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mathematics
calculus

Questions and Answers of Calculus

Use the method of support to estimate 95% confidence limits and compare your results with Exercise 35. Consider a data set where three out of three people are found with an allele.
A person wins the lottery the fifth time he plays. Does the value q = 0.001 for the probability of success lie within the 99% confidence limits? In the above situation, find the probability of a
Find the 95% confidence limits. A person places an advertisement to sell his car in the newspaper and settles down to await calls, which he expects will arrive with an exponential distribution. The
Find the 98% confidence limits. A person places an advertisement to sell his car in the newspaper and settles down to await calls, which he expects will arrive with an exponential distribution. The
How many calls might he expect to miss if he went out for a 2-h hike? A person places an advertisement to sell his car in the newspaper and settles down to await calls, which he expects will arrive
Write the equations for the 95% confidence limits, and solve them numerically if you have a computer. 14 mutations are counted in one million base pairs.
Write the equations for the approximate 95% confidence limits using the method of support, and solve them numerically if you have a computer. 14 mutations are counted in one million base pairs.
Find 95% confidence limits around the maximum likelihood estimate of q. How do you interpret these results? Recall the couple that has seven boys before having a girl (Section 8.1, Exercise 33).
Find 99% confidence limits around the maximum likelihood estimate of q. How do you interpret these results? Recall the couple that has seven boys before having a girl (Section 8.1, Exercise 33).
Three cosmic rays hit a detector in 1 yr. Does the value λ = 10.0 for the rate at which rays hit lie within the 98% confidence limits? In the above situation, find the probability of a result as
One cosmic ray hits a detector in 1 yr. Does the value λ = 5.0 for the rate at which rays hit lie within the 98% confidence limits? In the above situation, find the probability of a result as
Find the 95% confidence limits if a coin is flipped five times and comes up heads every time (Exercise 1). Find the exact confidence limits and check if the value for the earlier problem lies within
Find the 95% confidence limits if a coin is flipped seven times and comes up heads six out of seven times (Exercise 2). Find the exact confidence limits and check if the value for the earlier problem
Find the 99% confidence limits if a person wins the lottery the second time he plays (Exercise 3). Find the exact confidence limits and check if the value for the earlier problem lies within them.
The weight W. Consider the following data on 20 plants.Find the following for the given measurement. a. The sample mean. b. The sample median. c. The trimmed means tr(5), tr(10), and tr(20).
Find how many measurements lie (a) less than one sample standard deviation from the sample mean and (b) more than two sample standard deviations from the sample mean for the given measurement. Which
Find how many measurements lie (a) less than one sample standard deviation from the sample mean and (b) more than two sample standard deviations from the sample mean for the given measurement. Which
Find how many measurements lie (a) less than one sample standard deviation from the sample mean and (b) more than two sample standard deviations from the sample mean for the given measurement. Which
Find the 95% confidence limits around the sample mean for the given measurement, assuming that the sample variance s is a good estimate of the true variance Ïƒ.1. The weight W in Exercises 1 and
Find the 95% confidence limits around the sample mean for the given measurement, without assuming that the sample variance s is a good estimate of the true variance Ïƒ (thus using the t
The height H. Consider the following data on 20 plants.Find the following for the given measurement.a. The sample mean.b. The sample median.c. The trimmed means tr(5), tr(10), and tr(20).
98% confidence limits around the weight W in Exercise 1. Find the given confidence limits around the sample mean for the given measurement, assuming that the sample variance s is a good estimate of
90% confidence limits around the height H in Exercise 2. Find the given confidence limits around the sample mean for the given measurement, assuming that the sample variance s is a good estimate of
99.8% confidence limits around the yield Y in Exercise 3. Find the given confidence limits around the sample mean for the given measurement, assuming that the sample variance s is a good estimate of
99.9% confidence limits around the seed number S in Exercise 4. Find the given confidence limits around the sample mean for the given measurement, assuming that the sample variance s is a good
Find the normal approximation to the following.1. The average of 30 numbers chosen from the exponential p.d.f. g(x) = 2e-2x.2. The average of 30 numbers chosen from the p.d.f. f(x) = 1 for 0 ≤ x
Find 95% confidence intervals in the following cases, assuming that the standard deviations are known to match those in the earlier problem. Does the confidence interval include the true mean?1. A
A coin is flipped 100 times and comes out heads 44 times.Use the normal approximation to find 95% confidence limits around the estimated proportion in the above case.
The yield Y. Consider the following data on 20 plants.Find the following for the given measurement. a. The sample mean. b. The sample median. c. The trimmed means tr(5), tr(10), and tr(20).
Of 1000 people polled, 320 favor the use of mathematics in biology.Use the normal approximation to find 95% confidence limits around the estimated proportion in the above case.
