### Solutions Modeling Dynamics of Life 3ed Adler - Chapter 8.9

8.9.1 Consider the data in the following table. ... ... Plot yield (Y ) against weight (W ). Suppose we think that the line Y = W + 0.6 describes these data. Plot the line on your graph of yield against weight. Find and plot the residuals.
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8.9.2 Consider the data in the following table. ... ... Plot height (H) against weight (W). Suppose we think that the line H = 8W + 5 describes the data. Plot the line on your graph of height against weight. Find and plot the residuals.
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8.9.3 Consider the data in the following table. ... ... Find SSE for the model used in Exercise 1
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8.9.4 Consider the data in the following table. ... ... Find SSE for the model used in Exercise 2
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8.9.5 Consider the data in the following table. ... ... Find the null model that best fits Y as a function of W and find SST.
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8.9.6 Consider the data in the following table. ... ... Find the null model that best fits H as a function of W and find SST.
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8.9.7 Consider the data in the following table. ... ... Find ...for the model in Exercise 1.
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8.9.8 Consider the data in the following table. ... ... Find ...for the model in Exercise 2.
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8.9.9 Consider the data in the following table. ... ... Compute the best fitting line for yield as a function of weight. Graph the line.
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8.9.10 Consider the data in the following table. ... ... Compute the best fitting line for height as a function of weight. Graph the line.
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8.9.11 Consider the data in the following table. ... ... Find SSE for the line in Exercise 9, find ..., and compare with the model in Exercise 1.
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8.9.12 Consider the data in the following table. ... ... Find SSE for the line in Exercise 10, find ..., and compare with the model in Exercise 2.
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8.9.13 Consider the data in the following table. ... ... Find the correlation between weight and yield. Check that its square is equal to the value of ...found in Exercise 11.
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8.9.14 Consider the data in the following table. ... ... Find the correlation between weight and height. Check that its square is equal to the value of ...found in Exercise 12.
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8.9.15 Linear regression has important connections with other techniques in statistics, such as testing whether two populations differ. In the following data set, the independent variable takes on only two values. Find the best fitting line and ..., and then test whether the two sets of points differ in their mean distribution using the techniques in Section 8.6. Assume that the data are normally distributed with known variance of 25. In each case, graph the regression line and the data. ... Find the best fitting line and ...for replicate 1 and then test whether the diet has a significant effect.
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8.9.16 Linear regression has important connections with other techniques in statistics, such as testing whether two populations differ. In the following data set, the independent variable takes on only two values. Find the best fitting line and ..., and then test whether the two sets of points differ in their mean distribution using the techniques in Section 8.6. Assume that the data are normally distributed with known variance of 25. In each case, graph the regression line and the data. ... Find the best fitting line and ...for replicate 2 and then test whether the diet has a significant effect. Compare with the previous problem.
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8.9.18 Best fit regression lines have many nice properties. Show that the best linear fit from Theorem 8.6 passes through the center of the data in the sense that the sum of the residuals is 0.
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8.9.18 Best fit regression lines have many nice properties. Show that the best linear fit from Theorem 8.6 passes through the center of the data in the sense that the sum of the residuals is 0.
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8.9.19 Best fit regression lines have many nice properties. Consider models of the form Y = b. Show that the sum of the squares of the residuals is minimized when ...the sample mean of the ... .
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8.9.20 Best fit regression lines have many nice properties. Consider models of the form Y =aX. Find the slope that minimizes the sum of the squares of the residuals.
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8.9.21 Consider the following measurements. ... Find the best linear fit. Plot the line and find ...How good is the model?
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8.9.22 Consider the following measurements. ... Use the principle of least squares to write the expression you would use to fit a curve of the form .... One easy way to solve this is to think of a new measurement ...and find the linear regression of Y on Z . Plot the linear regression of Y against Z and the curved regression of Y against .... Which model does better?
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8.9.23 Consider the following measurements. ... Find the dimensions of the slope ...
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8.9.24 Consider the following measurements. ... Find the dimensions of the intercept ...and check that all the parts of the equation match.
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8.9.25 Consider the following data describing change in a bacterial population. ... Find .... Graph the data and the line.
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8.9.26 Consider the following data describing change in a bacterial population. ... Find the best fitting line, and compare with a mathematically idealized model. Which makes more sense ?
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8.9.27 Consider the following data on the growth of two bacterial populations. ... Find the best fitting line for population 1 as a function of time and ....
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8.9.28 Consider the following data on the growth of two bacterial populations. ... Find the best fitting line for population 2 as a function of time and compute ...
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8.9.29 Consider the following data on the growth of two bacterial populations. ... Find the best fitting line for the logarithm of population 1 as a function of time and compute .... Is this a better fit?
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8.9.30 Consider the following data on the growth of two bacterial populations. ... Find the best fitting line for the logarithm of population 2 as a function of time and compute .... Is this a better fit?
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8.9.31 Consider the following data which include one outlying point. Find the best fitting line with and without that point. How much difference does that point make? The idea of removing one point and testing how much the fit changes is an important tool in regression, and is sometimes called the leverage of that point. ... Find the best fitting line and ...for replicate 1 with and without the fourth point. Graph the two regression lines and the data.
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8.9.32 Consider the following data which include one outlying point. Find the best fitting line with and without that point. How much difference does that point make? The idea of removing one point and testing how much the fit changes is an important tool in regression, and is sometimes called the leverage of that point. ... Find the best fitting line and ...for replicate 2 with and without the last point. Graph the two regression lines and the data. Why do you think the outlier affects this regression line more?
