(In some of the exercises that follow, we must

Some nurses in county public health conducted a survey of women who had received inadequate prenatal care. They used information from birth certificates to select mothers for the survey. The mothers selected were divided into two groups: 14 mothers who said they had five or fewer prenatal visits and 14 mothers who said they had six or more prenatal visits. Let \(X\) and \(Y\) equal the respective birth weights of the babies from these two sets of mothers, and assume that the distribution of \(X\) is \(N\left(\mu_{X}, \sigma^{2}\right)\) and the distribution of \(Y\) is \(N\left(\mu_{Y}, \sigma^{2}\right)\). (a) Define the test statistic and critical region for testing \(H_{0}: \mu_{X}-\mu_{Y}=0\) against \(H_{1}: \mu_{X}-\mu_{Y}<0\). Let \(\alpha=0.05\). (b) Given that the observations of \(X\) were 49 108 110 82 93 114 134 114 96 52 101 114 120 116 and the observations of \(Y\) were 133 108 93 119 119 98 106 131 87 153 116 129 97 110 calculate the value of the test statistic and state your conclusion. (c) Approximate the \(p\)-value. (d) Construct box plots on the same figure for these two sets of data. Do the box plots support your conclusion? Equation Transcription: Text Transcription: X Y N(mu_X, sigma^2) N(mu_Y, sigma^2) H_0:mu_X-mu_Y=0 H_1:mu_X-mu_Y<0 alpha=0.05 p

Let \(X\) and \(Y\) denote the weights in grams of male and female common gallinules, respectively. Assume that \(X\) is \(N\left(\mu_{X}, \sigma_{X}^{2}\right)\) and \(Y\) is \(N\left(\mu_{Y}, \sigma_{Y}^{2}\right)\). (a) Given \(n=16\) observations of \(X\) and \(m=13\) observations of \(Y\), define a test statistic and a critical region for testing the null hypothesis \(H_{0}: \mu_{X}=\mu_{Y}\) against the one-sided alternative hypothesis \(H_{1}: \mu_{X}>\mu_{Y}\). Let $\alpha=0.01$. (Assume that the variances are equal.) (b) Given that \(\bar{x}=415.16, s_{x}^{2}=1356.75, \bar{y}=347.40\), and \(s_{y}^{2}=692.21\), calculate the value of the test statistic and state your conclusion. (c) Although we assumed that \(\sigma_{X}^{2}=\sigma_{Y}^{2}\), let us say we suspect that that equality is not valid. Thus, use the test proposed by Welch. Equation Transcription: Text Transcription: X Y N(mu_X, sigma_X^2) N(mu_Y, sigma_Y^2) n=16 m=13 H_0:mu_X=mu_Y H_1:mu_X>mu_Y alpha=0.01 Bar x=415.16, s_x^2=1356.75, bar y=347.40 s_y^2=692.21 sigma_X^2=sigma_Y^2

Let X equal the weight in grams of a Low-Fat Strawberry Kudo and Y the weight of a Low-Fat Blueberry Kudo. Assume that the distributions of X and Y are \(N(\mu_X , \sigma^2_X )\) and \(N(\mu_Y , \sigma^2_Y )\), respectively. Let 21.7 21.0 21.2 20.7 20.4 21.9 20.2 21.6 20.6 be n = 9 observations of X, and let 21.5 20.5 20.3 21.6 21.7 21.3 23.0 21.3 18.9 20.0 20.4 20.8 20.3 be m = 13 observations of Y. Use these observations to answer the following questions: (a) Test the null hypothesis \(H_0: \mu_X = \mu_Y\) against a two sided alternative hypothesis. You may select the significance level. Assume that the variances are equal. (b) Construct and interpret box-and-whisker diagrams to support your conclusions.

