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Introduction To Probability And Statistics For Science Engineering And Finance 1st Edition Walter A. Rosenkrantz - Solutions

Problem 10.25(This is a continuation of Problem 10.9.) (a) Display the results of your regression analysis in the format of Table 10.5. (b) Compute 95% confidence intervals for the regression coefficients. (c) Compute a 95% confidence interval for the mean gpm (gallons per hundred miles) of an
Problem 10.24(This is a continuation of Problem 10.8.) (a) Display the results of your regression analysis in the format of Table 10.5. (b) Compute 95% confidence intervals for the regression coefficients. (c) Compute a 95% confidence interval for the mean mpg of an automobile that weighs 3655 lbs.
Problem 10.23 Twelve batches of of plastic are made, and from each batch one test item was molded and its brinell hardness y was measured at time x. The results are recorded in the following table. Time(x) Hardness(y) Time(x) Hardness(y) 32 230 40 248 48 262 48 279 72 323 48 267 64 298 24 214 48
Problem 10.22(Continuation of Problem 10.21) (a) Draw the scatter plot for Model (Y = time |X = data bytes) (ARGUS DATA). (b) Display the results of the regression analysis in the format of Table 10.5. (c) What proportion of the variability is explained by the model? (d) Compute 95% confidence
Problem 10.21 The following data were obtained in a comparative study of a remote pro[1]cedure call (RPC) on two computer operating systems, UNIX and ARGUS.UNIX ARGUS DATA BYTES TIME DATA BYTES TIME 64 26.4 92 32.8 64 26.4 92 34.2 64 26.4 92 32.4 64 26.2 92 34.4 234 33.8 348 41.4 590 41.6 604 51.2
Problem 10.20(This is a continuation of the fuel efficiency Example 10.2.) The ANOVA table for the Model (Y = gpm|X = weight) follows. Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 1 0.59506 0.59506 23.916 0.0006 Error 10 0.24881 0.02488 C Total 11 0.84387 Root MSE
Problem 10.19(This is a continuation of the fuel efficiency Example 10.2.) (a) Using the computer output displayed in the Table 10.7 (and the result that x = 3.21, Sxx = 2.95), construct 95% confidence intervals for β0 and β1. (b) Compute a 95% confidence interval for the mpg when the automobile
Problem 10.18(a) Display the results of the regression analysis of Problem 10.4 in the format of an ANOVA table (Table 10.5). (b) What proportion of the variability is explained by the model? (c) Compute a 95% confidence interval for the regression coefficients. (d) Compute a 95% confidence
Problem 10.17(a) Display the results of the regression analysis of Problem 10.2 in the format of an ANOVA table (Table 10.5). (b) What proportion of the variability is explained by the model? (c) Compute a 95% confidence interval for the regression coefficients. (d) Compute a 95% confidence
Problem 10.16 A student gave the following expression for the fitted regression line of Figure 10.4: Y = −216.3271 + 3.990413 × x + . Explain why this expression is incorrect.
Problem 10.15 In Table 10.1 the weld diameter is the same for observations 4 and 6 (x = 165) but the shear strengths Y are different. How does the regression model (10.22) account for this?
Problem 10.14(a) Verify Equation 10.19, which asserts that (yi − yˆi)=0. (b) Verify Equation 10.20, which asserts that (yi − yˆi)xi = 0. (c) Verify Equation 10.21, which asserts that the arithmetic mean of the observed values equals the arithmetic mean of the fitted values; that is, 1 n yi =
Problem 10.13 Suppose yi = a + bxi, i = 1, . . . n. Show that r = +1 when b > 0 and r = −1 when b < 0.
