain how you could answer the question in part (a) by constructing a 95% two-sided confidence interval for 6. 9-80. Data from an Izod impact test was described in Exercise 8-28. The sample standard deviation was 0.25 and H = 20 specimens were tested. 95 TESTS ON A POPULATION PROPORTION 32.3 conclusion. State any necessary assumptions about the underlying distribution of the data. (b) What is the P-value for this test? (c) Could the question in part (a) have been answered by constructing a 99% tWO-sided confidence interval for 0'2? 9-81. Data for the life was described in Exercise 8-27. The sample standard deviation was 3645.94 kilometers and n = 16. (a) Can you conclude, using or = 0.05, that the standard devia- tion of tire life is less than 4000 kilometers? State any nec~ essary assumptions about the underlying distribution of the data. Find the P-value for this test. (b) Explain how you could answer the question in part (a) by constructing a 95% one—sided confidence interval for (r. 9—82. It the standard deviation of hole diameter exceeds 0.01 millimeters , there is an unacceptably high probability that the rivet will not fit. Suppose that n = 15 and s = 0.008 mil: limeter. (a) Is there snong evidence to indicate that the standard devi- ation of hole diameter exceeds 0.01 millimeter? Use or = 0.01. State any necessary assumptions about the underly- ing distribution of the data. Find the P-value for this test. (b) Suppose that the actual. standard deviation of hole diame— ter exceeds the hypothesized value by 50%. What is the probability that this difference will be detected by the test described in part (a)? (c) If U is really as large as 0.0125 millimeters, what sample size will be required to detect this with power of at least 0.8? 9-83. Recall the sugar content of the syrup in canned peaches from Exercise 8-46. Suppose that the variance is thought to be 02 = 18 (milligramsf. Recall that a random sample of n = 10 cans yields a sample standard deviation of s = 4.8 milligrams. (a) Test the hypothesis H0: cr2 = 18 versus H1: 02 at 18 using (1 = 0.05. Find the P—value for this test. (b) Suppose that the actual standard deviation is twice as large as the hypothesized value. What is the probability that this difierence will be detected by the test described in part (a)? ' ' (c) Suppose that the true 'variance is o2 = 40. How large a sample wouldbe required to detect this difference with probability at least 0.90? (a) Test the hypothesis that 0' = 0.10 against an alternative specifying that (I at 0.10, using or = 0.01, and draw a 9—5 TESTS ON A POPULATION PROPORTION It is often necessary to test hypotheses on a population proportion. For example, suppose that a random sample of size n has been taken from a large (possibly infinite) population and that X(S n) observations in this sample belong to a class of interest. Then P = X/n is a point estimator of the proportion of the population p that belongs to this class. Note that n. and p are the parameters of a binomial distribution. Furthermore, from Chapter 7 we know that the sampling distribution of P is approximately normal with mean p and variance p(l — p) / n, if are. 96 SUMMARY TABLE OF INFERENCE PROCEDURES FOR A SINGLE SAMPLE 3 29 9.86. Suppose that 1000 customers are surveyed and 850 are satisfied or very satisfied with a corporation’s products and services. (a) Test the hypothesis Ho: p -—— 0.9 against Hi: p at 0.9 at or = 0.05. Find the P-value. ([3) Explain how the question in part (a) could be answered by constructing a 95% two—sided confidence interval for p. Suppose that 500 parts are tested in manufacturing an 10 are rejected. (a) Test the hypothesis H0: p = 0.03 against H1: p < 0.03 at or = 0.05. Find the P—value. (b) Explain how the question in part (a) could be answered by constructing a 95% one-sided confidence interval for p. 988. A random sample of 300 circuits generated 13 defectives. (a) Use the data to test H0: p = 0.05 versus H ,1 p =# 0.05. Use or = 0.05. Find the P—value for the test. (b) Explain how the question in part (a) could be answered with a confidence interval. 