Among the data collected for the World Health Organization air quality monitoring project is a measure of suspended particles, in \(\mu \mathrm{g} / \mathrm{m}^{3}\). Let \(X\) and \(Y\) equal the concentration of suspended particles in \(\mu \mathrm{g} / \mathrm{m}^{3}\) in the city centers (commercial districts), of Melbourne and Houston, respectively. Using \(n=13\) observations of \(X\) and \(m=16\) observations of \(Y\), we shall test \(H_{0}: \mu_{X}=\mu_{Y}\) against \(H_{1}: \mu_{X}<\mu_{Y}\). (a) Define the test statistic and critical region, assuming that the variances are equal. Let \(\alpha=0.05\). (b) If \(\bar{x}=72.9, s_{x}=25.6, \bar{y}=81.7\), and \(s_{y}=28.3\), calculate the value of the test statistic and state your conclusion. (c) Give bounds for the \(p\)-value of this test. Equation Transcription: Text Transcription: mu g/m^3. X Y n=13 m=16 H_0:mu_X=mu_Y H_1:mu_X>mu_Y alpha =0.05 Bar x=72.9, s_x=25.6, bar y=81.7, s_y=28.3 p

An ecology laboratory studied tree dispersion patterns for the sugar maple, whose seeds are dispersed by the wind, and the American beech, whose seeds are dispersed by mammals. In a plot of area \(50 m\) by \(50 m\), they measured distances between like trees, yielding the following distances in meters for 19 American beech trees and 19 sugar maple trees: (a) Test the null hypothesis that the means are equal against the one-sided alternative that the mean for the distances between beech trees is greater than that between maple trees. (b) Construct two box plots to confirm your answer. Equation Transcription: Text Transcription: 50 m

The botanist in Example 8.2-1 is really interested in testing for synergistic interaction. That is, given the two hormones gibberellin \(\left(\mathrm{GA}_{3}\right)\) and indoleacetic acid ((\mathrm{IAA})\), let \(X_{1}\) and \(X_{2}\) equal the growth responses (in \(\mathrm{mm}\) ) of dwarf pea stem segments to \(\mathrm{GA}_{3}\) and \(\mathrm{IAA}\), respectively and separately. Also, let \(X=X_{1}+X_{2}\) and let \(Y\) equal the growth response when both hormones are present. Assuming that \(X\) is \(N\left(\mu_{X}, \sigma^{2}\right)\) and \(Y\) is \(N\left(\mu_{Y}, \sigma^{2}\right)\), the botanist is interested in testing the hypothesis \(H_{0}: \mu_{X}=\mu_{Y}\) against the alternative hypothesis of synergistic interaction \(H_{1}: \mu_{X}<\mu_{Y}\). (a) Using \(n=m=10\) observations of \(X\) and \(Y\), define the test statistic and the critical region. Sketch a figure of the \(t\) pdf and show the critical region on your figure. Let \(\alpha=0.05\). (b) Given \(n=10\) observations of \(X\), namely, 2.1 2.6 2.6 3.4 2.1 1.7 2.6 2.6 2.2 1.2 and \(m = 10\) observations of \(Y\), namely, 3.5 3.9 3.0 2.3 2.1 3.1 3.6 1.8 2.9 3.3 calculate the value of the test statistic and state your conclusion. Locate the test statistic on your figure. (c) Construct two box plots on the same figure. Does this confirm your conclusion? Equation Transcription: Text Transcription: (GA_3) X_1 X_2 mm GA_3 IAA X=X_1+X_2 Y X N(mu_X, sigma^2) N(mu_Y, sigma^2) H_0:mu_X=mu_Y H_1:mu_X<mu_Y n=m=10 t =0.05 n=10 m=10

Let \(X\) and \(Y\) denote the tarsus lengths of male and female grackles, respectively. Assume that \(X\) is \(N\left(\mu_{X}, \sigma_{X}^{2}\right)\) and \(Y\) is \(N\left(\mu_{Y}, \sigma_{Y}^{2}\right)\). Given that \(n=25, \bar{x}=33.80\), \(s_{x}^{2}=4.88, m=29, \bar{y}=31.66\), and \(s_{y}^{2}=5.81\), test the null hypothesis \(H_{0}: \mu_{X}=\mu_{Y}\) against \(H_{1}: \mu_{X}>\mu_{Y}\) with \(\alpha=0.01\). Equation Transcription: Text Transcription: X Y N(mu_X, sigma_X^2) N(mu_Y, sigma_Y^2) n=25, bar x=33.80, s_x^2=4.88,m=29, bar y=31.66 s_y^2=5.81 H_0:mu_X=mu_Y H_1:mu_X>mu_y alpha=0.01