The 99% confidence limits with the normal distribution. The 95% confidence limits with the t distribution are wider than those with the normal distribution. For approximately how many degrees of
The 98% confidence limits with the normal distribution. The 95% confidence limits with the t distribution are wider than those with the normal distribution. For approximately how many degrees of
Find the mean squared deviation from the mean for each and average them to find the expected sample variance. In simple cases, we can see why using a denominator of n in the equation for the sample
Compare with the true answer and show that using a denominator of n - 1 rather than n would give the right answer. Try to explain the bias. In simple cases, we can see why using a denominator of n in
Population a.Consider the following data on immigration into four populations over 20 yr. For each, find the sample mean and the sample standard deviation. Compare them with the mathematical mean and
Population b.Consider the following data on immigration into four populations over 20 yr. For each, find the sample mean and the sample standard deviation. Compare them with the mathematical mean and
Population c. Why are the mean and variance so different from the mathematical expectations?Consider the following data on immigration into four populations over 20 yr. For each, find the sample mean
The seed number S. Consider the following data on 20 plants.Find the following for the given measurement. a. The sample mean. b. The sample median. c. The trimmed means tr(5), tr(10), and tr(20).
Population d.Consider the following data on immigration into four populations over 20 yr. For each, find the sample mean and the sample standard deviation. Compare them with the mathematical mean and
Population a. Find the 95% confidence intervals around the mean number of immigrants using both the true variance and the sample variance. Does the true mean lie within the confidence limits?
Population b. Find the 95% confidence intervals around the mean number of immigrants using both the true variance and the sample variance. Does the true mean lie within the confidence limits?
Population c. Find the 95% confidence intervals around the mean number of immigrants using both the true variance and the sample variance. Does the true mean lie within the confidence limits?
Population d. Find the 95% confidence intervals around the mean number of immigrants using both the true variance and the sample variance. Does the true mean lie within the confidence limits?
Of 50 offspring, 35 are tall.Several plants are crossed, producing the following proportions. Find 99% confidence limits around the fraction of tall plants in each case.
Of 500 offspring, 350 are tall.Several plants are crossed, producing the following proportions. Find 99% confidence limits around the fraction of tall plants in each case.
Of 100 offspring, 52 are tall.Several plants are crossed, producing the following proportions. Find 99% confidence limits around the fraction of tall plants in each case.
Of 200 offspring, 13 are tall.Several plants are crossed, producing the following proportions. Find 99% confidence limits around the fraction of tall plants in each case.
Use the normal approximation to the Poisson distribution to estimate 95% confidence limits around the true mean number of mutations per million base pairs and compare with Section 8.2, Exercise
Find the sample variance, the sample standard deviation, and the standard error for the given measurement.1. The weight W in Exercise 1.2. The height H in Exercise 2.3. The yield Y in Exercise 3.4.
Thirty different sets of 1 million base pairs are measured, and the average number of mutations per million is found to be 13.5. Estimate the standard deviation, and find the standard error of the
Find how many measurements lie (a) less than one sample standard deviation from the sample mean and (b) more than two sample standard deviations from the sample mean for the given measurement. Which
The p-value associated with the null hypothesis is 0.2. Explain in words what the above statement mean, and restate the result in terms of type I or type II errors as appropriate.
What is the cutoff number of heads above which you reject the null hypothesis at the α = 0.1 significance level? A coin is flipped 10 times and comes up heads 9 times. You have reason to suspect
One cosmic ray hits a detector in 1 yr. The null hypothesis is that the rate at which rays hit is λ = 5/yr. Find the p-value with both one-tailed and a two-tailed tests.
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. Find the p-value with both one-tailed and a two-tailed tests.
You wait 4000 h for an exponentially distributed event to occur. The null hypothesis is that the mean wait is 1000 h with the alternative hypothesis that the mean wait is greater than 1000 h. Find
You wait 40 h for an exponentially distributed event to occur. The null hypothesis is that the mean wait is 1000 h with the alternative hypothesis that the mean wait is less than 1000 h. Find the
The first defective gasket is the 25th. The null hypothesis is that the first defect follows a geometric distribution with mean 10, and the alternative hypothesis is that the mean is greater than
The first defective gasket is the 50th. The null hypothesis is that the first defect follows a geometric distribution with mean 1000, and the alternative hypothesis is that the mean is less than
Certain screens for cancer work by examining many cells under a microscope and looking for abnormalities. Discuss how setting the threshold for cell abnormality can affect the number of type I and
Out of 1000 simulations of the null hypothesis, 70 produce a result as extreme as or more extreme than the actual observation. Explain in words what the above statement mean, and restate the result
Certain screens for cancer drugs work by examining many drugs and looking for those that suppress tumor growth. Discuss how setting the threshold for tumor growth reduction can affect the number of
Test the hypothesis that boys and girls are equally likely. Consider a couple that has seven boys and one girl in a family of eight.
