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Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions

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? A company develops a new method to reduce error rates in the polymerase chain reaction (PCR). With the old method, the number of
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? A company develops a new method to reduce error rates in the polymerase chain reaction (PCR). With the old method, the number of
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. Use the normal approximation to test the above hypotheses about growing populations. In each case, habitat improvements are tried and the population grows from
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? Use the normal approximation to test the above hypotheses about growing populations. In
The variance for seed number is 25.0, and plants outside the plot have mean seed number 15.0.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
Plants outside the plot have mean weight 10.0.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.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.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.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.
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. Find the smallest values of the sample mean for which the given hypothesis is rejected.
Use p-values to test the null hypothesis of equal means against an alternative that μ2 > μ1 when sample means of 1 and 2 are found from samples of size n1 and n2 with sample variances s21 and s22. Use a one-tailed test. State the significance level of the test. 1 = 70.0, 2 = 74.0, n1 = 50, n2
The situation in Exercise 9 with μ1 = 3.2 and μ2 = 2.1. Apply a two-tailed t test in the above case.
The situation in Exercise 10 with μ1 = 3.8 and μ2 = 2.1. Apply a two-tailed t test in the above case.
The situation in Exercise 11 with μ1 = 3.8 and μ2 = 2.1. Apply a two-tailed t test in the above case.
The situation in Exercise 12 with μ1 = 3.2 and μ2 = 2.1. Apply a two-tailed t test in the above case.
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.Recall the data in Section 8.5, Exercises 1-4 describing 10 plants in an experimental plot.Suppose that these
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.Recall the data in Section 8.5, Exercises 1-4 describing 10 plants in an experimental plot.Suppose that these
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.Recall the data in Section 8.5, Exercises 1-4 describing 10 plants in an experimental plot.Suppose that these plants
Use p-values to test the null hypothesis of equal means against an alternative that μ2 > μ1 when sample means of 1 and 2 are found from samples of size n1 and n2 with sample variances s21 and s22. Use a one-tailed test. State the significance level of the test. 1 = 70.0, 2 = 70.4, n1 = 500,
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.Recall the data in Section 8.5, Exercises 1-4 describing 10 plants in an experimental plot.Suppose that
The ten plants in the control plot have mean weight 10.0 and sample variance 8.80. Compare with the results in Exercise 17.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
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.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 15 plants in the control plot have mean yield 8.2 and sample variance of 8.2. Compare with the results in Exercise 19.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
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.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
Test the null hypothesis that the means from two populations are equal in the following case. 1 and 2 are sample means found from samples with size n1 and n2 drawn from normal distributions with known variances σ21 and σ22. State the significance level of the test. Use a two-tailed test. 1 =
Test the null hypothesis that the means from two populations are equal in the following case. 1 and 2 are sample means found from samples with size n1 and n2 drawn from normal distributions with known variances σ21 and σ22. State the significance level of the test. Use a two-tailed test. 1 =
Test the null hypothesis that the means from two populations are equal in the following case. 1 and 2 are sample means found from samples with size n1 and n2 drawn from normal distributions with known variances σ21 and σ22. State the significance level of the test. Use a two-tailed test. 1 =
Test the null hypothesis that the means from two populations are equal in the following case. 1 and 2 are sample means found from samples with size n1 and n2 drawn from normal distributions with known variances σ21 and σ22. State the significance level of the test. Use a two-tailed test. 1 =
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. Use the normal approximation to test the null hypothesis that men and women have the same opinions in the above case. State the significance level of a
Use p-values to test the null hypothesis of equal means against an alternative that μ2 > μ1 when sample means of 1 and 2 are found from samples of size n1 and n2 with sample variances s21 and s22. Use a one-tailed test. State the significance level of the test. 1 = 70.0, 2 = 74.0, n1 = 500,
Three hundred fifty out of 500 men and 400 out of 500 women.Use the normal approximation to test the null hypothesis that men and women have the same opinions in the above case. State the significance level of a two-tailed test.
Thirty-five out of 50 men and 400 out of 500 women.Use the normal approximation to test the null hypothesis that men and women have the same opinions in the above case. 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?Use the normal approximation to test the null hypothesis that men and women have the same opinions in the above case. State the significance level of a
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? Algorithm 8.4 uses 1 and 2 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?Algorithm 8.4 uses 1 and 2 to estimate the variance under the null hypothesis.