PROBLEM 6E (In some of the exercises that follow, we must make assumptions such as the existence of normal distributions with equal variances.) Let X and Y equal the forces required to pull stud No. 3 and stud No. 4 out of a window that has been manufactured for an automobile. Assume that the distributions of X and Y are N(?X, ?2X) and N(?Y, ?2Y ),respectively. (a) If m = n = 10 observations are selected randomly, define a test statistic and a critical region for testing H0: ?X ? ?Y = 0 against a two-sided alternative hypothesis. Let ? = 0.05. Assume that the variances are equal. (b) Given n = 10 observations of X, namely, 111 120 139 136 138 149 143 145 111 123 and m = 10 observations of Y, namely, 152 155 133 134 119 155 142 146 157 149 calculate the value of the test statistic and state your conclusion clearly. (c) What is the approximate p-value of this test? (d) Construct box plots on the same figure for these two sets of data. Do the box plots confirm your decision in part (b)?

Plants convert carbon dioxide \(\mathrm{(CO_2)}\) in the atmosphere, along with water and energy from sunlight, into the energy they need for growth and reproduction. Experiments were performed under normal atmospheric air conditions and in air with enriched \(\mathrm{CO_2}\) concentrations to determine the effect on plant growth. The plants were given the same amount of water and light for a four week period. The following table gives the plant growths in grams: Normal Air 4.67 4.21 2.18 3.91 4.09 5.24 2.94 4.71 4.04 5.79 3.80 4.38 Enriched Air 5.04 4.52 6.18 7.01 4.36 1.81 6.22 5.70 On the basis of these data, determine whether \(\mathrm{CO_2}\)-enriched atmosphere increases plant growth.

Let \(X\) equal the fill weight in April and \(X\) the fill weight in June for an \(8\)-pound box of bleach.We shall test the null hypothesis \(H_{0}: \mu_{X}-\mu_{Y}=0\) against the alternative hypothesis \(H_{1}: \mu_{X}-\mu_{Y}>0\\) given that \(n=90\) observations of \(X\) yielded \(\bar{x}=8.10\) and \(s_{x}=0.117\) and \(m=110\) observations of \(Y\) yielded \(\bar{y}=8.07\) and \(s_{y}=0.054\). (a) What is your conclusion if \(\alpha=0.05\)? HINT: Do the variances seem to be equal? (b) What is the approximate \(p\)-value of this test? Equation Transcription: Text Transcription: X Y 8 H_0:mu_X-mu_Y=0 H_1:mu_X-mu_Y>0 n=90 Bar x=8.10 s_x=0.117 m=110 Bar y=8.07 s_y=0.054 alpha=0.05

Let \(X\) and \(Y\) denote the respective lengths of male and female green lynx spiders. Assume that the distributions of \(X\) and \(Y\) are \(N\left(\mu_{X}, \sigma_{X}^{2}\right)\) and \(N\left(\mu_{Y}, \sigma_{Y}^{2}\right)\), respectively, and that \(\sigma_{Y}^{2}>\sigma_{X}^{2}\). Thus, use the modification of \(Z\) to test the hypothesis \(H_{0}: \mu_{X}-\mu_{Y}=0\) against the alternative hypothesis \(H_{1}: \mu_{X}-\mu_{Y}<0\). (a) Define the test statistic and a critical region that has a significance level of \(\alpha=0.025\). (b) Using the data given in Exercise 7.2-5, calculate the value of the test statistic and state your conclusion. (c) Draw two box-and-whisker diagrams on the same figure. Does your figure confirm the conclusion of this exercise? Equation Transcription: Text Transcription: X Y N(mu_X, sigma_X^2) N(mu_Y, sigma_Y^2) sigma_Y^2>sigma_X^2 Z H_0:mu_X-mu_Y=0 H_1:mu_X-m_Y<0 alpha=0.025