9'89. An article in the British Medical Journal [“Comparison of Treatment of Renal Calculi by Operative Surgery, Percutaneous Nephrolithotomy, and Extra-Corporeal Shock Wave Lithotrips,” (1986, Vol. 292, pp. 879—882)] found that percutaneous nephrolithotomy (PN) had a success rate in removing kidney stones'of 289 out of 350 patients. The tradi- tional method was 78% effective. (a) Is there evidence that the success rate for PN is greater than the historical success rate? Find the P-value. (b) Explain how the question in part (a) could be answered with a confidence interval. 9—90. A manufacturer of interocular lenses is qualifying a new grinding machine and will qualify the machine if there is evidence that the percentage of polished lenses that contain Surface defects does not exceed 2%. A random sample of 250 lenses contains six defective lenses. (3) Formulate and test an appropriate set of hypotheses to de- termine if the machine can be qualified Use or = 0.05. Find the P-value. 0)) Explain how the question in part (a) could be answered with a confidence interval. 9-91. A researcher claims that at least 10% of all football helmets have manufacturing flaws that could potentially cause injury to the wearer. A sample of 200 helmets revealed that 16 helmets contained such defects. (a) Does this finding support the researcher’s claim? Use at = 0.01. Find the P-value. (b) Explain how the question in part (a) could he answered with a confidence interval. 9—92. An article in Fortune (September 21, 1992) claimed that nearly one-half of all engineers continue academic studies beyond the BS. degree, ultimately receiving either an MS. or a PhD. degree. Data from an article in Engineering Horizons (Spring 1990) indicated that 117 of 484 new engineering graduates were planning graduate study. (a) Are the data from Engineering Horizons consistent with the claim reported by Fortune? Use or = 0.05 in reaching your conclusions. Find the P-value for this test. 03) Discuss how you could have answered the question in part (a) by constructing a two—sided confidence interval on p. 9-93. The advertised claim for batteries for cell phones is set at 48 operating hours, with proper charging procedures. A study of 5000 batteries is carried out and 15 stop operating prior to 48 hours. Do these experimental results support the claim that less than 0.2 percent of the company’s batteries will fail during the advertised time period, with proper charging procedures? Use a hypothesis-testing procedure with or = 0.01. 994. 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 315 voters respond posi- tively, we will conclude that at least 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? ' In a random sample of 85 automobile engine crank- s: . ' bearings, 10 have a surface finish roughness that exceeds the specifications. Does this data present strong evidence that the proportion of crankshaft bearings exhibiting excess surface roughness exceeds 0.10? (a) State and test the appropriate hypotheses using or = 0.05. (b) If it is really the situation that p = 0.15, how likely is it that the test procedure in part (a) will not reject the null hypothesis? (c) If p = 0.15, how large would the sample size have to be for us to have a probability of correctly rejecting the null hypothesis of 0.9‘? 9-6 SUMMARY TABLE OF INFERENCE PROCEDURES FOR A SINGLE SAMPLE The table in the end papers of this book (inside back cover) presents a summary of all the single-sample inference procedures from Chapters 8 and 9. The table contains the null hypothesis statement, the test statistic, the various alternative hypotheses and the criteria “1.4 Lina 360 CHAPTER 10 STATISTICAL INFERENCE FOR Two SAMPLES 10%. Consider the hypothesis test Ho‘: 11.1 = pt; against H] : u; > n2 with known variances 9‘1 = 10 and o; = 5. Suppose that sample sizes n1 = 10 and it; = 15 and that i, = 24.5 and E2 = 21.3. Use or = 0.01. (a) Test the hypothesis and find the P—value. (b) Explain how the test could be conducted with a confi- dence interval. (c) What is the power of the test in part (a) if ul is 2 units greater than M? (d) Assuming equal sample sizes, what sample size shouldbe used to obtain B = 0.05 if u] is 2 units greater than 1.1.2? Assume that or = 0.05. 