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8.9.33 The following table gives the winning Olympic times for men and women in the 400 m race. ...
a. Use your computer to find the best linear regression for men and for women.
b. Plot the residuals for each. Does the linear model fit well?
c. Predict the times in the 2000 Olympics for women and men. How well did it actually work?
d. Predict when women will outrun men. Do you believe this?
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### Solutions Modeling Dynamics of Life 3ed Adler - Chapter 8.8

8.8.1 Using the following data, use the method of support to evaluate the null hypothesis that the true probability of heads is 0.5. A coin is flipped 5 times and comes up heads every time (as in Section 8.4, Exercise 5).
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8.8.2 Using the following data, use the method of support to evaluate the null hypothesis that the true probability of heads is 0.5. A coin is flipped 7 times and comes up heads 6 out of 7 times (as in Section 8.4, Exercise 6).
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8.8.3 Using the following data, use the method of support to evaluate the null hypothesis that the true probability of heads is 0.5. A coin is flipped 10 times and comes up heads 9 times (as in Section 8.4, Exercise 7).
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8.8.4 Using the following data, use the method of support to evaluate the null hypothesis that the true probability of heads is 0.5. A coin is flipped 20 times and comes up heads 3 times (as in Section 8.4, Exercise 8).
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8.8.5 Use the method of support to evaluate the following null hypotheses. One cosmic ray hits a detector in 1 yr. The null hypothesis is that the rate at which rays hit is λ= 5/yr (as in Section 8.4, Exercise 13.)
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8.8.6 Use the method of support to evaluate the following null hypotheses. Three cosmic rays hit a larger detector in 1 yr. The null hypothesis is that the rate at which rays hit is λ= 10/yr (as in Section 8.4, Exercise 14.)
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8.8.7 Find the difference in support of the following hypotheses. Compare with the p-value in the earlier problem. You wait 4000 h for an exponentially distributed event to occur. The null hypothesis is that the mean wait is 1000 h with alternative that the mean wait is greater than 1000 h (as in Section 8.4, Exercise 15).
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8.8.8 Find the difference in support of the following hypotheses. Compare with the p-value in the earlier problem. You wait 40 h for an exponentially distributed event to occur. The null hypothesis is that the mean wait is 1000 h with alternative that the mean wait is less than 1000 h (as in Section 8.4, Exercise 16).
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8.8.9 Find the difference in support of the following hypotheses. Compare with the p-value in the earlier problem. The first defective gasket is the 25th. The null hypothesis follows a geometric distribution with mean wait 10, and the alternative is that the mean wait is greater than 10 (as in Section 8.4, Exercise 17).
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8.8.10 Find the difference in support of the following hypotheses. Compare with the p-value in the earlier problem. The first defective gasket is the 50th. The null hypothesis follows a geometric distribution with mean wait 1000, and the alternative is that the mean wait is less than 1000 (as in Section 8.4, Exercise 18).
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8.8.11 Use the method of support to check the following hypotheses. The hypothesis in Section 8.5, Exercise 1.
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8.8.12 Use the method of support to check the following hypotheses. The hypothesis in Section 8.5, Exercise 2.
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8.8.13 Use the method of support to check the following hypotheses. The hypothesis in Section 8.5, Exercise 3.
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8.8.14 Use the method of support to check the following hypotheses. The hypothesis in Section 8.5, Exercise 4.
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8.8.15 How many standard errors from the mean are the following? What are the corresponding p-values for a two-tailed test? The support for the null hypothesis is less than the maximum by 2.
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8.8.16 How many standard errors from the mean are the following? What are the corresponding p-values for a two-tailed test? The support for the null hypothesis is less than the maximum by 3.
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8.8.17 How many standard errors from the mean are the following? What are the corresponding p-values for a two-tailed test? The support for the null hypothesis is less than the maximum by 3.5.
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8.8.18 How many standard errors from the mean are the following? What are the corresponding p-values for a two-tailed test? The support for the null hypothesis is less than the maximum by 4.
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8.8.19 Follow the steps to show that the support has the simple quadratic form given in the text. Show that ... (expand the quadratic and plug in definitions of ...).
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8.8.20 Follow the steps to show that the support has the simple quadratic form given in the text. Remove the terms that do not depend on μ and show that the maximum occurs at μ = ....
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8.8.21 Consider the data in Section 8.4, Exercises 23 and 24. Find the difference in support of the null and alternative hypotheses. Day 1, when 7 calls arrive in 1 h while only 3.5 were expected (Section 8.4, Exercise 23).
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8.8.22 Consider the data in Section 8.4, Exercises 23 and 24. Find the difference in support of the null and alternative hypotheses. Day 2, when 8 calls arrive in 1 h while only 3.5 were expected (Section 8.4, Exercise 24).
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8.8.23 Consider again the data on 30 waiting times for 2 types of events used in Section 8.5, Exercises 21 and 22. ... ... Use maximum likelihood to estimate the rate λ from the waiting times for type
a. Compare the support for the null hypothesis that λ = 1.0 with the support for the maximum likelihood estimate.
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8.8.24 Consider again the data on 30 waiting times for 2 types of events used in Section 8.5, Exercises 21 and 22. ... ... Use maximum likelihood to estimate the rate λ from the waiting times for type
b. Compare the support for the null hypothesis that λ = 1.0 with the support for the maximum likelihood estimate.
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8.8.26 In Exercises 23 and 24, the mean and standard deviation are strongly affected by extreme values. Exclude the outlier or outliers and recompute the maximum likelihood estimator of λ. Compare the support for the null hypothesis that λ= 1.0 with the support for the maximum likelihood estimate. Does the estimator change a great deal? Why does the support become so much larger? For type b, exclude the extreme values 4.16 and 4.83.