Let \(X\) and \(Y\) equal the number of milligrams of tar in filtered and non-filtered cigarettes, respectively. Assume that the distributions of \(X\) and \(Y\) are \(N\left(\mu_{X}, \sigma_{X}^{2}\right)\) and \(N\left(\mu_{\gamma}, \sigma_{y}^{2}\right)\), respectively. We shall test the null hypothesis \(H_{0}: \mu_{X}-\mu_{Y}=0\) against the alternative hypothesis \(H_{1}: \mu_{X}-\mu_{y}<0\), using random samples of sizes \(n=9\) and \(m=11\) observations of \(X\) and \(Y\), respectively. (a) Define the test statistic and a critical region that has an \(\alpha=0.01\) significance level. Sketch a figure illustrating this critical region. (b) Given \(n=9\) observations of \(X\), namely, 0.9 1.1 0.1 0.7 0.4 0.9 0.8 1.0 0.4 and m = 11 observations of Y, namely, 1.5 0.9 1.6 0.5 1.4 1.9 1.0 1.2 1.3 1.6 2.1 calculate the value of the test statistic and state your conclusion clearly. Locate the value of the test statistic on your figure. Equation Transcription: Text Transcription: X Y N(mu_X, sigma_X^2) N(mu_Y, sigma_Y^2) H_0:mu_X-mu_Y=0 H_1:mu_X-mu_y<0 n=9 m=11 alpha=0.01

When a stream is turbid, it is not completely clear due to suspended solids in the water. The higher the turbidity, the less clear is the water. A stream was studied on 26 days, half during dry weather (say, observations of X) and the other half immediately after a significant rainfall (say, observations of Y). Assume that the distributions of X and Y are \(N(\mu_X , \sigma^2)\) and \(N(\mu_Y , \sigma^2)\), respectively. The following turbidities were recorded in units of NTUs (nephelometric turbidity units): x: 2.9 14.9 1.0 12.6 9.4 7.6 3.6 3.1 2.7 4.8 3.4 7.1 7.2 y: 7.8 4.2 2.4 12.9 17.3 10.4 5.9 4.9 5.1 8.4 10.8 23.4 9.7 (a) Test the null hypothesis \(H_0: \mu_X = \mu_Y\) against \(H_1: \mu_X < \mu_Y\). Give bounds for the p-value and state your conclusion. (b) Draw box-and-whisker diagrams on the same graph. Does this figure confirm your answer?

PROBLEM 13E (In some of the exercises that follow, we must make assumptions such as the existence of normal distributions with equal variances.) Students looked at the effect of a certain fertilizer on plant growth. The students tested this fertilizer on one group of plants (Group A) and did not give fertilizer to a second group (Group B). The growths of the plants, in mm, over six weeks were as follows: Group A: 55 61 33 57 17 46 50 42 71 51 63 Group B: 31 27 12 44 9 25 34 53 33 21 32 (a) Test the null hypothesis that the mean growths are equal against the alternative that the fertilizer enhanced growth. Assume that the variances are equal. (b) Construct box plots of the two sets of growths on the same graph. Does this confirm your answer to part (a)?