10-4. Two machines are used for filling plastic bottles with a net volume of 16.0 ounces. The fill volume can be assumed normal, with standard deviation 0', = 0.020 and oz = 0.025 ounces. A member of the quality engineering staff suspects that both machines fill to the same mean net volume, whether or not this volume is 16.0 ounces. A random sample of 10 hot— tles is taken from the output of each machine. 7, Machlnez 16:03 7 , . . '136102- _. 116.03. _16.04 15.96 _ _ i_5.97_ 16.04 116.051 " ,-;.1,5:9s '_ 15396; 16.02 ' 16.05 16.02 7 16.01 16.01 .-:1:6_L'02'--_""1715.907 r...'---1__5f99._’2_f- 16.00 (a) Do you think the engineer is correct? Use or = 0.05. What is the P—value for this test? (b) Calculate a 95% confidence interval 011 the difi'erence in means. Provide a practical interpretation of this interval. (0) What is the power of the test in part (a) for a true diifer- ence in means of 0.04? (d) Assuming equal sample sizes, what sample size should be used to assure that B = 0.05 if the true difference in cans is 0.04? Assume that Gt = 0.05. Two types of plastic are suitable for use by an elec4 tronics component manufacturer. The breaking strength of this plastic is important. It is known that 01 = 0'; = 1.0 psi. From a random sample of size r11 = 10 and r22 = 12, we obtain 31 = 162.5 and 32 = 155.0. The company will not adopt plas— tic 1 unless its mean breaking strength exceeds that of plastic 2 by at least 10 psi. (a) Based on the sample information, should it use plastic 1? Use or = 0.05 in reaching a decision. Find the P—value. (b) Calculate a 95% confidence interval on the difference in means. Suppose that the true difference in means is really 12 psi. (c) Find the power of the test assuming that or = 0.05. (d) If it is really important to detect a difference of 12 psi, are the sample sizes employed in part (a) adequate, in your opinion? 10-6. The burning rates of two different solid-fuel propel- ' lants used in aircrew escape systems are being studied. It is known that both propellants have approximately the same standard deviation of burning rate; that is 0'] = 0'2 =3 centimeters per second. Two random samples of n] = 20 and a2 = 20 specimens are tested; the sample mean burning rates are 3?, = 18 centimeters per second and $2 = 24 centimeters per second. (a) Test the hypothesis that both propellants have the same mean burning rate. Use (1'. = 0.05. What is the P—Value'? (b) Construct a 95% confidence interval on the difference in‘ means 3)., — [4.2. What is the practical meaning of this interval? (c) What is the B—error of the test in part (a) if the true differ- ence in mean burning rate is 2.5 centimeters per second? (CD Assuming equal sample sizes, what sample size is needed to obtain power of 0.9 at a true diiTerence in means of 14 cm/S? 10—7. Two different formulations of an oxygenated motor fuel are being tested to study their road octane numbers. The variance of road octane number for formulation 1 is of = 1.5, and for formulation 2 it is 0% = 1.2. Two random satu- ples of size HI = 15 and n; = 20 are tested, and the mean road octane numbers observed are 31 = 89.6 and E2 = 92.5. Assume normality. (a) If formulation 2 produces a higher road octane number than formulation 1, the manufacturer would like to detect it. Formulate and test an appropriate hypothesis, using or = 0.05. What is the P—value? (b) Explain how the question in part (a) could be answered with a 95% confidence interval on the difference in mean road octane number. (c) What sample size would be required in each population if we wanted to be 95% confident that the error in esti- mating the difference in mean road octane number is less than 1? 108. A polymer is manufactured in a hatch chemical process. Viscosity measurements are normally made on each batch, and long experience with the process has indicated that — the Variability in the process is fairly stable with [T = 20. Fifteen batch viscosity measurements are given as follows: 724, 718, 776, 760, 745, 759, 795, 756, 742, 740, 761, 749, 739, 747, 742 A process change is made which involves switching the type of catalyst used the process. Following the process change, eight batch viscosity measurements are taken: 735, 775, 729, 755, 783, 760, 738, 780 Assume that process variability is unaffected by the catalyst change. If the difference in mean batch viscosity is 10 or less, the manufacturer would like to detect it with a high probability. (a) Formulate and test an appropriate hypothesis using or = 0.10. What are your conclusions? Find the P-value. 