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8.8.26 In Exercises 23 and 24, the mean and standard deviation are strongly affected by extreme values. Exclude the outlier or outliers and recompute the maximum likelihood estimator of λ. Compare the support for the null hypothesis that λ= 1.0 with the support for the maximum likelihood estimate. Does the estimator change a great deal? Why does the support become so much larger? For type b, exclude the extreme values 4.16 and 4.83.
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8.8.27 Use the method of support to test whether the following samples differ. One player makes 5 out of 10 shots, another makes 9 out of 10.
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8.8.28 Use the method of support to test whether the following samples differ. One player makes 5 out of 10 shots, another makes 16 out of 20.
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8.8.29 Use the method of support to test whether the following samples differ. A 1 ... region in Utah is hit by 4 cosmic rays in 1 yr, and a 1 ... region at the North Pole is hit by 10 cosmic rays in 1 yr.
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8.8.30 Use the method of support to test whether the following samples differ. Two 1 ... regions in Utah are hit by 3 and 5 cosmic rays in 1 yr, and a 1 ...region at the North Pole is hit by 10 cosmic rays in 1 yr.
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8.8.31 Use the G test to test the following by building a table complete with observed and expected values. Compare the G statistic with the difference in support found in the earlier problem. As in Exercise 27, one player makes 5 out of 10 shots, another makes 9 out of 10.
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8.8.32 Use the G test to test the following by building a table complete with observed and expected values. Compare the G statistic with the difference in support found in the earlier problem. As in Exercise 28, one player makes 5 out of 10 shots, another makes 16 out of 20.
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8.8.33 One organism has 8 mutations in 1 million base pairs, a second has 18 in 1 million, and a third has 28 in 1 million. Use the method of support to test the following differences. Check whether organisms 1 and 2 differ and compare with Section 8.6, Exercise 39.
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8.8.34 One organism has 8 mutations in 1 million base pairs, a second has 18 in 1 million, and a third has 28 in 1 million. Use the method of support to test the following differences. Check whether organisms 2 and 3 differ and compare with Section 8.6, Exercise 40.
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8.8.35 It has been proposed that a particular salubrious bath extends cell lifespan. Suppose that cell mortality follows an exponential model. Use the method of support to evaluate the following cases. A cell in the salubrious bath survives 30 min, and a cell in standard culture survives only 5 min. Is there reason to think that the salubrious bath lengthens cell life?
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8.8.36 It has been proposed that a particular salubrious bath extends cell lifespan. Suppose that cell mortality follows an exponential model. Use the method of support to evaluate the following cases. In a repeated experiment, the cell in the salubrious bath survives 60 min, and a cell in standard culture survives only 3 min. Is there reason to think that the salubrious bath lengthens cell life?
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8.8.37 It has been proposed that a particular salubrious bath extends cell lifespan. Suppose that cell mortality follows an exponential model. Use the method of support to evaluate the following cases. Combine the data from Exercises 35 and 36, and evaluate the difference in support between the null and alternative hypotheses.
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8.8.38 It has been proposed that a particular salubrious bath extends cell lifespan. Suppose that cell mortality follows an exponential model. Use the method of support to evaluate the following cases. What would happen to the result in Exercise 37 if a third experiment were done and both cells survived 10 min? Why the change?
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### Solutions Modeling Dynamics of Life 3ed Adler - Chapter 8.7

8.7.1 Suppose that the number N of molecules of toxin left in a cell after 10.0 min is thought to follow the probability distribution with Pr(N = 0) = 0.4, Pr(N = 1) = 0.3, Pr(N = 2) = 0.2, and Pr(N = 3) = 0.1 (as in Example 6.3.11). Test whether the following data fit the expectation from this extrinsic hypothesis. There are 35 cells with no molecules, 25 with one molecule, 25 with two molecules, and 15 with three molecules.
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8.7.2 Suppose that the number N of molecules of toxin left in a cell after 10.0 min is thought to follow the probability distribution with Pr(N = 0) = 0.4, Pr(N = 1) = 0.3, Pr(N = 2) = 0.2, and Pr(N = 3) = 0.1 (as in Example 6.3.11). Test whether the following data fit the expectation from this extrinsic hypothesis. There are 25 cells with no molecules, 21 with one molecule, 19 with two molecules, and 15 with three molecules.
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8.7.4 Suppose that the number N of molecules of toxin left in a cell after 10.0 min is thought to follow the probability distribution with Pr(N = 0) = 0.4, Pr(N = 1) = 0.3, Pr(N = 2) = 0.2, and Pr(N = 3) = 0.1 (as in Example 6.3.11). Test whether the following data fit the expectation from this extrinsic hypothesis. Consider again the data in Exercise 2, but suppose that we can only distinguish cells with no molecules from those with at least one. Find how many cells are in each of these two categories and compare with the appropriate extrinsic hypothesis. Why might the test give a different result than with the unpooled data?
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8.7.4 Suppose that the number N of molecules of toxin left in a cell after 10.0 min is thought to follow the probability distribution with Pr(N = 0) = 0.4, Pr(N = 1) = 0.3, Pr(N = 2) = 0.2, and Pr(N = 3) = 0.1 (as in Example 6.3.11). Test whether the following data fit the expectation from this extrinsic hypothesis. Consider again the data in Exercise 2, but suppose that we can only distinguish cells with no molecules from those with at least one. Find how many cells are in each of these two categories and compare with the appropriate extrinsic hypothesis. Why might the test give a different result than with the unpooled data?