Show that the two-sample test turns into the one-sample test as n1 approaches infinity.1. What is the null hypothesis about the difference between means?2. What is the distribution of sample means in the treatment population under the null hypothesis?
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? A cell is placed in a medium with volume equal to that
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? A cell is placed in a medium with volume equal to that
The difference between the first and second organisms.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 above differences are significant.
Use p-values to test the null hypothesis of equal means against an alternative that μ2 > μ1 when sample means of 1 and 2 are found from samples of size n1 and n2 with sample variances s21 and s22. Use a one-tailed test. State the significance level of the test. 1 = 70.0, 2 = 70.4, n1 = 500,
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?One organism has 8 mutations in 1 million base pairs, a second has 18 in 1 million, and a third has 28
Use an unpaired test to look for an effect from treatment A.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.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.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.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.
Find the pooled variance for two populations with the following sample sizes and sample variances. n1 = 9, n2 = 15, s21 = 2.5, s22 = 3.5. Compare with the value in Exercise 5.
Find the pooled variance for two populations with the following sample sizes and sample variances. n1 = 16, n2 = 10, s21 = 2.7, s22 = 3.9. Compare with the value in Exercise 6.
Find the standard error of the difference of the means in each case.1. The situation in Exercise 5.2. The situation in Exercise 6.3. The situation in Exercise 7.4. The situation in Exercise 8.
There are 35 cells with no molecules, 25 with one molecule, 25 with two molecules, and 15 with three molecules. 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
The situation in Exercise 6. 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. Exercise 6 Suppose there are 4 molecules, and that the probability of a
Test the extrinsic hypothesis that Î› = 4.5 for experiment 1.Consider the following data, which were supposedly generated from 200 replicates of a Poisson process.
Test the extrinsic hypothesis that Î› = 4.0 for experiment 2.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.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 2.Consider the following data, which were supposedly generated from 200 replicates of a Poisson process.
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?Consider again the data on mites and lice from Example 8.7.14.Find the probabilities for each term
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?Consider again the data on mites and lice from Example 8.7.14.Find the probabilities for each term
Consider the following data on mating in birds.Do matings deviate from independence, and what might it mean? Test the above table for independence.
Consider the following data on student class attendance.Is attendance independent, and if not, what might it mean? Test the above table for independence.
Suppose we tested only 50 diseased people. Find the significance of the result and compare to the results with a sample size of 100. The significance of deviations from the null hypothesis depends on the sample size. Conduct a x2 test for the following samples based on Example 8.7.1. Suppose that
There are 25 cells with no molecules, 21 with one molecule, 19 with two molecules, and 15 with three molecules. 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
Suppose we tested 200 diseased people. Find the significance and compare to the results with a sample size of 100. The significance of deviations from the null hypothesis depends on the sample size. Conduct a x2 test for the following samples based on Example 8.7.1. Suppose that 20% of diseased
Suppose we tested n diseased people. Compute x2 as a function of n. Does it increase proportionally to the sample size? The significance of deviations from the null hypothesis depends on the sample size. Conduct a x2 test for the following samples based on Example 8.7.1. Suppose that 20% of
Find the expectation of C1. A random variable Cv follows a x2 distribution with v degrees of freedom if Cv = X21 + X22 + ... + X2v where X1, X2, ... , Xv follow the standard normal distribution.
Compute the critical value for p = 0.05 with 1 degree of freedom. A random variable Cv follows a x2 distribution with v degrees of freedom if Cv = X21 + X22 + ... + X2v where X1, X2, ... , Xv follow the standard normal distribution.
Remarkably enough, C2 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. A random variable Cv follows a x2 distribution with v degrees of freedom if Cv = X21 + X22 + ... + X2v where X1, X2, ... ,
Consider the following data on the behavior of 50 wild type and 100 mutant worms.Use the x2 test to check whether the control and treatment differ in the above contingency table.
Consider the following data on the behavior of 100 wild type and 150 mutant worms.Use the x2 test to check whether the control and treatment differ in the above contingency table.
