8.2.1 In
the following situations, find the probability of a result as extreme
as or more extreme than the actual result for the given value of the
parameter. A coin is flipped five times and comes up heads every time. Does the value p = 0.5 for the probability of heads lie within the 95% confidence limits?
Get solution
8.2.2 In
the following situations, find the probability of a result as extreme
as or more extreme than the actual result for the given value of the
parameter. A coin is flipped seven times and comes up heads six out of seven times. Does the value p = 0.5 for the probability of heads lie within the 95% confidence limits?
Get solution
8.2.3 In
the following situations, find the probability of a result as extreme
as or more extreme than the actual result for the given value of the
parameter. A person wins the lottery the second time he plays. Does the value q = 0.001 for the probability of success lie within the 99% confidence limits?
Get solution
8.2.4 In
the following situations, find the probability of a result as extreme
as or more extreme than the actual result for the given value of the
parameter. 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?
Get solution
8.2.5 In
the following situations, find the probability of a result as extreme
as or more extreme than the actual result for the given value of the
parameter. 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?
Get solution
8.2.6 In
the following situations, find the probability of a result as extreme
as or more extreme than the actual result for the given value of the
parameter. 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?
Get solution
8.2.8
Find the exact confidence limits and check if the value for the
earlier problem lies within them. Find the 95% confidence limits if a
coin is flipped seven times and comes up heads six out of seven times
(Exercise 2).
Get solution
8.2.8
Find the exact confidence limits and check if the value for the
earlier problem lies within them. Find the 95% confidence limits if a
coin is flipped seven times and comes up heads six out of seven times
(Exercise 2).
Get solution
8.2.9
Find the exact confidence limits and check if the value for the
earlier problem lies within them. Find the 99% confidence limits if a
person wins the lottery the second time he plays (Exercise 3).
Get solution
8.2.10
Find the exact confidence limits and check if the value for the
earlier problem lies within them. Find the 98% confidence limits if a
one cosmic ray hits a detector in 1 yr (Exercise 6).
Get solution
8.2.11
Find the approximate 95% confidence limits using the method of
support and compare with earlier exercises. A coin is flipped five
times and comes up heads every time (Exercise 7).
Get solution
8.2.12
Find the approximate 95% confidence limits using the method of
support and compare with earlier exercises. A coin is flipped seven
times and comes up heads six out of seven times (Exercise 8).
Get solution
8.2.13 Find the approximate 95% confidence limits using the method of support and compare with earlier exercises. A
person wins the lottery the second time he plays (Exercise 9, but
recall that the earlier exercise found 99% confidence limits).
Get solution
8.2.14
Find the approximate 95% confidence limits using the method of
support and compare with earlier exercises. One cosmic ray hits a
detector in 1 yr (Exercise 6, but recall that the earlier exercise found
99% confidence limits).
Get solution
8.2.15
Explain how you would use the Monte Carlo method to estimate
confidence limits in the following cases. A coin is flipped five times
and comes up heads every time, and we wish to find the upper and lower
95% confidence limits.
Get solution
8.2.16 Explain how you would use the Monte Carlo method to estimate confidence limits in the following cases. A
coin is flipped seven times and comes up heads six out of seven times,
and we wish to find upper and lower 95% confidence limits.
Get solution
8.2.17
Explain how you would use the Monte Carlo method to estimate
confidence limits in the following cases. A person wins the lottery the
second time he plays, and we wish to find upper and lower 99%
confidence limits.
Get solution
8.2.18
Explain how you would use the Monte Carlo method to estimate
confidence limits in the following cases. One cosmic ray hits a
detector in 1 yr, and we wish to find upper and lower 98% confidence
limits.
Get solution
8.2.19 Check whether the expected value of p ( p = 1/2 for a fair coin and p = 1/6
for a fair die) lies within the approximate95% confidence limits given
by the method of support. Flipping 2 out of 4 heads with a fair coin
(as in Section 8.1, Exercise 9).
