8.4.1 Explain in words what the following statements mean, and
restate the result in terms of type I or type II errors as appropriate.
The p-value associated with the null hypothesis is 0.2.
Get solution
8.4.2
Explain in words what the following statements mean, and restate the
result in terms of type I or type II errors as appropriate. Out of 1000
simulations of the null hypothesis, 70 produce a result as extreme as
or more extreme than the actual observation.
Get solution
8.4.3
Explain in words what the following statements mean, and restate the
result in terms of type I or type II errors as appropriate. The null
hypothesis is false, and a test (at significance level 0.05) rejects the
null hypothesis with probability 0.6.
Get solution
8.4.4
Explain in words what the following statements mean, and restate the
result in terms of type I or type II errors as appropriate. The null
hypothesis is false, and a test (at significance level 0.01) rejects the
null hypothesis with probability 0.2.
Get solution
8.4.5 Using
the following data, find the p-value associated with the null
hypothesis that the true probability of heads is 0.5. Is the result
significant? A coin is flipped 5 times and comes up heads every time (as in Section 8.2, Exercise 1).
Get solution
8.4.6 Using
the following data, find the p-value associated with the null
hypothesis that the true probability of heads is 0.5. Is the result
significant? A coin is flipped 7 times and comes up heads 6 out of 7 times (as in Section 8.2, Exercise 2).
Get solution
8.4.7 Using
the following data, find the p-value associated with the null
hypothesis that the true probability of heads is 0.5. Is the result
significant? 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).
Get solution
8.4.8 Using
the following data, find the p-value associated with the null
hypothesis that the true probability of heads is 0.5. Is the result
significant? 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).
Get solution
8.4.9
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), as in Exercise 7. What is the cutoff
number of heads above which you reject the null hypothesis at the α = 0.05 significance level?
Get solution
8.4.10
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), as in Exercise 7. What is the cutoff
number of heads above which you reject the null hypothesis at the α = 0.1 significance level?
Get solution
8.4.11 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), as in Exercise 7. Find the power of your test with α = 0.05 if the true probability of a head is 0.6.
Get solution
8.4.12 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), as in Exercise 7. Find the power of your test with α = 0.05 if the true probabilityof a head is 0.9.
Get solution
8.4.13
Find the p-value with both one-tailed and a two-tailed tests. 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 8.2, Exercise 6.)
Get solution
8.4.14
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 (as in Section 8.2, Exercise 5.)
Get solution
8.4.15 Find the p-value associated with the following hypotheses. 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.
Get solution
8.4.16 Find the p-value associated with the following hypotheses. 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.
Get solution
8.4.17 Find the p-value associated with the following hypotheses. 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 10.
Get solution
8.4.18 Find the p-value associated with the following hypotheses. 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 1000.
Get solution
8.4.19 Think about how thresholds are set in the following situations. 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 type II errors. What
factors would go into deciding where to set the threshold?
Get solution
8.4.20 Think about how thresholds are set in the following situations. 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 type I and type II
errors. What factors would go into deciding where to set the threshold?
Get solution
8.4.21
Consider a couple that has seven boys and one girl in a family of
eight. Test the hypothesis that boys and girls are equally likely.
Get solution
8.4.22
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.
Get solution
8.4.23 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 each day,
a. State null and alternative hypotheses.
b. Use the Poisson distribution to compute the probability of this event if the null hypothesis is true.
c. Compute the probability of an event at least this extreme if the null hypothesis is true.
d. Is this result significant? How would you interpret it? You receive 7 calls in 1 h on the first day.
Get solution
8.4.24 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 each day,
a. State null and alternative hypotheses.
b. Use the Poisson distribution to compute the probability of this event if the null hypothesis is true.
c. Compute the probability of an event at least this extreme if the null hypothesis is true.
d. Is this result significant? How would you interpret it? You receive 8 calls in 1 hr on the second day.
Get solution
8.4.25 Consider
the data in Exercise 23, where calls arrive at a rate of 3.5/h before
posting your phone number on your Web page, but 7 arrive in 1 h on the
next day. 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.
Get solution
8.4.26 Consider
the data in Exercise 23, where calls arrive at a rate of 3.5/h before
posting your phone number on your Web page, but 7 arrive in 1 h on the
next day. 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?
Get solution
8.4.27 The
survival time for one type of cell in culture is exponentially
distributed with a mean of 200 h. After applying a new treatment, one
cell lasts 800 h. For a second type of cell, the survival time is
exponentially distributed with a mean of 100 h. After applying the same
new treatment, one cell lasts 30 h. State null and alternative
hypotheses for the first type of cell. At what level can you reject the
null hypothesis? Is it significant?
