A transect is an archaeological study area
that is 1/5 mile wide and 1 mile long. A site in
a transect is the location of a significant archaeological find.
Let x represent the number of sites per
transect. In a section of Chaco Canyon, a large number of transects
showed that x has a population
variance σ2 = 42.3. In a different
section of Chaco Canyon, a random sample of 28 transects
gave a sample
variance s2 = 46.1 for the
number of sites per transect. Use a 5% level of significance to
test the claim that the variance in the new section is greater than
42.3. Find a 95% confidence interval for the population
variance.
(a) What is the level of significance?
State the null and alternate hypotheses.
Ho: σ2 = 42.3; H1: σ2 > 42.3
Ho: σ2 = 42.3; H1: σ2 < 42.3
Ho: σ2 = 42.3; H1: σ2 ≠ 42.3
Ho: σ2 > 42.3; H1: σ2 = 42.3
(b) Find the value of the chi-square statistic for the sample.
(Round your answer to two decimal places.)
What are the degrees of freedom?
What assumptions are you making about the original
distribution?
We assume a uniform population distribution.
We assume a exponential population distribution.
We assume a binomial population distribution.
We assume a normal population distribution.
(c) Find or estimate the P-value of the sample test
statistic.
P-value > 0.100
0.050 < P-value < 0.100
0.025 < P-value < 0.050
0.010 < P-value < 0.025
0.005 < P-value < 0.010
P-value < 0.005
(d) Based on your answers in parts (a) to (c), will you reject or
fail to reject the null hypothesis?
Since the P-value > α, we fail to
reject the null hypothesis.
Since the P-value > α, we reject
the null hypothesis.
Since the P-value ≤ α, we
reject the null hypothesis.
Since the P-value ≤ α, we fail
to reject the null hypothesis.
(e) Interpret your conclusion in the context of the
application.
At the 5% level of significance, there is insufficient evidence
to conclude that the variance is greater in the new
section.
At the 5% level of significance, there is sufficient evidence to
conclude that the variance is greater in the new
section.
(f) Find the requested confidence interval for the population
variance. (Round your answers to two decimal places.)
lower limit
upper limit
Interpret the results in the context of the application.
We are 95% confident that σ2 lies
outside this interval.
We are 95% confident
that σ2 lies below this
interval.
We are 95% confident
that σ2 lies above this interval.
We are 95% confident that σ2 lies within
this interval.
2. Let x represent the average annual
salary of college and university professors (in thousands of
dollars) in the United States. For all colleges and universities in
the United States, the population variance
of x is
approximately σ2 = 47.1. However, a
random sample of 16 colleges and universities in Kansas
showed that x has a sample
variance s2 = 86.8. Use a 5%
level of significance to test the claim that the variance for
colleges and universities in Kansas is greater than 47.1. Find a
95% confidence interval for the population variance.
(a) What is the level of significance?
(b) Find the value of the chi-square statistic for the sample.
(Round your answer to two decimal places.)
What are the degrees of freedom?
(f) Find the requested confidence interval for the population
variance. (Round your answers to two decimal places.)
lower limit
upper limit
3. A new kind of typhoid shot is being developed by a
medical research team. The old typhoid shot was known to protect
the population for a mean time of 36 months, with a standard
deviation of 3 months. To test the time variability of the new
shot, a random sample of 29 people were given the new
shot. Regular blood tests showed that the sample standard deviation
of protection times was 1.9 months. Using a 0.05 level of
significance, test the claim that the new typhoid shot has a
smaller variance of protection times.
(a) What is the level of significance?
(b) Find the value of the chi-square statistic for the sample.
(Round your answer to two decimal places.)
What are the degrees of freedom?
(f) Find a 90% confidence interval for the population standard
deviation. (Round your answers to two decimal places.)
lower limit
upper limit
4. Jim Mead is a veterinarian who visits a Vermont farm to
examine prize bulls. In order to examine a bull, Jim first gives
the animal a tranquilizer shot. The effect of the shot is supposed
to last an average of 65 minutes, and it usually does. However, Jim
sometimes gets chased out of the pasture by a bull that recovers
too soon, and other times he becomes worried about prize bulls that
take too long to recover. By reading journals, Jim has found that
the tranquilizer should have a mean duration time of 65 minutes,
with a standard deviation of 15 minutes. A random sample
of 8 of Jim's bulls had a mean tranquilized duration time
of close to 65 minutes but a standard deviation
of 25 minutes. At the 1% level of significance, is Jim
justified in the claim that the variance is larger than that stated
in his journal? Find a 95% confidence interval for the population
standard deviation.