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8.7.5 The number of molecules remaining in a cell is thought to follow a binomial distribution with the given parameter. In each case, find whether there is reason to reject this model. Suppose there are three molecules, and that the probability of remaining is thought to be p = 0.6. In a sample of 80 cells, we find 10 with 0 molecules, 20 with 1 molecule, 30 with 2 molecules, and 20 with 3 molecules.
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8.7.6 The number of molecules remaining in a cell is thought to follow a binomial distribution with the given parameter. In each case, find whether there is reason to reject this model. Suppose there are 4 molecules, and that the probability of a molecule’s remaining is thought to be p = 0.6. In a sample of 80 cells, we find 5 with no molecules, 20 with one molecule, 20 with two molecules, 20 with three molecules, and 15 with four molecules.
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8.7.7 Compute the statistic ... in the earlier exercise using the continuity correction. Does it alter the conclusions? The situation in Exercise 5.
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8.7.8 Compute the statistic ... in the earlier exercise using the continuity correction. Does it alter the conclusions? The situation in Exercise 6.
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8.7.9 Suppose that the data in Exercises 5 and 6 are thought to follow a binomial distribution with an unknown parameter. Estimate this parameter and test whether the data fit the resulting model. The situation in Exercise 5.
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8.7.10 Suppose that the data in Exercises 5 and 6 are thought to follow a binomial distribution with an unknown parameter. Estimate this parameter and test whether the data fit the resulting model. The situation in Exercise 6.
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8.7.11 Consider the following data, which were supposedly generated from 200 replicates of a Poisson process. ... Test the extrinsic hypothesis that Λ= 4.5 for experiment 1.
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8.7.12 Consider the following data, which were supposedly generated from 200 replicates of a Poisson process. ... Test the extrinsic hypothesis that Λ = 4.0 for experiment 2
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8.7.13 Consider the following data, which were supposedly generated from 200 replicates of a Poisson process. ... Test the intrinsic hypothesis that the data follow a Poisson distribution for experiment 1.
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8.7.15 Consider again the data on mites and lice from Example 8.7.14. ... Find the probabilities for each term in the table, and find the conditional distributions. The distribution of the number of lice conditional on 0, 1, and 2 mites. How different are the conditional distributions, and would they lead you to suspect that the two pests do not act independently?
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8.7.15 Consider again the data on mites and lice from Example 8.7.14. ... Find the probabilities for each term in the table, and find the conditional distributions. The distribution of the number of lice conditional on 0, 1, and 2 mites. How different are the conditional distributions, and would they lead you to suspect that the two pests do not act independently?
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8.7.16 Consider again the data on mites and lice from Example 8.7.14. ... Find the probabilities for each term in the table, and find the conditional distributions. The distribution of the number of mites conditional on 0, 1, and 2 lice. How different are the conditional distributions, and would they lead you to suspect that the two pests do not act independently?
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8.7.17 Test the following tables for independence. Consider the following data on mating in birds. ... Do matings deviate from independence, and what might it mean?
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8.7.18 Test the following tables for independence. Consider the following data on student class attendance. ... Is attendance independent, and if not, what might it mean?
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8.7.19 The significance of deviations from the null hypothesis depends on the sample size. Conduct a ... test for the following samples based on Example 8.7.1. Suppose that 20% of diseased people tested have a particular allele (this would, of course, vary in a series of real experiments), and that 13% of healthy people are known to have the allele. Suppose we tested only 50 diseased people. Find the significance of the result and compare to the results with a sample size of 100.
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8.7.20 The significance of deviations from the null hypothesis depends on the sample size. Conduct a ... test for the following samples based on Example 8.7.1. Suppose that 20% of diseased people tested have a particular allele (this would, of course, vary in a series of real experiments), and that 13% of healthy people are known to have the allele. Suppose we tested 200 diseased people. Find the significance and compare to the results with a sample size of 100.
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8.7.21 The significance of deviations from the null hypothesis depends on the sample size. Conduct a ... test for the following samples based on Example 8.7.1. Suppose that 20% of diseased people tested have a particular allele (this would, of course, vary in a series of real experiments), and that 13% of healthy people are known to have the allele. Suppose we tested n diseased people. Compute ... as a function of n. Does it increase proportionally to the sample size?
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8.7.22 The significance of deviations from the null hypothesis depends on the sample size. Conduct a ... test for the following samples based on Example 8.7.1. Suppose that 20% of diseased people tested have a particular allele (this would, of course, vary in a series of real experiments), and that 13% of healthy people are known to have the allele. Suppose we tested n diseased people. How many people would we need to test to find a result significant at the 0.01 level?
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8.7.23 A random variable ...follows a ... distribution with ν degrees of freedom if ... where ...follow the standard normal distribution. Find the expectation of ....
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8.7.24 A random variable ...follows a ... distribution with ν degrees of freedom if ... where ...follow the standard normal distribution. Find the expectation of ... .
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8.7.25 A random variable ...follows a ... distribution with ν degrees of freedom if ... where ...follow the standard normal distribution. Compute the critical value for p = 0.05 with 1 degree of freedom.
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8.7.26 A random variable ...follows a ... distribution with ν degrees of freedom if ... where ...follow the standard normal distribution. Remarkably enough, ... is an exponential distribution. Using the mean found in Exercise 24, find the parameter of this distribution, and compute the critical value for p = 0.05.
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8.7.27 Use the ...test to check whether the control and treatment differ in the following contingency tables. Consider the following data on the behavior of 50 wild type and 100 mutant worms. ...
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8.7.28 Use the ...test to check whether the control and treatment differ in the following contingency tables. Consider the following data on the behavior of 100 wild type and 150 mutant worms. ...
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8.7.29 Use the ...test to check whether the control and treatment differ in the following contingency tables. Consider the following data on the behavior of 80 wild type and 120 mutant worms. ...