At birth, boys are slightly more common than girls. Test the hypothesis that 55% of births are boys. Consider a couple that has seven boys and one girl in a family of eight
You receive 7 calls in 1 h on the first day. Phone calls used to arrive at an average rate of 3.5/h, but after posting your number on your Web page, you receive more calls on subsequent days. For
You receive 8 calls in 1 hr on the second day. Phone calls used to arrive at an average rate of 3.5/h, but after posting your number on your Web page, you receive more calls on subsequent days. For
Find the cutoff value for a test with α = 0.05. Find the power with λ = 4.0, λ = 7.0, and λ = 10.0. Explain why the power is higher for larger values of Λ. Consider the data in Exercise 23,
Find the cutoff value for a test with α = 0.01. Find the power with λ = 4.0, λ = 7.0, and λ = 10.0. Why does a higher significance level reduce the power? Consider the data in Exercise 23, where
State null and alternative hypotheses for the first type of cell. At what level can you reject the null hypothesis? Is it significant? The survival time for one type of cell in culture is
State null and alternative hypotheses for the second type of cell. At what level can you reject the null hypothesis? Is it significant? The survival time for one type of cell in culture is
Suppose you adopt the cutoff from Exercise 29. Find and graph the power as a function of the true mean. What is the power of the test if the true mean is 500? What is the power of the test if the
Suppose you adopt the cutoff from Exercise 30. Find and graph the power as a function of the true mean. What is the power of the test if the true mean is 50? What is the power of the test if the true
Solve for the smallest value of the mean for which the power to detect an improvement in cells of the first type is equal to 0.95. Interpret your answer. The survival time for one type of cell in
Solve for the largest value of the mean for which the power to detect harm to cells of the second type is equal to 0.95. Interpret your answer. The survival time for one type of cell in culture is
95% confidence limits where the upper and lower confidence limits are 1.0 cm from the sample mean. Consider measuring n plants with a known standard deviation of 3.2 cm. How many plants would have to
99% confidence limits where the upper and lower confidence limits are 1.0 cm from the sample mean. Consider measuring n plants with a known standard deviation of 3.2 cm. How many plants would have to
A coin is flipped 5 times and comes up heads every time. Using the above data, find the p-value associated with the null hypothesis that the true probability of heads is 0.5. Is the result
A coin is flipped 7 times and comes up heads 6 out of 7 times. Using the above data, find the p-value associated with the null hypothesis that the true probability of heads is 0.5. Is the result
A coin is flipped 10 times and comes up heads 9 times. You have reason to suspect however, that the coin produces an excess of heads (and have told this to a friend in advance). Using the above data,
A coin is flipped 20 times and comes up heads 3 times. You have reason to suspect however, that the coin produces an excess of tails (and have told this to a friend in advance). Using the above data,
What is the cutoff number of heads above which you reject the null hypothesis at the α = 0.05 significance level? A coin is flipped 10 times and comes up heads 9 times. You have reason to suspect
The variance for weight is 9.0, and plants outside the plot have mean weight 10.0.The weights, heights, yields, and seed number for 10 plants grown in an experimental plot are given in the table.
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. Find the smallest
The true mean weight is 13.0 in Exercise 9. Find the power of the test assuming the given true mean.
The true mean seed height is 18.0 in Exercise 12. Find the power of the test assuming the given true mean.
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.1. Show that a measured sample mean of 40.0 is highly significant if
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? Use the
The variance for height is 16.0, and plants outside the plot have mean height 38.0.The weights, heights, yields, and seed number for 10 plants grown in an experimental plot are given in the table.
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
Find the p-value associated with the null hypothesis that the mean of type a is 1.0.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.Consider the following data on 30 waiting times for 2 types of events.
The outlier is extreme value 6.33 at time 16.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
The outliers are the extreme values 4.16 and 4.83.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
Of 25 of 50 patients tested with the new medication, 30 improve. Is this significant? A chronic condition improves spontaneously in 45% of people. A new medication is being tested to try to increase
Of 26 of 100 patients tested with the new medication, 60 improve. Is this significantly better? A chronic condition improves spontaneously in 45% of people. A new medication is being tested to try to
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? A chronic condition improves spontaneously in
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? A chronic
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. A company develops a new method to reduce
The variance for yield is 6.25, and plants outside the plot have mean yield 9.0.The weights, heights, yields, and seed number for 10 plants grown in an experimental plot are given in the table. Each
Find the significance level if the DNA with the new method has only 23 errors. A company develops a new method to reduce error rates in the polymerase chain reaction (PCR). With the old method, the

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