10,2 INFERENCE ON THE DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES UNKNOWN 369 ; Therefore, the pooled standard deviation estimate is Sp = V1952 = 4.4. The 95% confidence interval is found using Equation 10-19: 3. 1 1 7“ * — :‘c -t s — + — s — i I; 2 0025,23 p HI R: M P-z ‘ < _ _ I l — x! — x2 + r0,025,233,» :1 + IT: or upon substimfing the sample values and using raw, = 2.069, ' l 1 1 i — . — . 4.4 — + —- S — 90.0 87 0 2 069( ) 10 15 a, p, ' s 90.0 — 37.0 + 2.069(4.4)1ii + i ‘ 10 15 Case 2.: 0% at: 0% which reduces to —0.72 5 ll] — pa 5 6.72 Practical Interpretation: Notice that the 95% confidence interval includes zero; therefore, at this level of confidence we cannot conclude that there is a difference in the means. Put another way, there is no evidence that doping the cement with lead affected the mean weight percent of calcium; therefore, we cannot claim that the presence of lead affects this aspect of the hydration mechanism at the 95% level of confidence. In many situations it is not reasonable to assume that 011 = 6%. When this assumption is unwarrantediwe niay still find a 10 that T3 = [X] — X; * (pd — u2)]/ Shi/n] + Syn; is distributed approximately as twith 0(1 — 0L % confidence interval on m i 51.2 using the fact \/_)__ degrees of freedom v given by Equation 10-16. The CI expression follows. Case 2: Approximate Confidence Interval on the Difference in : Means, I Variances- Unknown Are Not Assumed I Equal ' gm'esssp, .:— p;ng '- _- EXERCISES FOR SECTION 102 _.'If_ifi,’5‘c;,_s%, fails-are the insane andivariances aim ia‘ndoiri assessors-see a, "an . _ _' a2, respectiveljg fi‘Om two independent normal populations with unknown and unequal ' _ '_ Variances; an_approxirnate-._100(_1._—. 00% 'confidenc.e::iriterva1_on the difference in; . - ‘ _' wheie {is given 10— 1 Grand theupper (it/2 percentagepoint- of .i. -' - idem-115.11% with? désfeés offreédomt _-. ' l " ' i '= ' - - - .0949?- - ( 10-10.i Consider the computer output below. (c) This test was done assuming that the two population vari— . ances were equal. Does this seem reasonable? Two-Sample T—Test and CI Sample N Mean StDev SE Mean 1 12 10.94 1.26 0.36 2 16 12.15 1.99 0.50 Difference = mu (1) H mu (2) Estimate for difference: —l.210 95% CI for difference: (—2.560, 0.140) T-Test of difference r“: 0 (vs not =) : T—Value == ? P-Value = ? DF 5 ? Both use Pooled StDev 5 '? (61) Fill in the missing values. Is this a one-sided or a two—sided test? Use lower and upper bounds for the P-value. (b) What are your conclusions ifOL = 0.05? What ifs: = 0.01? (0) Suppose that the hypothesis had been H0: in = pa versus H0: is] < pa. What would your conclusions be if OL = 0.05? 10-1 1. Consider the computer output below. Two-Sample T—Test and Cl Sample N Mean StDev SE Mean 1 15 54.73 2.13 0.55 2 20 58.64 5.28 1.2 Difference = mu (1) - mu (2) Estimate for difference: —3.91 95% upper bound for difference: '9 T—Test of difference = 0(vs <): T-Value = —3.00 P~Value = ? DF = ? i l vi 2 l l .1 '1 l f 0 l 370 (a) Fill in the missing values. Is this a one-sided or a two-sided test? Use lower and upper bounds for the P-value. (b) What are your conclusions if 0t = 0.05? What if or = 0.01? (c) This test was done assuming that the two population vari- ances were different. Does this seem reasonable? (d) Suppose that the hypotheses had been H0: |J..1 = 11.2 versus H0: u, 7": 114. What would your conclusions be if or = 0.05? @ Consider the hypothesis test H0 : u! = u; against 1 : it, is uz.‘ Suppose that sample sizes are 721 = 15 and n, 2 15, that r, 2 4.7 and r, 2 7.8, and that s? 2 4 and 3% = 6.25. Assume that 621 = 0% and that the data are drawn fi'om normal distributions. Use or = 0.05. (a) Test the hypothesis and find the P-value. (b) Explain how the test could be conducted with a confi- dence interval. (c) What is the power of the test in part (a) for a true differ- ence in means of 3? ' (d) Assuming equal sample sizes, what sample size should be used to obtain B = 0.05 if the true difference in means is —2? Assume that or = 0.05. 