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8.7.30 Use the ...test to check whether the control and treatment differ in the following contingency tables. Consider the following data on the behavior of 100 wild type and 125 mutant worms. ...
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8.7.31 A recessive allele is expected to be expressed in 25% of offspring from a cross of heterozygous plants. Check whether the following data are consistent with this hypothesis. Ten out of 60 plants are homozygous for the recessive allele.
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8.7.32 A recessive allele is expected to be expressed in 25% of offspring from a cross of heterozygous plants. Check whether the following data are consistent with this hypothesis. Twenty-one out of 120 plants are homozygous for the recessive allele.
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8.7.33 Suppose that plants with genotype WW have white flowers, those with genotype WR or RW have pink flowers, and those with genotype RR have red flowers. Two RW plants are crossed. Check whether the following data are consistent with the expected ratios. If not, try to explain why. Out of 90 offspring, there are 18 white, 40 pink, and 32 red.
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8.7.34 Suppose that plants with genotype WW have white flowers, those with genotype WR or RW have pink flowers, and those with genotype RR have red flowers. Two RW plants are crossed. Check whether the following data are consistent with the expected ratios. If not, try to explain why. Suppose 10 additional plants had been measured in Exercise 33, and there were 3 pink ones and 7 red ones.
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8.7.35 Suppose two traits are controlled by two unlinked loci (so the phenotypes are independent), one for flower color and one for height. Check whether the following data are consistent with the expected numbers in the following scenarios. Suppose that both yellow flower color and shortness are recessive, with white flower color and tallness expressed in the dominant plants. Two parents that are heterozygous for these two traits are crossed, and 80 offspring are checked. Of these, 3 have yellow flowers and are short, 12 have yellow flowers and are tall, 17 have white flowers and are short, and 48 have white flowers and are tall.
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8.7.36 Suppose two traits are controlled by two unlinked loci (so the phenotypes are independent), one for flower color and one for height. Check whether the following data are consistent with the expected numbers in the following scenarios. Suppose that both yellow flower color and shortness are recessive, with white flower color and tallness expressed in the dominant plants. Two parents that are heterozygous for these two traits are crossed, and 87 offspring are checked. Of these, 11 have yellow flowers and are short, 8 have yellow flowers and are tall, 13 have white flowers and are short, and 55 have white flowers and are tall.
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8.7.37 An ecologist counts the numbers of jack rabbits and eagles observed, and wishes to know whether they are independent (as in Section 6.4, Exercises 27 and 28). E represents the number of eagles seen, and J the number of jackrabbits. Use the ... test to check. Eighty counts are made, with the following results. ... Are the results significant? Compare with Section 7.1,
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8.7.38 An ecologist counts the numbers of jack rabbits and eagles observed, and wishes to know whether they are independent (as in Section 6.4, Exercises 27 and 28). E represents the number of eagles seen, and J the number of jackrabbits. Use the ... test to check. Eighty counts are made, with the following results. ... Are the results significant? Compare with Section 7.1, Exercise 28.
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8.7.39 Recall the falcon data studied in Example 8.7.15, where 44 families of two birds were studied, and 14 had no males, 14 had one male, and 16 had 2 males. However, now assume that the order of birth is taken into account, so that there are four possible families (the first offspring could be male or female as could the second). Write a table and evaluate for lack of independence in the following cases, and compare with the results in Example 8.7.15. Of the 14 females with one male, 7 had a male first.
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8.7.40 Recall the falcon data studied in Example 8.7.15, where 44 families of two birds were studied, and 14 had no males, 14 had one male, and 16 had 2 males. However, now assume that the order of birth is taken into account, so that there are four possible families (the first offspring could be male or female as could the second). Write a table and evaluate for lack of independence in the following cases, and compare with the results in Example 8.7.15. Of the 14 females with one male, 3 had a male first.
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### Solutions Modeling Dynamics of Life 3ed Adler - Chapter 8.6

8.6.1 ...
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8.6.2 ...
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8.6.3 ...
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8.6.4 ...
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8.6.5 Find the pooled variance for two populations with the following sample sizes and sample variances. ...
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8.6.6 Find the pooled variance for two populations with the following sample sizes and sample variances. ...
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8.6.7 Find the pooled variance for two populations with the following sample sizes and sample variances. ... Compare with the value in Exercise 5.
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8.6.8 Find the pooled variance for two populations with the following sample sizes and sample variances. ... Compare with the value in Exercise 6.
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8.6.9 Find the standard error of the difference of the means in each case. The situation in Exercise 5.
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8.6.10 Find the standard error of the difference of the means in each case. The situation in Exercise 6.
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8.6.11 Find the standard error of the difference of the means in each case. The situation in Exercise 7.
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8.6.12 Find the standard error of the difference of the means in each case. The situation in Exercise 8.
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8.6.13 Apply a two-tailed t test in the following cases. The situation in Exercise 9 with ...
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8.6.14 Apply a two-tailed t test in the following cases. The situation in Exercise 10 with ...
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8.6.15 Apply a two-tailed t test in the following cases. The situation in Exercise 11 with ...
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8.6.16 Apply a two-tailed t test in the following cases. The situation in Exercise 12 with ...
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8.6.17 Recall the data in Section 8.5, Exercises 1–4 describing 10 plants in an experimental plot. ... Suppose that these plants are being compared with populations in a control plot. Use a two-tailed test, and compare the p-values with those found in the earlier problem. There are ten plants in the control plot with mean weight 10.0, and the variance for weight in both populations is known to be 9.0. Compare with the results in Section 8.5, Exercise 1.