The botanist in Example 8.2-1 is really interested in testing for synergistic interaction. That is, given the two hormones gibberellin (GA3) and indoleacetic acid (IAA), let X1 and X2 equal the growth responses (in mm) of dwarf pea stem segments to GA3 and IAA, respectively and separately. Also, let X = X1 + X2 and let Y equal the growth response when both hormones are present. Assuming that X is N(X , 2) and Y is N(Y , 2), the botanist is interested in testing the hypothesis H0: X = Y against the alternative hypothesis of synergistic interaction H1: X < Y . (a) Using n = m = 10 observations of X and Y, define the test statistic and the critical region. Sketch a figure of the t pdf and show the critical region on your figure. Let = 0.05. (b) Given n = 10 observations of X, namely, 2.1 2.6 2.6 3.4 2.1 1.7 2.6 2.6 2.2 1.2 and m = 10 observations of Y, namely, 3.5 3.9 3.0 2.3 2.1 3.1 3.6 1.8 2.9 3.3 calculate the value of the test statistic and state your conclusion. Locate the test statistic on your figure. (c) Construct two box plots on the same figure. Does this confirm your conclusion?

Let X and Y denote the weights in grams of male and female common gallinules, respectively. Assume that X is \(N\left(\mu_{X}, \sigma_{X}^{2}\right)\) and Y is \(N\left(\mu_{Y}, \sigma_{Y}^{2}\right)\). (a) Given n = 16 observations of X and m = 13 observations of Y, define a test statistic and a critical region for testing the null hypothesis \(H_{0}: \mu_{X}=\mu_{Y}\) against the one-sided alternative hypothesis \(H_{1}: \mu_{X}>\mu_{Y}\). Let \(\alpha = 0.01\). (Assume that the variances are equal.) (b) Given that \(\bar{x}=415.16, s_{x}^{2}=1356.75, \bar{y}=347.40\), and \(s_{y}^{2}=692.21\), calculate the value of the test statistic and state your conclusion. (c) Although we assumed that \(\sigma_{X}^{2}=\sigma_{Y}^{2}\), let us say we suspect that that equality is not valid. Thus, use the test proposed by Welch.

Let X equal the weight in grams of a Low-Fat Strawberry Kudo and Y the weight of a Low-Fat Blueberry Kudo. Assume that the distributions of X and Y are N(X , 2 X ) and N(Y , 2 Y ), respectively. Let 21.7 21.0 21.2 20.7 20.4 21.9 20.2 21.6 20.6 be n = 9 observations of X, and let 21.5 20.5 20.3 21.6 21.7 21.3 23.0 21.3 18.9 20.0 20.4 20.8 20.3 be m = 13 observations of Y. Use these observations to answer the following questions: (a) Test the null hypothesis H0: X = Y against a twosided alternative hypothesis. You may select the significance level. Assume that the variances are equal. (b) Construct and interpret box-and-whisker diagrams to support your conclusions

Among the data collected for the World Health Organization air quality monitoring project is a measure of suspended particles, in \(\mu g/m^3\). Let X and Y equal the concentration of suspended particles in \(\mu g/m^3\) in the city centers (commercial districts), of Melbourne and Houston, respectively. Using n = 13 observations of X and m = 16 observations of Y, we shall test \(H_0: \mu_X = \mu_Y\) against \(H_1: \mu_X < \mu_Y\). (a) Define the test statistic and critical region, assuming that the variances are equal. Let \(\alpha=0.05\). (b) If \(\bar x = 72.9, s_x = 25.6, \bar y = 81.7\), and \(s_y = 28.3\), calculate the value of the test statistic and state your conclusion. (c) Give bounds for the p-value of this test.

Some nurses in county public health conducted a survey of women who had received inadequate prenatal care. They used information from birth certificates to select mothers for the survey. The mothers selected were divided into two groups: 14 mothers who said they had five or fewer prenatal visits and 14 mothers who said they had six or more prenatal visits. Let X and Y equal the respective birth weights of the babies from these two sets of mothers, and assume that the distribution of X is N(X , 2) and the distribution of Y is N(Y , 2).(a) Define the test statistic and critical region for testing H0: X Y = 0 against H1: X Y < 0. Let = 0.05. (b) Given that the observations of X were 49 108 110 82 93 114 134 114 96 52 101 114 120 116 and the observations of Y were 133 108 93 119 119 98 106 131 87 153 116 129 97 110 calculate the value of the test statistic and state your conclusion. (c) Approximate the p-value. (d) Construct box plots on the same figure for these two sets of data. Do the box plots support your conclusion?