10-13. Consider the hypothesis test H0 : |J..1 = it; against H] : “.1 < 11.2. Suppose that sample sizes n] = 15 and n; = 15, that r, 2 6.2 and s, 2 7.8, and that 3% 2 4 and 33 2 6.25. Assume that 0'] : 0% and that the data are drawn from normal distributions. Useor = 0.05. (a) Test the hypothesis and find the P—value. (b) Explain how the test could be conducted with a confi- dence interval. (c) Whatis the power of the test in part (a) if it] is 3 units less than 11.2? (d) Assuming equal sample sizes, what sample size should be used to obtain [3 = 0.05 if 11.1 is 2.5 units less than 1L2? Assume that at = 0.05. 10-14. Consider the hypothesis test H0 : u1 = it; against H1 : '11 > [1.2. Suppose that sample sizes 121 = 10 and 722 = 10, that rt, 2 7.3 and i, 2 5.6, and that 52, 2 4 and SE 2 9. Assume that 0% 2 0% and that the data are drawn fiom normal distributions. Use or = 0.05. 2 (a) Test the hypothesis and find the P—value. (b) Explain how the test could be conducted with a confi- dence interval. (c) What is the power of the test in part (a) if it] is 3 units greater than 13.2? (d) Assuming equal sample sizes, what sample size should be used to obtain [3 = 0.05 if M is 3 units greater than 112? Assume that at = 0.05. 10-15. The diameter of steel rods manufactured on two dif- ferent extrusion machines is being investigated. Two random samples of sizes 11] = 15 and h2 = 17 are selected, and the sample means and sample variances are it] = 8.73, s, = 0.35, 22 = 8.68, and s; = 0.40, respectively. Assume that o] = 6% and that the data are drawn from a normal distribution. (a) Is there evidence to support the claim that the two ma- chines produce rods with different mean diameters? CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES Use or = 0.05 in arriving at this conclusion. Find the P-value. (b) Construct a 95% confidence interval for the difierence in mean rod diameter. Interpret this interval. 10-16. An article in Fire Technology investigated two dif. ferent foam expanding agents that can be used in the nozzles of fire-fighting spray equipment. A random sample of five ob- servations with an aqueous film-forming foam (AFFF) had a sample mean of 4.7 and a standard deviation of 0.6. A random sample of five observations with alcohol-type concentrates (AT C) had a sample mean of 6.9 and a standard deviation 0.8_ (a) Can you draw any conclusions about differences in mean foam expansion? Assume that both populations are well represented by normal distributions with the same stan— dard deviations. (b) Find a 95% confidence interval on the difference in mean foam expansion of these two agents. 10-17. Two catalysts may be used in a batch chemical process. Twelve batches were prepared using catalyst l , resulting in an average yield of 86 and a sample standard deviation of 3, Fifteen batches- were prepared using catalyst 2, and they resulted in an average yield of 89 with a standard deviation of 2. Assume that yield measurements are approximately normally distributed with the same standard deviation. (a) Is there evidence to support a claim that catalyst 2 pro- duces a higher mean yield than catalyst 17 Use or = 0.01. (b) Find a 99% confidence interval on the difference in mean yields that can be used to test the claim in part (a). 10-18. The deflection temperature under load for two dif—_ ferent types of plastic pipe is being investigated. Two random- samples of 15 pipe specimens are tested, and the deflection temperatures observed are as follows (in 0E): Type 1: 206, 188, 205, 187, 194, 193, 207, 185, 189, 213, 192, 210, 194, 178, 205 Type 2: 177, 197, 206, 201, 180, 176, 185, 200, 197, 192, 198, 188, 189, 203, 192 (a) Construct box plots and normal probability plots for the two samples. Do these plots provide support of the asstunp-' tions of normality and equal variances? Write a practical interpretation for these plots. (b) Do the data support the claim that the deflection tempera- ture under load for type 1 pipe exceeds that of type 2? [u reaching your conclusions, use or = 0.05. Calculate a P—value. (c) Ifthe mean deflection temperature for type 1 pipe exceeds that of type 2 by as much as 5°F, it is important to detect this difi'erence with probability at least 0.90. Is the choice of m = 712 = 15 adequate? Use a = 0.05. 