Get solution
8.4.28 The
survival time for one type of cell in culture is exponentially
distributed with a mean of 200 h. After applying a new treatment, one
cell lasts 800 h. For a second type of cell, the survival time is
exponentially distributed with a mean of 100 h. After applying the same
new treatment, one cell lasts 30 h. State null and alternative
hypotheses for the second type of cell. At what level can you reject the
null hypothesis? Is it significant?
Get solution
8.4.29 The
survival time for one type of cell in culture is exponentially
distributed with a mean of 200 h. After applying a new treatment, one
cell lasts 800 h. For a second type of cell, the survival time is
exponentially distributed with a mean of 100 h. After applying the same
new treatment, one cell lasts 30 h. What is the shortest
survival over 200 h for which you might claim a significant result for
the first cell type (at the 0.05 level)?
Get solution
8.4.30 The
survival time for one type of cell in culture is exponentially
distributed with a mean of 200 h. After applying a new treatment, one
cell lasts 800 h. For a second type of cell, the survival time is
exponentially distributed with a mean of 100 h. After applying the same
new treatment, one cell lasts 30 h. What is the longest survival
under 30 h for which you might claim a significant result for the
second cell type (at the 0.05 level)?
Get solution
8.4.31 The
survival time for one type of cell in culture is exponentially
distributed with a mean of 200 h. After applying a new treatment, one
cell lasts 800 h. For a second type of cell, the survival time is
exponentially distributed with a mean of 100 h. After applying the same
new treatment, one cell lasts 30 h. 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 true mean is 1000?
Get solution
8.4.32 The
survival time for one type of cell in culture is exponentially
distributed with a mean of 200 h. After applying a new treatment, one
cell lasts 800 h. For a second type of cell, the survival time is
exponentially distributed with a mean of 100 h. After applying the same
new treatment, one cell lasts 30 h. 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 mean is 10?
Get solution
8.4.33 The
survival time for one type of cell in culture is exponentially
distributed with a mean of 200 h. After applying a new treatment, one
cell lasts 800 h. For a second type of cell, the survival time is
exponentially distributed with a mean of 100 h. After applying the same
new treatment, one cell lasts 30 h. 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.
Get solution
8.4.34 The
survival time for one type of cell in culture is exponentially
distributed with a mean of 200 h. After applying a new treatment, one
cell lasts 800 h. For a second type of cell, the survival time is
exponentially distributed with a mean of 100 h. After applying the same
new treatment, one cell lasts 30 h. 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.
Get solution
8.4.35 Consider measuring n plants
with a known standard deviation of 3.2 cm. How many plants would have
to be measured to achieve the following? 95% confidence limits where
the upper and lower confidence limits are 1.0 cm from the sample mean.
Get solution
8.4.36 Consider measuring n plants
with a known standard deviation of 3.2 cm. How many plants would have
to be measured to achieve the following? 99% confidence limits where
the upper and lower confidence limits are 1.0 cm from the sample mean.
Get solution
8.4.37 Consider measuring n plants
with a known standard deviation of 3.2 cm. How many plants would have
to be measured to achieve the following? 95% confidence limits where
the upper and lower confidence limits are 0.25 cm from the sample mean.
Get solution
8.4.38 Consider measuring n plants
with a known standard deviation of 3.2 cm. How many plants would have
to be measured to achieve the following? 99% confidence limits where
the upper and lower confidence limits are 0.25 cm from the sample mean.
Get solution
8.4.39 Repeat the calculations in Table 8.3. Is the hypothesis q = 0.13 rejected? Try with different numbers of simulations and compare with the theoretical p-values. Table 8.3 ...
Get solution
8.4.40 Generate nine independent random numbers from a normal distribution with mean 10 and variance of 9 and find their average ...If your value of ... is
within 0.5 of 10.0, try again to guarantee nice pictures and
interesting results (real scientists are not allowed to do this sort of
thing, of course). The average of your nine measurements comes from a
normal distribution with mean and some standard error. Find the standard
error and the 95% confidence interval around ... , calling the lower limit a...nd the upper limit ... .
Simulate the following four experiments 100 times: (1) Average nine
values from a normal distribution with true mean 10. (2) Average nine
values from a normal distribution with true mean ... . (3) Average nine values from a normal distribution with true mean ... . (4) Average nine values from a normal distribution with true mean ... . Count how many values in each experiment lie below ..., between ... and 10, between 10 and ..., and above ... .Define functions ...to
be the p.d.f.’s describing the distribution of elements in the four
experiments. Print a graph of each function, and list below it the
number of elements from the appropriate experiment lying in the various
intervals. Indicate which ones have values predicted by the theory of
confidence intervals and what those values should be. Are you bothered
by the fact that more than 5% of the elements of experiment 1 lie
outside your confidence interval? Why or why not?
Get solution