Let X and Y equal the forces required to pull stud No. 3 and stud No. 4 out of a window that has been manufactured for an automobile. Assume that the distributions of X and Y are N(X , 2 X ) and N(Y , 2 Y ), respectively. (a) If m = n = 10 observations are selected randomly, define a test statistic and a critical region for testing H0: X Y = 0 against a two-sided alternative hypothesis. Let = 0.05. Assume that the variances are equal. (b) Given n = 10 observations of X, namely, 111 120 139 136 138 149 143 145 111 123 and m = 10 observations of Y, namely, 152 155 133 134 119 155 142 146 157 149 calculate the value of the test statistic and state your conclusion clearly. (c) What is the approximate p-value of this test? (d) Construct box plots on the same figure for these two sets of data. Do the box plots confirm your decision in part (b)?

Let X and Y equal the number of milligrams of tar in filtered and nonfiltered cigarettes, respectively. Assume that the distributions of X and Y are N(X , 2 X ) and N(Y , 2 Y ), respectively. We shall test the null hypothesis H0: X Y = 0 against the alternative hypothesis H1: X Y < 0, using random samples of sizes n = 9 and m = 11 observations of X and Y, respectively. (a) Define the test statistic and a critical region that has an = 0.01 significance level. Sketch a figure illustrating this critical region. (b) Given n = 9 observations of X, namely, 0.9 1.1 0.1 0.7 0.4 0.9 0.8 1.0 0.4 and m = 11 observations of Y, namely, 1.5 0.9 1.6 0.5 1.4 1.9 1.0 1.2 1.3 1.6 2.1 calculate the value of the test statistic and state your conclusion clearly. Locate the value of the test statistic on your figure.

Let X and Y denote the tarsus lengths of male and female grackles, respectively. Assume that X is \(N\left(\mu_X, \sigma_X^2\right)\) and Y is \(N\left(\mu_Y, \sigma_Y^2\right)\). Given that \(n=25, \bar{x}=33.80\), \(s_x^2=4.88, m=29, \bar{y}=31.66\), and \(s_y^2=5.81\), test the null hypothesis \(H_0: \mu_X=\mu_Y\) against \(H_1: \mu_X>\mu_Y\) with \(\alpha=0.01\).

When a stream is turbid, it is not completely clear due to suspended solids in the water. The higher the turbidity, the less clear is the water. A stream was studied on 26 days, half during dry weather (say, observations of X) and the other half immediately after a significant rainfall (say, observations of Y). Assume that the distributions of X and Y are N(X , 2) and N(Y , 2), respectively. The following turbidities were recorded in units of NTUs (nephelometric turbidity units): x: 2.9 14.9 1.0 12.6 9.4 7.6 3.6 3.1 2.7 4.8 3.4 7.1 7.2 y: 7.8 4.2 2.4 12.9 17.3 10.4 5.9 4.9 5.1 8.4 10.8 23.4 9.7 (a) Test the null hypothesis H0: X = Y against H1: X < Y . Give bounds for the p-value and state your conclusion. (b) Draw box-and-whisker diagrams on the same graph. Does this figure confirm your answer?

Plants convert carbon dioxide (CO2) in the atmosphere, along with water and energy from sunlight, into the energy they need for growth and reproduction. Experiments were performed under normal atmospheric air conditions and in air with enriched CO2 concentrations to determine the effect on plant growth. The plants were given the same amount of water and light for a fourweek period. The following table gives the plant growths in grams: Normal Air 4.67 4.21 2.18 3.91 4.09 5.24 2.94 4.71 4.04 5.79 3.80 4.38 Enriched Air 5.04 4.52 6.18 7.01 4.36 1.81 6.22 5.70 On the basis of these data, determine whether CO2- enriched atmosphere increases plant growth.