10-19. In semiconductor manufacturing, wet chemical etch— ing is often used to-remove silicon from the backs of wafers prior to metallization. The etch rate is an important characteris- tic in this process and known to follow a normal distribution- Two different etching solutions have been compared, using two 10,2 {NFERENCE ON THE DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS. VARIANCES UNKNOWN 3 7 1 random samples of 10 wafers for each solution. The observed etch rates are as follows (in mils per minute): minimal. Salaam 2 ‘ 9's ' 10.6 r 10.2 . 10.0 9.4 10.3 10.5 10.2 7 9.3' 10.0. I ' 10.7 ' jicj 9.6 10.3 10.4 10.4 7710.2" ' 10.1 : ' 710.5' ' 10.3. (a) Construct normal probability plots for the two samples. Do these plots provide support for the assmnptions of nor- mality and equal variances? Write a practical interpreta- tion for these plots. (b) Do the data support the claim that the mean etch rate is the same for both solutions? In reaching your conclusions, use 01 = 0.05 and assume that both population variances are equal. Calculate a P-value. (c) Find a 95% confidence interval on the difference in mean etch rates. _ Two suppliers manufacture a plastic gear used in a ser printer. The impact strength of these gears measured in foot-pounds is an important characteristic. A random sample of 10 gears from supplier 1 reSults in i1 = 290 and s] = 12= while another random sample of 16 gears from the second supplier results min = 321 and s2 = 22. '(a) Is there evidence to support the claim that supplier 2 ' provides gears with'higher mean impact strength? Use or : 0.05, and assume that both populations are normally distributed but the variances are not equal. What is the P—value for this test? (1)) Do the data support the claim that the mean impact strength of gears from supplier 2 is at least 25 foot—pounds higher than that of supplier 1? Make the same assump— tions as in part (a). (C) Construct a confidence interval estimate for the difference in mean impact strength, and explain how this interval could be used to answer the question posed regarding supplier- to-supplier differences. 10-21. The melting points of two alloys used in formulating SOlder were investigated by melting 21 samples of each mate- rial. The sample mean and standard deviation for alloy 1 was 31 = 420°F and S1 = 4°12, while for alloy 2 they were 7:2 = 426°F anus2 = 31°F. (3) Do the sample data support the claim that both alloys have M the same melting point? Use or = 0.05 and assume that both populations are normally distributed and have the same standard deviation. Find the P-value for the test. (b) Suppose that the true mean difference in melting points is 3°F. How large a sample would be required to detect this . difference using an or = 0.05 level test with probability at least 0.9? Use 0'1 = 02 = 4 as an initial estimate of the common standard deviation. 1042. A photoconductor film is manufactured at a nominal thickness of 25 mils. The product engineer wishes to increase the mean speed of the film, and believes that this can be achieved by reducing the thickness of the film to 20 mils. Eight samples of each film thickness are manufactured in a pi- lot production process, and the film speed (in microj oules per square inch) is measured For the 25—mil film the sample data result is £1 = 1.15 and s] = 0.11, while for the 20-mil film the data yield 5:; = 1.06 and s2 = 0.09. Note that an increase in film speed would lower the value of the observation in microjoules per square inch. (a) Do the data support the claim that reducing the film thick- ness increases the mean speed of the film? Use or = 0.10 and assume that the two population variances are equal and the underlying population of film speed is normally distributed. What is the P-value for this test? (b) Find a 95% confidence interval on the difference in the two means that can be used to test the claim in part (a). 1043. Two companies manufacture a rubber material in- tended for use in an automotive application. The part will be subjected to abrasive wear in the field application, so we decide to compare the material produced by each company in a test. Twenty-five samples of material from each company are tested in an abrasion test, and the amount of wear after 1000 cycles is observed. For company 1, the sample mean and standard de- viation of wear are i1 : 20 miliigramsflOOO cycles and s1 = 2 milligrams/ 1000 cycles, while for company 2 we obtain E2 = 15 milligrams/ 1000 cycles and .92 = 8 milligrams! 