Problem 10.12 The following table lists 10 measurements of three ground water quality parameters from city wells in Fresno, California: specific electrical conductivity (SEC), total dissolved solids (TDS), and silica (SiO2). SEC SiO2 TDS (ms/cm) (mg/l) (mg/l) 422 56 275 470 48 305 841 48 535 714 55
Problem 10.11(Continuation of Problem 10.9) Refer to the automobile fuel efficiency data of Problem 1.76. (a) Draw the scatter plot for the regression Model(Y = gpm |X = disp).(b) Calculate the least squares estimates βˆ0 and βˆ1 and draw the graph of the estimated regression line y = βˆ0 +
Problem 10.10(Continuation of Problem 10.8) Refer to the automobile fuel efficiency data of Problem 1.76. (a) Draw the scatter plot for the regression Model(Y = mpg |X = disp). (b) Calculate the least squares estimates of βˆ0 and βˆ1 and draw the graph of the estimated regression line y = βˆ0
Problem 10.9 (Continuation of Problem 10.8) Refer to the automobile fuel efficiency data of Problem 1.76. (a) Draw the scatter plot for the regression Model(Y = gpm |X = weight). (b) Calculate the least squares estimates βˆ0 and βˆ1 for the regression Model(Y = gpm |X = weight). (c) Compute the
Problem 10.8 Refer to the fuel efficiency data for 18 2004-05 model year automobiles of Problem 1.76. (a) Draw the scatter plot for the regression Model(Y = mpg |X = weight). (b) Calculate the least squares estimates of βˆ0 and βˆ1 and draw the graph of the estimated regression line y = βˆ0 +
Problem 10.7 Refer to the fuel efficiency data (Table 10.2). (a) Draw the scatter plot for the regression Model(Y = gpm |X = disp). (b) Calculate the least squares estimates βˆ0 and βˆ1 and draw the graph of the estimated regression line y = βˆ0 + βˆ1x. Use the same axes on which you
Problem 10.6 Refer to the fuel efficiency data (Table 10.2). (a) Calculate the least squares estimates βˆ0 and βˆ1 for the regression Model(Y = gpm |X = weight). Copy the scatter plot from Figure 10.3 and draw the graph of the estimated regression line y = βˆ0 + βˆ1x. Use the same axes on
Problem 10.5 Refer to the fuel efficiency data (Table 10.2). (a) Draw the scatter plot for the regression Model(Y = mpg |X = disp). (b) Calculate the least squares estimates of βˆ0 and βˆ1 and draw the graph of the estimated regression line y = βˆ0 + βˆ1x. Use the same axes on which you
Problem 10.4 The following data represent the number of disk I/Os (input-output) and CPU times for seven programs. Disk I/Os: 14, 16, 27, 42, 39, 50, 83 CPU times: 2, 5, 7, 9, 10, 13, 20 (Source: R. Jain (1991), The Art of Computer Systems Performance Analysis, John Wiley & Sons, New York. Used
Problem 10.3 The following data record the amount of water (x), in centimeters, and the yield of hay (y), in metric tons per hectare, on an experimental farm. water (x) 30 45 60 75 90 105 120 yield (y) 2.11 2.27 2.5 2.88 3.21 3.48 3.37 (a) Draw the scatter plot (xi, yi). (b) Calculate the least
Problem 10.2 (a) Draw the scatter plot corresponding to the shear strength data for the 0.064 gauge steel given in Table 10.1. (b) Calculate the least squares estimates of βˆ0 and βˆ1 and draw the graph of the estimated regression line y = βˆ0 + βˆ1x. Use the same axes on which you graphed
Problem 10.1 (a) Draw the scatter plot for the following artificial data. x: 0 0 1122 y: 0 1 1323(b) Sketch a straight line that seems to satisfactorily fit these points. Denoting the equation of your line by y = b0 + b1x find the values of b0, b1. (c) Calculate the least squares line y = βˆ0 +
Problem 9.18 In a comparative study of the effectiveness of two brands of motion sickness pills, brand A pills were given to 45 persons randomly selected from a group of 90 air travelers while the other 45 persons were given brand B pills. The responses were classified as: None (no motion
Problem 9.17 To test the effectiveness of drug treatment for the common cold, 164 patients with colds were selected, half were given the drug treatment (the treatment group), and the other half (the control group) were given a sugar pill. The patient’s reactions to the treatment and sugar pill
Problem 9.16 In a comparison study of two processes for manufacturing steel plates each plate is classified into one of three categories: no defects, minor defects, major defects. The results of the study are displayed in the following 2 × 3 contingency table.Process No defects Minor defects Major
Problem 9.15 The following data were collected in a study of the relationship between the blood pressures of children and their fathers. Blood pressures of the fathers were clasified as A1 (below average), A2 (average), and A3 (above average). The childrens’ blood pressures were classified
Problem 9.14 A survey was conducted to evaluate the effectiveness of a new flu vaccine. The vaccine was provided in a two-shot sequence over a period of two weeks. Some people received the two-shot sequence, some appeared for only the first shot, and others received neither. A survey of 1000
Problem 9.13 Each of 6805 pieces of moulded vulcanite made from a resinous powder was classified according two criteria: porosity and dimension. The results are displayed in the following 2 × 2 contingency table. Porous Nonporous Total With defective dimensions 142 331 473 Without defective