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8.6.18 Recall the data in Section 8.5, Exercises 1–4 describing 10 plants in an experimental plot. ... Suppose that these plants are being compared with populations in a control plot. Use a two-tailed test, and compare the p-values with those found in the earlier problem. There are ten plants in the control plot with mean height 36.5, and the variance for height in both populations is known to be 16.0. Compare with the results in Section 8.5, Exercise 2.
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8.6.19 Recall the data in Section 8.5, Exercises 1–4 describing 10 plants in an experimental plot. ... Suppose that these plants are being compared with populations in a control plot. Use a two-tailed test, and compare the p-values with those found in the earlier problem. There are 15 plants in the control plot with mean yield 8.2, and the variance for yield in both populations is known to be 6.25. Compare with the results in Section 8.5, Exercise 3.
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8.6.20 Recall the data in Section 8.5, Exercises 1–4 describing 10 plants in an experimental plot. ... Suppose that these plants are being compared with populations in a control plot. Use a two-tailed test, and compare the p-values with those found in the earlier problem. There are 20 plants in the control plot with mean seed number 15.0, and the variance for seed number in both populations is known to be 25.0. Compare with the results in Section 8.5, Exercise 4.
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8.6.21 Consider again the data in Exercises 17–20, but suppose that variances are unknown. Use the sample variance for the experimental population found in the earlier problem and the given sample variance for the control population to perform a t test on these unpaired populations. The ten plants in the control plot have mean weight 10.0 and sample variance 8.80. Compare with the results in Exercise 17.
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8.6.22 Consider again the data in Exercises 17–20, but suppose that variances are unknown. Use the sample variance for the experimental population found in the earlier problem and the given sample variance for the control population to perform a t test on these unpaired populations. The ten plants in the control plot have mean height 36.5 and sample variance of 17.2. Compare with the results in Exercise 18.
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8.6.23 Consider again the data in Exercises 17–20, but suppose that variances are unknown. Use the sample variance for the experimental population found in the earlier problem and the given sample variance for the control population to perform a t test on these unpaired populations. The 15 plants in the control plot have mean yield 8.2 and sample variance of 8.2. Compare with the results in Exercise 19.
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8.6.24 Consider again the data in Exercises 17–20, but suppose that variances are unknown. Use the sample variance for the experimental population found in the earlier problem and the given sample variance for the control population to perform a t test on these unpaired populations. The 20 plants in the control plot have mean seed number 15.0 and sample variance of 14.2. Compare with the results in Exercise 20.
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8.6.25 Test the null hypothesis that the means from two populations are equal in the following cases. ...are sample means found from samples with size ...drawn from normal distributions with known variances .... State the significance level of the test. Use a two-tailed test. ...
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8.6.27 Test the null hypothesis that the means from two populations are equal in the following cases. ...are sample means found from samples with size ...drawn from normal distributions with known variances .... State the significance level of the test. Use a two-tailed test. ...
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8.6.27 Test the null hypothesis that the means from two populations are equal in the following cases. ...are sample means found from samples with size ...drawn from normal distributions with known variances .... State the significance level of the test. Use a two-tailed test. ...
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8.6.28 Test the null hypothesis that the means from two populations are equal in the following cases. ...are sample means found from samples with size ...drawn from normal distributions with known variances .... State the significance level of the test. Use a two-tailed test. ...
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8.6.29 Use the normal approximation to test the null hypothesis that men and women have the same opinions in the following cases. State the significance level of a two-tailed test. Thirty-five out of 50 men believe that if dolphins were so smart they could find their way out of nets, whereas 40 out of 50 women believe this.
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8.6.30 Use the normal approximation to test the null hypothesis that men and women have the same opinions in the following cases. State the significance level of a two-tailed test. Three hundred fifty out of 500 men and 400 out of 500women.
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8.6.31 Use the normal approximation to test the null hypothesis that men and women have the same opinions in the following cases. State the significance level of a two-tailed test. Thirty-five out of 50 men and 400 out of 500 women.
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8.6.32 Use the normal approximation to test the null hypothesis that men and women have the same opinions in the following cases. State the significance level of a two-tailed test. Seventy out of 1000 men and 40 out of 500 women. Why do you think the difference is not significant even though the samples are very large?
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8.6.33 Algorithm 8.4 uses ...to estimate the variance under the null hypothesis. Why might it make more sense to use ..., the proportion in the pooled sample? What is the pooled proportion if 96 out of 200 events occur in the control and 54 out of 100 events occur in the treatment?
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8.6.34 Algorithm 8.4 uses ...to estimate the variance under the null hypothesis. Redo the test using ...How different are the results? Under what circumstances might it make a larger difference which proportion was used?
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8.6.35 Algorithm 8.4 uses ...to estimate the variance under the null hypothesis. Show that the two-sample test turns into the one-sample test as ... approaches infinity. What is the null hypothesis about the difference between means?
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8.6.36 Algorithm 8.4 uses ...to estimate the variance under the null hypothesis. Show that the two-sample test turns into the one-sample test as ... approaches infinity. What is the distribution of sample means in the treatment population under the null hypothesis?
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8.6.37 A cell is placed in a medium with volume equal to that of the cell. Then 100 marked molecules are placed inside, and after 1 h, 40 are found inside and 60 are found outside. In a control, protein in the membrane thought to be involved in transporting the molecule has been removed and 50 out of 100 of the molecules are found inside after the same amount of time. What is the null hypothesis if the cell with the transporter in place is compared with the control? What is the null hypothesis if the treatment is compared with the expectation that molecules end up inside and outside with equal probability?