1000 cycles. (a) Do the data support the claim that the two companies pro- duce material with different mean wear? Use or = 0.05, and assume each population is normally distributed but that their variances are not equal. What is the P-value for this test? (b) Do the data support a claim that the material from com- pany 1 has higher mean wear than the material from com- pany 2? Use the same assumptions as in part (a). (c) Construct confidence intervals that will address the ques- tions in parts (a) and (b) above. 10—24. The thickness of a plastic film (in mils) on a sub- strate material is thought to be influenced by the temperature at which the coating is applied. A completely randomized experiment is carried out. Eleven substrates are coated at 125°F, resulting in a sample mean coating thickness of it] = 103.5 and a sample standard deviation of s; t 10.2. Another 13 Substrates are coated at 150°F, for which it; = 99.7 and 32 = 20.1 are observed. It was originally suspected that raising the process temperature would reduce mean coating thickness. (a) Do the data support this claim? Use or = 0.01 and assume that the two population standard deviations are not equal. Calculate an approximate P—value for this test. (b) How could you have answered the question posed regard- ing the effect of temperature on coating thickness by using a confidence interval? Explain your answer. Hale Design Design Language Language Programmer ' . l ' 2 (a) Is the assumption that the difference in coding time is nor— mally distributed reasonable? 0:) Find a 95% confidence interval on the difference in mean coding times. ls there any indication that one design lan— guage is preferable? 10-41. Fifteen adult males between the ages of 35 and 50 participated in a study to evaluate the effect of diet and exercise on blood cholesterol levels. The total cholesterol was meas- ured in each subject initially and then three months afier participating in an aerobic exercise program and switching to a low-fat diet. The data are shown in the accompanying table _______—1—«— Blood cugtesteror Level 10-4 PAIRED t’TEST 381 (a) Do the data support the claim that low-fat diet and aerobic exercise are of value in producing a mean reduction in blood cholesterol levels? Use on = 0.05. Find the P-value. (b) Calculate a one-sided confidence limit that can be used c answer the question in part (a). An article in the Journal ofAircmfi (Vol. 23, 1986, p. 859—864) described a new equivalent plate analysis method formulation that is capable of modeling aircrafi struc- tures such as cranked wing boxes= and that produces results similar to the more computationally intensive finite element analysis method. Natural vibration frequencies for the cranked wing box structure are calculated using both methods, and results for the first seven natural frequencies follow: ' Finite, 7 Equivalent ' . ' Element ' ' Plate, Freq._ 'Cycler's ' 7' ejs 2 48.52 49.11) 4 113.99 117.53 6 " 212.72 “7220.14 (a) Do the data suggest that the two methods provide the same mean value for natural vibration frequency? Use or = 0.05. Find the P-value. (b) Find a 95% confidence interval on the mean difference between the two methods. 10—43. Ten individuals have participated in a diet-modification program to stimulate weight loss. Their weight both before and after participation in the program is shown in the following list. .fiybieot ' ' 1710'" ' ' 3'10 285 (a) Is there evidence to support the claim that this particular diet-modification program is effective in producing a mean weight reduction? Use OL = 0.05. (b) Is there evidence to support the claim that this particular diet-modification program will result in a mean weight loss of at least 10 pounds? Use or = 0.05. ...
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This note was uploaded on 03/13/2012 for the course ISE 130 taught by Professor Patel,n during the Spring '08 term at San Jose State.

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HW#11 - 9,75 Consider the test of H0 0'2 5 against H1...

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