Get solution
8.2.20 Check whether the expected value of p ( p = 1/2 for a fair coin and p = 1/6
for a fair die) lies within the approximate95% confidence limits given
by the method of support. Rolling 2 out of 4 sixes with a fair die (as
in Section 8.1, Exercise 10).
Get solution
8.2.21 Check whether the expected value of p ( p = 1/2 for a fair coin and p = 1/6
for a fair die) lies within the approximate95% confidence limits given
by the method of support. Flipping 2 out of 12 heads with a fair coin
(as in Section 8.1, Exercise 11).
Get solution
8.2.22 Check whether the expected value of p ( p = 1/2 for a fair coin and p = 1/6
for a fair die) lies within the approximate95% confidence limits given
by the method of support. Rolling 2 out of 12 sixes with a fair die
(as in Section 8.1, Exercise 12).
Get solution
8.2.23 Check whether the given value of Λ lies
within the approximate 95% confidence limits given by the method of
support. Twenty events occur in 1 min with a given value of Λ = 10.0 (as in Section 8.1, Exercise 17).
Get solution
8.2.24 Check whether the given value of Λ lies
within the approximate 95% confidence limits given by the method of
support. Ten high energy cosmic rays hit detector over the course of 1
yr, with a given value of Λ= 8.0 (as in Section 8.1, Exercise 18).
Get solution
8.2.25
Use experimentation and Newton’s method to solve the equations for
the approximate confidence limits with the method of support. The
confidence limits if 20 out of 100 individuals are measured with a
particular allele (Example 8.2.13).
Get solution
8.2.26
Use experimentation and Newton’s method to solve the equations for
the approximate confidence limits with the method of support. The
confidence limits if 23 seeds are found in 1 ... (Example 8.2.14).
Get solution
8.2.27
prove that the support takes on its maximum where the likelihood
does. Using the likelihood function in Section 8.1, Exercise 9.
Get solution
8.2.28
prove that the support takes on its maximum where the likelihood
does. Using the likelihood function in Section 8.1, Exercise
17.Reference Section 8.1, Exercise 17 Find
the likelihood as a function of the Poisson parameter Λ, find the
maximum likelihood estimator, and evaluate the likelihood at the maximum
and at the other given value of Λ. Twenty events occur in 1 min.
Compare the likelihood with the maximum likelihood estimator of Λ with
the likelihood if Λ= 10.0.
Get solution
8.2.29
prove that the support takes on its maximum where the likelihood
does. Using the likelihood function in Section 8.1, Exercise 19.
Section 8.1, Exercise 19.Find
the likelihood as a function of the Poisson parameter Λ, find the
maximum likelihood estimator, and evaluate the likelihood at the maximum
and at the other given value of Λ. The number of events that
occur are counted for 3 min. Twenty events occur the first minute, 16
events occur the second minute, and 21 events occur the third minute.
Compare the likelihood with the maximum likelihood estimator of Λ with the likelihood if Λ= 20.0.
Get solution
8.2.30 prove that the support takes on its maximum where the likelihood does. For a general likelihood function.
Get solution
8.2.31 Consider a tiny data set where one out of two people is found with an allele. Find the exact 95% confidence limits.
Get solution
8.2.32 Consider a tiny data set where one out of two people is found with an allele. Find the exact 99% confidence limits.
Get solution
8.2.33
Consider a tiny data set where one out of two people is found with an
allele. How would you use the Monte Carlo method to estimate the 99%
confidence limits?
Get solution
8.2.34
Consider a tiny data set where one out of two people is found with an
allele. Use the method of support to estimate 95% confidence limits
and compare your results with Exercise 31.
Get solution
8.2.35 Consider a data set where three out of three people are found with an allele. Find the exact 95% confidence limits.
Get solution
8.2.36 Consider a data set where three out of three people are found with an allele. Find the exact 99% confidence limits.
Get solution
8.2.37 Consider a data set where three out of three people are found with an allele. Why is the upper confidence limit strange?
Get solution
8.2.38
Consider a data set where three out of three people are found with an
allele. Use the method of support to estimate 95% confidence limits
and compare your results with Exercise 35.