Problem 9.12 For each of 75 rockets fired the lateral deflection and range in yards were measured with the results displayed in the following 3 × 3 contingency table. Lateral Deflection (Yards) Range −250 to −51 −50 to 49 50 to 199 Total 0 − 1199 5 9 7 21 1200 − 1799 7 3 9 19 1800 −
Problem 9.11 To determine the possible effects of the length of a heating cycle on the brittleness of nylon bars, 400 bars were subjected to a 30 second heat treatment and 400 were subjected to a 90 second heat treatment. The nylon bars were then classified as brittle or non-brittle. Length of
Problem 9.10 The results of a newspaper poll of 1,000 voters to the question, “The Congress has passed and the President has signed a new tax cut. Overall, do you think this is a good thing or a bad thing?” are listed below. Do these data indicate any differences in the re[1]sponses of the
Problem 9.9 A political consultant surveyed 275 urban, 250 suburban, and 100 rural voters and obtained the following data concerning the public policy issue that is the most important to them.Do these data indicate any differences in the responses of the three classes of voters with respect to the
Problem 9.8 The following 2 × 2 contingency table gives the pass/fail results for the boys and girls who took the advanced level test in pure mathematics with statistics in Great Britain for the 1974-75 year. Sex Pass Fail Total Boy 891 569 1460 Girl 666 290 956 Totals 1557 859 2416 (Source: J.R.
Problem 9.7 Show that the multinomial distribution reduces to the binomial distribution when k = 2. Here Y1 and Y2 denote the number of successes and failures, respectively; use the notation p1 = p, p2 = 1 − p.
Problem 9.6 Monte Carlo simulations of complex random processes are widely used in science, engineering, and finance. These simulations rely on a uniform random number generator (also called pseudorandom number generators), which is a computer program that produces, or is supposed to produce, an
Problem 9.5 The following table gives the results of a high-energy physics experiment that counts the number of electron-positron pairs observed in a hydrogen bubble chamber exposed to a beam of photons. The number of electron-positron pairs are recorded on a bubble chamber photograph. Theoretical
Problem 9.4 The number X of chocolate chips per cookie produced by a large commercial bakery is assumed to have a Poisson distribution with λ = 5. To test this hypothesis a random sample of 100 cookies is chosen and the frequency distribution of the number of chocolate chips per cookie is recorded
Problem 9.3 Let X denote the number of heads that appear when 5 coins are tossed. It is clear that X has a binomial distribution with P(X = i) = b(i; 5, p),(i = 0,..., 5). The following data record the frequency distribution of the number of heads in n = 50 tosses of the 5 coins. Number of Observed
Problem 9.2 We define a fair die to mean that each face is equally likely; thus, there are six categories, and the null hypothesis we wish to test is H0 : pi = 1/6, i = 1,..., 6. To determine if a die is fair, as opposed to the alternative that it is not, a die was thrown n = 60 times, with the
Problem 9.1 A biology student repeating Mendel’s experiment described in Example 9.1 obtained the following results: Seed type ry wy rg wg Frequency 273 94 88 25 Are these data consistent with Mendel’s theory that the frequency counts of seeds of each type produced from this experiment occur in
Problem 8.70 Shingles is a virus infection of the nerves that causes a painful, blistering rash. It is the same virus that causes chickenpox. A new vaccine for shingles, Zostovax, was approved in May 2006 by the Food and Drug Administration (FDA). The approval was based on a large scale clinical
Problem 8.69 Patients suffering cardiac arrest were divided into two groups: Group 1 consisted of n1 patients who were given cardiac pulmonary resuscitation (CPR) by trained civilians until the arrival of an ambulance crew. Group 2 patients consisted of n2 patients whose CPR was delayed until the
Problem 8.68 The number of defective items out of a total production of 1125 is 28. The manufacturing process is then modified and the number of defective items out of a total production of 1250 is 22. Can you reasonably conclude that the modifications reduced the proportion of defective items?
Problem 8.67 A random sample of 250 items from lot A contains 10 defective items and a random sample of 300 items from lot B is found to contain 18 defective items. What conclusions can you draw concerning the quality of the two lots?
Problem 8.66 In a comparison study of two machines for manufacturing memory chips the following data were collected: Of 500 chips produced by machine 1, 20 were defective; of 600 chips produced by machine 2, 40 were defective. Can you conclude from these results that there is no significant
Problem 8.65 The natural recovery rate from a disease is 75%. A new drug is developed that, according to its manufacturer, increases the recovery rate to 90%. Suppose we test this hypothesis at the level α = 0.01. What is the minimal sample size needed to detect this improvement with probability
Problem 8.64 Suppose that we have to test H0 : p = 0.2 against H1 : p > 0.2 at the level α = 0.05. What is the minimum sample size needed so that the type II error at the alternative p = 0.3 is 0.1?