Let X equal the fill weight in April and Y the fill weight in June for an 8-pound box of bleach. We shall test the null hypothesis H0: XY = 0 against the alternative hypothesis H1: X Y > 0 given that n = 90 observations of X yielded x = 8.10 and sx = 0.117 and m = 110 observations of Y yielded y = 8.07 and sy = 0.054. (a) What is your conclusion if = 0.05? Hint: Do the variances seem to be equal? (b) What is the approximate p-value of this test?

Let X and Y denote the respective lengths of male and female green lynx spiders. Assume that the distributions of X and Y are N(X , 2 X ) and N(Y , 2 Y ), respectively, and that 2 Y > 2 X . Thus, use the modification of Z to test the hypothesis H0: X Y = 0 against the alternative hypothesis H1: X Y < 0. (a) Define the test statistic and a critical region that has a significance level of = 0.025. (b) Using the data given in Exercise 7.2-5, calculate the value of the test statistic and state your conclusion. (c) Draw two box-and-whisker diagrams on the same figure. Does your figure confirm the conclusion of this exercise?

Students looked at the effect of a certain fertilizer on plant growth. The students tested this fertilizer on one group of plants (Group A) and did not give fertilizer to a second group (Group B). The growths of the plants, in mm, over six weeks were as follows: Group A: 55 61 33 57 17 46 50 42 71 51 63 Group B: 31 27 12 44 9 25 34 53 33 21 32 (a) Test the null hypothesis that the mean growths are equal against the alternative that the fertilizer enhanced growth. Assume that the variances are equal. (b) Construct box plots of the two sets of growths on the same graph. Does this confirm your answer to part (a)?

An ecology laboratory studied tree dispersion patterns for the sugar maple, whose seeds are dispersed by the wind, and the American beech, whose seeds are dispersed by mammals. In a plot of area 50 m by 50 m, they measured distances between like trees, yielding the following distances in meters for 19 American beech trees and 19 sugar maple trees: American beech: 5.00 5.00 6.50 4.25 4.25 8.80 6.50 7.15 6.15 2.70 2.70 11.40 9.70 6.10 9.35 2.85 4.50 4.50 6.50 sugar maple: 6.00 4.00 6.00 6.45 5.00 5.00 5.50 2.35 2.35 3.90 3.90 5.35 3.15 2.10 4.80 3.10 5.15 3.10 6.25 (a) Test the null hypothesis that the means are equal against the one-sided alternative that the mean for the distances between beech trees is greater than that between maple trees. (b) Construct two box plots to confirm your answer.

Say X and Y are independent random variables with distributions that are N(X , 2 X ) and N(Y , 2 Y ). We wish to test H0: 2 X = 2 Y against H1: 2 X > 2 Y . (a) Argue that, if H0 is true, the ratio of the two variances of the samples of sizes n and m, S2 X /S2 Y , has an F(n1, m1) distribution. (b) If n = m = 31, x = 8.153, s2 x = 1.410, y = 5.917, s2 y = 0.4399, s2 x/s2 y = 3.2053, and = 0.01, show that H0 is rejected and H1 is accepted since 3.2053 > 2.39. (c) Where did the 2.39 come from?

To measure air pollution in a home, let X and Y equal the amount of suspended particulate matter (in g/m3) measured during a 24-hour period in a home in which there is no smoker and a home in which there is a smoker, respectively. We shall test the null hypothesis H0: 2 X /2 Y = 1 against the one-sided alternative hypothesis H1: 2 X /2 Y > 1. (a) If a random sample of size n = 9 yielded x = 93 and sx = 12.9 while a random sample of size m = 11 yielded y = 132 and sy = 7.1, define a critical region and give your conclusion if = 0.05. (b) Now test H0: X = Y against H1: X < Y if = 0.05.

Consider the distributions N(X , 400) and N(Y , 225). Let = X Y . Say x and y denote the observed means of two independent random samples, each of size n, from the respective distributions. Say we reject H0: = 0 and accept H1: > 0 if x y c. Let K() be the power function of the test. Find n and c so that K(0) = 0.05 and K(10) = 0.90, approximately.