Get solution
8.2.39 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 first call arrives in 20 min. Find the maximum likelihood estimator of the rate.
Get solution
8.2.40 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 first call arrives in 20 min. Find the 95% confidence limits.
Get solution
8.2.41 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 first call arrives in 20 min. Find the 98% confidence limits.
Get solution
8.2.42
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 first call arrives in 20 min. How many
calls might he expect to miss if he went out for a 2-h hike?
Get solution
8.2.43
14 mutations are counted in one million base pairs. Write the
equations for the 95% confidence limits, and solve them numerically if
you have a computer.
Get solution
8.2.44
Recall the couple that has seven boys before having a girl (Section
8.1, Exercise 33). Find 95% confidence limits around the maximum
likelihood estimate of q. How do you interpret these results?
Get solution
8.2.45
Recall the couple that has seven boys before having a girl (Section
8.1, Exercise 33). Find 95% confidence limits around the maximum
likelihood estimate of q. How do you interpret these results?
Get solution
8.2.46
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?
Get solution
8.2.48 Consider
again the data in Section 8.1, Exercise 50. Use the method of support
to find approximate 95% confidence limits around your estimated value.
Do they include the true value x = 3.0? Reference Section 8.1, Exercise 50. Cells
are placed for 1 min in an environment where they are hit by X-rays,
some of which are damaging. Cells not hit by the damaging rays are
healthy, those hit exactly once are damaged, and those hit more than
once are dead. By measuring the states of a number of cells, we wish to
infer the rate at which cells are hit by damaging rays. Let x denote
the unknown parameter of the Poisson distribution. Use the formula for
the Poisson distribution to compute the probabilities ...of more than one hit in 1 min as functions of x.
a. Suppose the true value of x is 3.0. Plot the histogram. b.
Simulate 50 cells, and count how many you have of each type. (To keep
things interesting, keep sampling until you get at least one cell of
each type.)
c. Compare the results of your simulation with the idealized histogram.
d. Now pretend that x is unknown. We can use the method of maximum likelihood to analyze our data. Find the likelihood function L of these data (it is the product of the likelihoods for each of the 50 cells) as a function of the unknown parameter y, and let S be the natural log of L. Plot S(y) over a reasonable range.
e. Find the maximum of S and mark it on your graph.
f. Find the S(x) for x = ..., x = 2, x = 4, and the “truth” x = 3, and indicate each on your graph.
Get solution
8.2.48 Consider
again the data in Section 8.1, Exercise 50. Use the method of support
to find approximate 95% confidence limits around your estimated value.
Do they include the true value x = 3.0? Reference Section 8.1, Exercise 50. Cells
are placed for 1 min in an environment where they are hit by X-rays,
some of which are damaging. Cells not hit by the damaging rays are
healthy, those hit exactly once are damaged, and those hit more than
once are dead. By measuring the states of a number of cells, we wish to
infer the rate at which cells are hit by damaging rays. Let x denote
the unknown parameter of the Poisson distribution. Use the formula for
the Poisson distribution to compute the probabilities ...of more than one hit in 1 min as functions of x.
a. Suppose the true value of x is 3.0. Plot the histogram. b.
Simulate 50 cells, and count how many you have of each type. (To keep
things interesting, keep sampling until you get at least one cell of
each type.)
c. Compare the results of your simulation with the idealized histogram.
d. Now pretend that x is unknown. We can use the method of maximum likelihood to analyze our data. Find the likelihood function L of these data (it is the product of the likelihoods for each of the 50 cells) as a function of the unknown parameter y, and let S be the natural log of L. Plot S(y) over a reasonable range.
e. Find the maximum of S and mark it on your graph.
f. Find the S(x) for x = ..., x = 2, x = 4, and the “truth” x = 3, and indicate each on your graph.
Get solution
8.2.49 Use
the Monte Carlo method to estimate the 95% confidence limits in
Exercise 18. How close are your results to those obtained with the other
methods?
Get solution