Problem 8.63 Before proposition A was put to the voters for approval, a random sample of 1300 voters produced the following data: 625 in favor of proposition A and 675 opposed. Test the null hypothesis that a majority of the voters will approve proposition A. Use α = 0.05. What is the alternative
Problem 8.62 A garden supply company claims that its flower bulbs have an 80% ger[1]mination rate. To challenge this claim, a consumer protection agency decides to test the following hypothesis: H0 : p = 0.80 against H1 : p < 0.80. The following data were collected: n = 100 flower bulbs were
Problem 8.61 Suppose the natural recovery rate from a disease is p0 = 0.4. A new drug is developed that, it is claimed, significantly improves the recovery rate. Consequently, the null hypothesis and its alternative are given by H0 : p = 0.4 against H1 : p > 0.4. Consider the following test: A
Problem 8.60 An urn contains two red balls, two white balls, and a fifth ball that is either red or white. Let p denote the probability of drawing a red ball. The null hypothesis is that the fifth ball is red. (a) Describe the parameter space Θ. (b) To test this hypothesis we draw 10 balls at
Problem 8.59 Let X be the sample total from a random sample of size n = 15 taken from a Bernoulli distribution with parameter p. To test H0 : p = 0.90 against H1 : p < 0.90, we choose the rejection region C = {x : x 0.4. Consider the following test: A random sample of 15 patients are given the
Problem 8.58 Let X be the sample total from a random sample of size n = 20 from a Bernoulli distribution with parameter p. To test H0 : p ≤ 0.40 against H1 : p > 0.40, we choose the rejection region C = {x : x>c}. (a) Find the value of c such that the type I error probability α ≤ 0.05. What is
Problem 8.57 Let X be the sample total from a random sample of size n = 10 from a Bernoulli distribution with parameter p. To test H0 : p ≤ 0.25 against H1 : p > 0.25, we choose the rejection region C = {x : x>c}. (a) What is the significance level of this test for c = 4? Determine the type II
Problem 8.56 Refer to the weekly 2002 return data for PTTRX listed in Table 1.20. Use a statistical software package to: (a) Draw the graph of the Q–Q plot (normal probability plot) of the returns. (b) Test for normality by computing the Shapiro–Wilk test statistic. State the alternatives,
Problem 8.54 Refer to the weekly 2002 return data for GE listed in Table 1.20. Use a statistical software package to: (a) Draw the graph of the Q–Q plot (normal probability plot) of the returns. (b) Test for normality by computing the Shapiro–Wilk test statistic. State the alternatives,
Problem 8.53 The efficient market hypothesis, discussed in Section 5.5 implies that stock market returns are normally distributed. Refer to Example 1.17 and Figure 1.11 (histogram of the S&P returns (1999)), which suggest that the returns are normally distributed. Test the data for normality by
Problem 8.52 Refer to the capacitor data set of Problem 1.20. Test the data for normality by (i) constructing a Q–Q plot of the data and (ii) applying the Shapiro–Wilk test. What is the P-value of the test?
Problem 8.51 Refer to the cadmium levels data of Problem 1.19. Test the data for nor[1]mality by (i) constructing a Q–Q plot of the data and (ii) applying the Shapiro–Wilk test. What is the P-value of the test?
Problem 8.50(a) Refer to the radiation data of Table 1.3. The histogram (Figure 1.10) suggests that these data are not normally distributed. Construct a Q–Q plot of the data and use it as an heuristic tool to assess the plausibility of the hypothesis of normality. (b) Test the data for normality
Problem 8.49 Refer to the lead absorption data set of Problem 1.18. The paired sample t-test assumes that the data in the difference column are normally distributed. Test the third column (difference) for normality by applying the Shapiro–Wilk test. State the alternatives, decision rule, and
Problem 8.48 Refer to the lead absorption data set of Problem 1.18. Construct a Q–Q plot to assess, informally, the plausibility of the normality hypothesis for the data in the second column (control). Comment on whether or not the points (zj , x(j)) appear to lie along a line. (b) Test the data
Problem 8.47 Refer to the lead absorption data set of Problem 1.18. Construct a Q–Q plot to assess, informally, the plausibility of the normality hypothesis for the data in the first column (the exposed group). Comment on whether or not the points (zj , x(j)) appear to lie along a straight line.