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8.6.38 A cell is placed in a medium with volume equal to that of the cell. Then 100 marked molecules are placed inside, and after 1 h, 40 are found inside and 60 are found outside. In a control, protein in the membrane thought to be involved in transporting the molecule has been removed and 50 out of 100 of the molecules are found inside after the same amount of time. Find the p-value associated with the comparison of the treatment with the control, and the comparison of the treatment with the expectation that molecules end up inside and outside with equal probability. Why do the p-values differ as they do?
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8.6.39 One organism has 8 mutations in 1 million base pairs, a second has 18 in 1 million, and a third has 28 in 1 million. Use the normal approximation to test whether the following differences are significant. The difference between the first and second organisms.
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8.6.40 One organism has 8 mutations in 1 million base pairs, a second has 18 in 1 million, and a third has 28 in 1 million. Use the normal approximation to test whether the following differences are significant. The difference between the second and third organisms. Why is the significance level different from that in Exercise 39 even though the observed difference of 10 mutations is the same in each case?.
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8.6.41 Consider the following data on ten patients with viral loads measured under control conditions, after treatment A, and then again after treatment B. Use the given test to check whether the treatment has an effect. ... Use an unpaired test to look for an effect from treatment A.
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8.6.42 Consider the following data on ten patients with viral loads measured under control conditions, after treatment A, and then again after treatment B. Use the given test to check whether the treatment has an effect. ... Use an unpaired test to look for an effect from treatment B.
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8.6.43 Consider the following data on ten patients with viral loads measured under control conditions, after treatment A, and then again after treatment B. Use the given test to check whether the treatment has an effect. ... Use a paired test to look for an effect from treatment A.
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8.6.44 Consider the following data on ten patients with viral loads measured under control conditions, after treatment A, and then again after treatment B. Use the given test to check whether the treatment has an effect. ... Use a paired test to look for an effect from treatment B.
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### Solutions Modeling Dynamics of Life 3ed Adler - Chapter 8.5

8.5.1 The weights, heights, yields, and seed number for 10 plants grown in an experimental plot are given in the table. Each measurement is approximately normally distributed. We wish to determine whether plants in the plot differ from those outside. In each case, state one- and two-tailed alternative hypotheses and find their significance. The variance for weight is 9.0, and plants outside the plot have mean weight 10.0. ...
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8.5.2 The weights, heights, yields, and seed number for 10 plants grown in an experimental plot are given in the table. Each measurement is approximately normally distributed. We wish to determine whether plants in the plot differ from those outside. In each case, state one- and two-tailed alternative hypotheses and find their significance. The variance for height is 16.0, and plants outside the plot have mean height 38.0....
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8.5.3 The weights, heights, yields, and seed number for 10 plants grown in an experimental plot are given in the table. Each measurement is approximately normally distributed. We wish to determine whether plants in the plot differ from those outside. In each case, state one- and two-tailed alternative hypotheses and find their significance. The variance for yield is 6.25, and plants outside the plot have mean yield 9.0. ...
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8.5.4 The weights, heights, yields, and seed number for 10 plants grown in an experimental plot are given in the table. Each measurement is approximately normally distributed. We wish to determine whether plants in the plot differ from those outside. In each case, state one- and two-tailed alternative hypotheses and find their significance. The variance for seed number is 25.0, and plants outside the plot have mean seed number 15.0....
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8.5.6 Consider again the weights, heights, yields, and seed number for 10 plants given in Exercises 1–4. Find the sample variance for each and use the t distribution to perform a two-tailed test of the hypothesis. Plants outside the plot have mean height 38.0.
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8.5.6 Consider again the weights, heights, yields, and seed number for 10 plants given in Exercises 1–4. Find the sample variance for each and use the t distribution to perform a two-tailed test of the hypothesis. Plants outside the plot have mean height 38.0.
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8.5.7 Consider again the weights, heights, yields, and seed number for 10 plants given in Exercises 1–4. Find the sample variance for each and use the t distribution to perform a two-tailed test of the hypothesis. Plants outside the plot have mean yield 9.0.
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8.5.8 Consider again the weights, heights, yields, and seed number for 10 plants given in Exercises 1–4. Find the sample variance for each and use the t distribution to perform a two-tailed test of the hypothesis. Plants outside the plot have mean seed number 15.0.
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8.5.9 Find the smallest values of the sample mean for which the given hypothesis is rejected. Under the conditions in Exercise 1, find the smallest weight that can reject the null hypothesis that the mean weight is 10.0 with a one-tailed test at the 0.01 level.
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8.5.10 Find the smallest values of the sample mean for which the given hypothesis is rejected. Under the conditions in Exercise 2, find the smallest height that can reject the null hypothesis that the mean height is 38.0 with a two-tailed test at the 0.01 level.
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8.5.11 Find the smallest values of the sample mean for which the given hypothesis is rejected. Under the conditions in Exercise 3, find the smallest yield that can reject the null hypothesis that the mean yield is 9.0 with a two-tailed test at the 0.001 level.
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8.5.12 Find the smallest values of the sample mean for which the given hypothesis is rejected. Under the conditions in Exercise 4, find the smallest seed number that can reject the null hypothesis that the mean seed number is 15.0 with a one-tailed test at the 0.001 level.
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8.5.13 Find the power of the test assuming the given true mean. The true mean weight is 13.0 in Exercise 9.
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8.5.14 Find the power of the test assuming the given true mean. The true mean height is 43.0 in Exercise 10.
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8.5.15 Find the power of the test assuming the given true mean. The true mean yield is 11.0 in Exercise 11.
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8.5.16 Find the power of the test assuming the given true mean. The true mean seed height is 18.0 in Exercise 12.