(a) What is the level of significance?
(b) Find the value of the chi-square statistic for the sample.
(Round your answer to two decimal places.)
What are the degrees of freedom?
(f) Find the requested confidence interval for the population
standard deviation. (Round your answers to two decimal place.)
lower limit
upper limit
5. Rothamsted Experimental Station (England) has studied
wheat production since 1852. Each year, many small plots of equal
size but different soil/fertilizer conditions are planted with
wheat. At the end of the growing season, the yield (in pounds) of
the wheat on the plot is measured. For a random sample of years,
one plot gave the following annual wheat production (in
pounds).
3.72
4.08
3.84
4.05
3.72
3.79
4.09
4.42
3.89
3.87
4.12
3.09
4.86
2.90
5.01
3.39
Use a calculator to verify that, for this plot, the sample
variance is s2 ≈ 0.303.
Another random sample of years for a second plot gave the following
annual wheat production (in pounds).
3.61
3.79
3.49
4.09
3.73
3.72
4.13
4.01
3.59
4.29
3.78
3.19
3.84
3.91
3.66
4.35
Use a calculator to verify that the sample variance for this
plot is s2 ≈ 0.091.
Test the claim that the population variance of annual wheat
production for the first plot is larger than that for the second
plot. Use a 1% level of significance.
(a) What is the level of significance?
(b) Find the value of the sample F statistic.
(Use 2 decimal places.)
What are the degrees of freedom?
dfN
dfD
6. You don't need to be rich to buy a few shares in a mutual
fund. The question is, how reliable are mutual
funds as investments? This depends on the type of fund you buy. The
following data are based on information taken from a mutual fund
guide available in most libraries.
A random sample of percentage annual returns for mutual funds
holding stocks in aggressive-growth small companies is shown
below.
-1.1
14.1
41.5
17.4
-16.6
4.4
32.6
-7.3
16.2
2.8
34.3
-10.6
8.4
-7.0
-2.3
-18.5
25.0
-9.8
-7.8
-24.6
22.8
Use a calculator to verify
that s2 ≈ 347.567 for the
sample of aggressive-growth small company funds.
Another random sample of percentage annual returns for mutual funds
holding value (i.e., market underpriced) stocks in large companies
is shown below.
16.7
0.6
7.7
-1.2
-3.1
19.4
-2.5
15.9
32.6
22.1
3.4
-0.5
-8.3
25.8
-4.1
14.6
6.5
18.0
21.0
0.2
-1.6
Use a calculator to verify
that s2 ≈ 136.207 for value
stocks in large companies.
Test the claim that the population variance for mutual funds
holding aggressive-growth in small companies is larger than the
population variance for mutual funds holding value stocks in large
companies. Use a 5% level of significance. How could your test
conclusion relate to the question
of reliability of returns for each type of
mutual fund?
(a) What is the level of significance?
(b) Find the value of the sample F statistic.
(Use 2 decimal places.)
What are the degrees of freedom?
dfN
dfD
7. A new thermostat has been engineered for the frozen food
cases in large supermarkets. Both the old and new thermostats hold
temperatures at an average of 25°F. However, it is hoped that the
new thermostat might be more dependable in the
sense that it will hold temperatures closer to 25°F. One frozen
food case was equipped with the new thermostat, and a random sample
of 21 temperature readings gave a sample variance
of 5.3. Another similar frozen food case was equipped with the
old thermostat, and a random sample of 19 temperature
readings gave a sample variance of 12.7. Test the claim that
the population variance of the old thermostat temperature readings
is larger than that for the new thermostat. Use a 5% level of
significance. How could your test conclusion relate to the question
regarding the dependability of the temperature
readings? (Let population 1 refer to data from the old
thermostat.)
(a) What is the level of significance?
(b) Find the value of the sample F statistic.
(Round your answer to two decimal places.)
What are the degrees of freedom?
dfN
dfD