Problem 8.45 In a comparison study of two independent normal populations the sample variances were s2 1 = 1.1 and s2 2 = 3.4, and the corresponding sample sizes were n1 = 6, n2 = 10. Is one justified in using the two sample t test? Justify your conclusions.Problem 8.46 (a) Refer to the data set of
Problem 8.44 Refer to the data in Problem 7.41. Let σ2 1 and σ2 2 denote the variances of the heights of the cross-fertilized and self-fertilized seedlings, respectively. Test for the equality of the variances at the 1% level; that is, test the following hypothesis: H0 : σ2 1 σ2 2 = 1 against
Problem 8.43 Refer to Lord Rayleigh’s data that led to his discovery of argon (cf. Table 1.4 and Example 7.10). Are the data consistent with with Lord Rayleigh’s claim that nitrogen gas obtained from air is heavier than nitrogen gas obtained from chemical decomposition? Use the two sample
Problem 8.42 In a study comparing the breaking strengths (measured in psi) of two types of fiber, random samples of sizes n1 = n2 = 9 of both types of fiber were tested, and the following summary statistics were obtained: x1 = 40.1, s2 1 = 2.5; x2 = 41.5, s2 2 = 2.9. Assume both samples come from
Problem 8.41 Refer to the data in Problem 1.64. Use the two sample t test to determine the effect, if any, of the word processor program on the length of time it takes to type a text file. Assume the data are normally distributed with a common variance; use α = 0.05. In detail: (a) Compare the
Problem 8.40 Refer to the data in Problem 1.40. (a) Do the data indicate a significant difference in the DBH activity between the non[1]psychotic and psychotic groups? Use the two sample t test and summarize your resu by stating the P-value. (b) Repeat part (a) but use the Wilcoxon rank sum test.
Problem 8.39 In a weight loss experiment 10 men followed diet A and 7 men followed diet B. The weight losses after one month are displayed in the following table. Diet A Diet B Diet A Diet B 15.3 3.4 10 5.2 2.1 10.9 8.3 2.5 8.8 2.8 9.4 5.1 7.8 12.5 8.3 0.9 11.1(a) Use the two sample t test to
Problem 8.38 Refer to Charles Darwin’s data in Problem 7.41. Do the data point to a significant difference between the heights of the cross-fertilized and self-fertilized seedlings? Summarize your results by giving the P-value.
Problem 8.37 In a comparison study of a standard versus a premium blend of gasoline, the miles per gallon (mpg) for each blend was recorded for five compact automobiles. Car Standard (mpg) Premium (mpg) 1 26.9 27.7 2 22.5 22.2 3 24.5 25.3 4 26.0 26.8 5 28.5 29.7 Test, at the 5% significance level,
Problem 8.36 Refer to the data in Problem 1.62. (a) Do the data indicate a significant difference in the mean concentration of thiol between the normal group and the rheumatoid arthritis group? Use the t test with significance level α = 0.05. Summarize your analysis by giving the P-value. (b)
Problem 8.35 In a code size comparison study, five determinations of the number of bytes required to code five workloads on two different processors were recorded; the data are dis- played in the following table. System 1 System 2 101 130 144 180 211 141 288 374 72 302
Problem 8.34 To determine the difference, if any, between two brands of radial tires, 12 tires of each brand were tested and the following mean lifetimes and sample standard deviations for the two brands were obtained: x1 = 38500 and s1 = 3100; x2 = 41000 and s2 = 3500. Assuming that the tire
Problem 8.33 Refer to Example 1.8 and Table 1.7. Denote the mean N-1-THB-ADE levels of the control and exposed group of workers by µ1, µ2, respectively. Use the Wilcoxon rank sum statistic to test H0 : µ1 ≥ µ2 against H1 : µ1 < µ2. Use α = 0.05. What is the P-value of the test?