Consider the following data on the behavior of 80 wild type and 120 mutant worms.Use the x2 test to check whether the control and treatment differ in the above contingency table.
Consider again the data in Exercise 1, but suppose that we can only distinguish cells with two or more molecules from those with one or fewer. 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
Consider the following data on the behavior of 100 wild type and 125 mutant worms.Use the x2 test to check whether the control and treatment differ in the above contingency table.
Ten out of 60 plants are homozygous for the recessive allele. A recessive allele is expected to be expressed in 25% of offspring from a cross of heterozygous plants. Check whether the above data are consistent with this hypothesis.
Twenty-one out of 120 plants are homozygous for the recessive allele. A recessive allele is expected to be expressed in 25% of offspring from a cross of heterozygous plants. Check whether the above data are consistent with this hypothesis.
Out of 90 offspring, there are 18 white, 40 pink, and 32 red. 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 above data are consistent with the
Suppose 10 additional plants had been measured in Exercise 33, and there were 3 pink ones and 7 red ones. 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
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
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
Eighty counts are made, with the following results.Are the results significant? Compare with Section 7.1, Exercise 27. An ecologist counts the numbers of jack rabbits and eagles observed, and wishes to know whether they are independent. E represents the number of eagles seen, and J the number of
Eighty counts are made, with the following results.Are the results significant? Compare with Section 7.1, Exercise 28. An ecologist counts the numbers of jack rabbits and eagles observed, and wishes to know whether they are independent. E represents the number of eagles seen, and J the number of
Of the 14 females with one male, 7 had a male first. 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
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
Of the 14 females with one male, 3 had a male first. 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
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. The number of molecules remaining in a cell is thought to follow a binomial
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. The number of molecules remaining in a
The situation in Exercise 5. Compute the statistic x2 in the earlier exercise using the continuity correction. Does it alter the conclusions? Exercise 5 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
The situation in Exercise 6. Compute the statistic x2 in the earlier exercise using the continuity correction. Does it alter the conclusions? Exercise 6 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
The situation in Exercise 5. 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. Exercise 5 Suppose there are three molecules, and that the probability of
A coin is flipped 5 times and comes up heads every time. Using the following data, use the method of support to evaluate the null hypothesis that the true probability of heads is 0.5.
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. Find the difference in support of the above hypotheses. Compare with the p-value in the earlier problem.
Use the method of support to check the following hypotheses.1. The hypothesis in Section 8.5, Exercise 1.2. The hypothesis in Section 8.5, Exercise 2.3. The hypothesis in Section 8.5, Exercise 3.4. The hypothesis in Section 8.5, Exercise 4.
How many standard errors from the mean are the following? What are the corresponding p values for a two-tailed test?1. The support for the null hypothesis is less than the maximum by 2.2. The support for the null hypothesis is less than the maximum by 3.3. The support for the null hypothesis is
Follow the steps to show that the support has the simple quadratic form given in the text.1. Show that(expand the quadratic and plug in definitions of and Î±2).2. Remove the terms that do not depend on Î¼ and show that the maximum occurs at Î¼ = .
A coin is flipped 7 times and comes up heads 6 out of 7 times. Using the following data, use the method of support to evaluate the null hypothesis that the true probability of heads is 0.5.
Remove the terms that do not depend on μ and show that the maximum occurs at μ = . Follow the steps to show that the support has the simple quadratic form given in the text.
Day 1, when 7 calls arrive in 1 h while only 3.5 were expected. 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. Consider the data in Section 8.4, Exercises 23 and 24. Find the difference in support of the null and alternative hypotheses.
Use maximum likelihood to estimate the rate Î» from the waiting times for type a. Compare the support for the null hypothesis that Î» = 1.0 with the support for the maximum likelihood estimate.Consider again the data on 30 waiting times for 2 types of events used in Section
Use maximum likelihood to estimate the rate Î» from the waiting times for type b. Compare the support for the null hypothesis that Î» = 1.0 with the support for the maximum likelihood estimate.Consider again the data on 30 waiting times for 2 types of events used in Section
For type a, exclude the extreme value 6.33 at time 16. 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
For type b, exclude the extreme values 4.16 and 4.83. 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
One player makes 5 out of 10 shots, another makes 9 out of 10. Use the method of support to test whether the above samples differ.

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