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8.5.17 Consider again plants with the null hypothesis that mean height is 39.0. Assume that the standard deviation is known to be 3.2 cm. Show that a measured sample mean of 40.0 is highly significant if the sample size is n = 88.
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8.5.18 Consider again plants with the null hypothesis that mean height is 39.0. Assume that the standard deviation is known to be 3.2 cm. Why is the power with this sample size only 90% (as found in the text), rather than more than 99%?
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8.5.19 Use the normal approximation to test the given hypothesis. A coin is flipped 100 times and comes out heads 44 times. It is thought that the coin is fair (has probability of heads is equal to 0.5). Do the data provide evidence that the coin is unfair?
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8.5.20 Use the normal approximation to test the given hypothesis. Of 1000 people polled in one state, 320 favor the use of mathematics in biology. The legislature has passed a bill mandating that at least 36% of people must be in favor. Does the poll provide evidence that the proportion is smaller than 0.36? Is the state in violation of the law?
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8.5.21 Consider the following data on 30 waiting times for 2 types of events. ... Find the p-value associated with the null hypothesis that the mean of type a is 1.0.
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8.5.22 Consider the following data on 30 waiting times for 2 types of events. ... Consider the following data on 30 waiting times for 2 types of events. Find the p-value associated with the null hypothesis that the mean of type b is 1.0.
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8.5.23 Consider again the data in Exercises 21 and 22. Each type has one or more outliers that strongly affect the mean and standard deviation. Exclude the outlier or outliers and recompute the p-value associated with the null hypothesis that the mean is 1.0. What do you think of this procedure if you were told that the data were generated from an exponential distribution with mean 1.0? The outlier is extreme value 6.33 at time 16.
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8.5.24 Consider again the data in Exercises 21 and 22. Each type has one or more outliers that strongly affect the mean and standard deviation. Exclude the outlier or outliers and recompute the p-value associated with the null hypothesis that the mean is 1.0. What do you think of this procedure if you were told that the data were generated from an exponential distribution with mean 1.0? The outliers are the extreme values 4.16 and 4.83.
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8.5.25 A chronic condition improves spontaneously in 45% of people. A new medication is being tested to try to increase this percentage. Of 25 of 50 patients tested with the new medication, 30 improve. Is this significant?
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8.5.26 A chronic condition improves spontaneously in 45% of people. A new medication is being tested to try to increase this percentage. Of 26 of 100 patients tested with the new medication, 60 improve. Is this significantly better?
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8.5.27 A chronic condition improves spontaneously in 45% of people. A new medication is being tested to try to increase this percentage. Suppose that the true fraction that improves with the medication is 0.6. What is the power to detect this at the 0.05 level with a sample of 50 patients?
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8.5.28 A chronic condition improves spontaneously in 45% of people. A new medication is being tested to try to increase this percentage. Suppose that the true fraction that improves with the medication is 0.6. What is the power to detect this at the 0.05 level with a sample of 100 patients? How much greater is the power?
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8.5.29 A company develops a new method to reduce error rates in the polymerase chain reaction (PCR). With the old method, the number of errors in a well-studied piece of DNA is known to have a Poisson distribution with mean 35.0. Use a one-tailed test in each case. Find the significance level if the DNA with the new method has only 27 errors. Make sure to start by finding the normal approximation to the null hypothesis.
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8.5.30 A company develops a new method to reduce error rates in the polymerase chain reaction (PCR). With the old method, the number of errors in a well-studied piece of DNA is known to have a Poisson distribution with mean 35.0. Use a one-tailed test in each case. Find the significance level if the DNA with the new method has only 23 errors.
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8.5.31 A company develops a new method to reduce error rates in the polymerase chain reaction (PCR). With the old method, the number of errors in a well-studied piece of DNA is known to have a Poisson distribution with mean 35.0. Use a one-tailed test in each case. What is the largest number of errors that would reject the null hypothesis at the 0.05 level?
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8.5.32 A company develops a new method to reduce error rates in the polymerase chain reaction (PCR). With the old method, the number of errors in a well-studied piece of DNA is known to have a Poisson distribution with mean 35.0. Use a one-tailed test in each case. What is the largest number of errors that would reject the null hypothesis at the 0.01 level?
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8.5.33 A company develops a new method to reduce error rates in the polymerase chain reaction (PCR). With the old method, the number of errors in a well-studied piece of DNA is known to have a Poisson distribution with mean 35.0. Use a one-tailed test in each case. Instead of measuring only a single piece of DNA with the new method, 10 pieces are measured and 300 errors are found. Does the new method reduce the number of errors?
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8.5.34 A company develops a new method to reduce error rates in the polymerase chain reaction (PCR). With the old method, the number of errors in a well-studied piece of DNA is known to have a Poisson distribution with mean 35.0. Use a one-tailed test in each case. Instead of measuring only a single piece of DNA with the new method, 20 pieces are measured and 650 errors are found. Does the new method reduce the number of errors?
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8.5.35 Use the normal approximation to test the following hypotheses about growing populations. In each case, habitat improvements are tried and the population grows from 1 to 250 individuals in 50 years. Is there reason to think that the habitat improvements helped? The population in Section 7.8, Exercise 39, where per capita production is a random variable with p.d.
f. g(x)=5.0 for 1.0 ≤ x ≤ 1.2.
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8.5.36 Use the normal approximation to test the following hypotheses about growing populations. In each case, habitat improvements are tried and the population grows from 1 to 250 individuals in 50 years. Is there reason to think that the habitat improvements helped? The population in Section 7.8, Exercise 40, where per capita production is a random variable with p.d.
f. g(x)= 1.25 for 0.7 ≤ x ≤ 1.5. Can you explain the difference from the result in the previous problem?
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