Problem 8.32 Refer to the data set of Problem 7.37. (a) By constructing a suitable two sided 95% confidence interval, test the hypothesis that there is no difference between the mean skein strengths of the two types of yarn. (b) What is the P-value of your test? (c) Use the Wilcoxon rank sum
Problem 8.31 The data set of Problem 7.36 comes from an experiment comparing the electrical resistances (measured in ohms) of two types of wire. (a) Test the null hypothesis that µ1 = µ2 against the alternative that µ1= µ2 by constructing an appropriate two sided 95% confidence interval. (b)
Problem 8.30 Suppose you want to test H0: µ ≥ 30 against H1 : µ < 30 using a random sample of size n taken from a normal distribution N(µ, 16). Find the minimum sample size n and the cutoff value c if the probability of a type I error is to be α = 0.05 and the probability of a type II error
Problem 8.29 Suppose you want to test H0: µ ≤ 50 against H1 : µ > 50 using a random sample of size n taken from a normal distribution N(µ, 81). Find the minimum sample size n and the cutoff value c if the probability of a type I error is to be α = 0.05 and the probability of a type II error
Problem 8.28 Compute and sketch the graph of the power function of the test with rejec[1]tion region C = {x : |x − 10| > 1} of Problem 8.7.
Problem 8.27 Compute and sketch the graph of the power function of the test with rejec[1]tion region C = {x : x < 28} in part (a) of Problem 8.6.
Problem 8.26 Compute and sketch the graph of the power function of the test with rejec[1]tion region C = {x : x > 210} in part (c) of Problem 8.4.
Problem 8.25 In order to test H0 : µ ≥ 10 against H1 : µ < 10 a researcher computes the observed values x and s of the sample mean and sample standard deviation of random sample of size n = 12 taken from a normal distribution N(µ, σ2). Compute the P-value of the test if the observed values x
Problem 8.24 In order to test H0 : µ ≤ 35 against H1 : µ > 35 a researcher records the observed value x of the sample mean of a random sample of size n = 9 taken from a normal distribution N(µ, 49). Compute the P-value of the test if the observed value x is: (a) x = 36. (b) x = 37. (c) x = 39.
Problem 8.23 The graph of the power function in Figure 8.8 is symmetric about the null value µ0, that is, π(µ0 +d) = π(µ0 − d). Show that the power function defined by Equation 8.16 is symmetric about the null value. Hint: Use the fact that Φ(−x)=1 − Φ(x).
Problem 8.22 Give the details of the derivation of Equation 8.16.
Problem 8.21 Derive the Equation 8.19 for the sample size n by solving the Equations 8.20 and 8.21.
Problem 8.20 Statistical analysis of experimental data suggest that the compressive strengths of bricks can be approximated by a normal distribution N(µ, σ2). A new (and cheaper) pro[1]cess for manufacturing these bricks is developed whose output is also assumed to be normall distributed. The
Problem 8.19 Suppose the lifetime of a tire is advertised to be 40000 miles. The lifetimes of 100 tires are recorded and the following results are obtained: x = 39360 miles and s = 3200 miles. (a) If we do not assume that the data come from a normal distribution what can you say about the
Problem 8.18 In a study of the effectiveness of a weight loss program the net weight loss, in lbs, of ten male subjects were recorded: 8.4, 15.9, 7.8, 12.8, 5.9, 10.2, 7.5, 15.5, 12.1, 12.5. The weight loss program is judged ineffective if the mean weight loss is less than or equal to 10 pounds.
Problem 8.17 Seven measurements of arsenic in copper yielded a sample mean of x = 0.17% and sample standard deviation s = 0.03%. Assuming the data come from a normal distribution N(µ, σ2): (a) Describe the test statistic and rejection region that is appropriate for testing (at the level α) H0 :
Problem 8.16 Twelve measurements of the ionization potential, measured in electron volts, of toluene were made and the following data were recorded: 10.6, 9.51, 9.83, 10.11, 10.05, 10.31, 9.37, 10.44, 10.09, 10.55, 9.19, 10.09. The published value of the ionization potential is 9.5. Assume that the
Problem 8.15(a) The graph of the power function in Figure 8.2 is an increasing function of µ. Show that this is a consequence of the fact that the power function defined by Equationπ(µ)=1 − Φ √n(c − µ) σ is an increasing function of µ. Hint: Use the fact that Φ(x) is an increasing
Problem 8.14 Suppose π(µ)=Φ((26.71 − µ)/2) is the power function of a test of H0 : µ ≥ 30 against H1 : µ < 30. Using this information find: (a) The significance level of the test. (b) The probability of a type II error at the alternative µ = 24.5.
Problem 8.13 Suppose π(µ)=1 − Φ(2(10.98 − µ)) is the power function of a test of H0 : µ ≤ 10 against H1 : µ > 10. Using this information find: (a) The significance level of the test. (b) The probability of a type II error at the alternative µ = 11.5.
Problem 8.12 Same problem as the previous one but this time assume σ is unknown and s = 10.

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