Approximately eight miles north of Salisbury,Wiltshire, England, stands a
large circular stone monument surrounded by an earthwork. This prehistoric
structure is known throughout the world as Stonehenge. Its name is
derived from the old English word hengen, referring to something hung
up. In the case of the monument, this name refers to the large horizontal
lintel stones. The monument consists of an outer ring of Sarsen
stones, surrounding two inner circles of bluestones. The first and
third circles are adorned with the familiar stone lintels. The Sarsen
stones show signs of having been carefully shaped, suggesting a Mycenaean
Greece and Minoan Crete influence. The entire structure is surrounded
by a ditch and bank. Just inside the bank are 56 pits, named the
Aubrey Holes, after their discoverer.These holes appear to have been filled
shortly after their excavation.
Stonehenge is a mysterious, magnificent monument, which popularly
has been associated with the Druids. However, there is no direct evidence
supporting this connection. Recently, it has been discovered that a number
of the stone alignments are associated with important solar and lunar risings
and settings, suggesting that the site served as some sort of massive astronomical
calendar. If this conclusion is accurate, then it seems likely that
the monument might have been used as a temple for sky worshipers, although
the exact nature of their religion is unknown.
Corinn Dillion is interested in dating the construction of the structure.
Excavations at the site uncovered a number of unshed antlers, antler tines,
and animal bones. Carbon-14 dating methods were used to estimate the
ages of the Stonehenge artifacts. Carbon-14 is one of three carbon isotopes
found in the Earth’s atmosphere. Carbon-12 makes up 99% of all the carbon
dioxide in the air. Virtually all of the remaining one percent is composed
of Carbon-13. By far, the rarest form of carbon isotope found in the
atmosphere is Carbon-14. The ratio of Carbon-14 to Carbon-12 remains
constant in living organisms. However, once the organism dies, the amount
of Carbon-14 in the remains of the organism begins to decline, because it is
radioactive, with half-life 5730 years (the “Cambridge half-life”). So, the
decay of Carbon-14 into ordinary nitrogen makes possible a reliable estimate
about the time of death of the organism. The counted Carbon-14
decay events are known to be normally distributed.
Dillion’s team used two different Carbon-14 dating methods to arrive
at age estimates for the numerous Stonehenge artifacts.The Liquid Scintillation
Counting (LSC) method utilizes benzene, acetylene, ethanol,
methanol, or a similar chemical. Unlike the LSC method, the Accelerator
Mass Spectrometry (AMS) technique offers direct Carbon-14 isotope
counting.The AMS method’s greatest advantage is that it requires only milligram-
sized samples for testing.The AMS method was used only on recovered
artifacts that were of extremely small size.
Stonehenge’s main ditch was dug in a series of segments. Excavations at
the base of the ditch uncovered a number of antlers, which bore signs of heavy
use.These antlers could have been used by the builders as picks or rakes.The
fact that no primary silt was discovered beneath the antlers suggests that they
were buried in the ditch shortly after its completion. Another researcher,
Phillip Corbin, using an archaeological markings approach, had previously
claimed that the mean date for the construction of the ditch was 2950 B.C. A
sample of nine age estimates from unshed antlers excavated from the ditch CASE STUDY
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produced a mean of 3033.1 B.C., with standard deviation 66.9 years. Assume
that the ages are normally distributed with no obvious outliers. At an
significance level, is there any reason to dispute Corbin’s claim?
Four animal bone samples were discovered in the ditch terminals.
These bones bore signs of attempts at artificial preservation and might have
been in use for a substantial period of time prior to their being placed at
Stonehenge.When dated, these bones had mean age 3187.5 B.C. and standard
deviation 67.4 years. Assume that the ages are normally distributed
with no obvious outliers. Use an significance level to test the claim
that the population mean age of the site is different from 2950 B.C.
In the center of the monument are two concentric circles of igneous
rock pillars, called bluestones. The construction of these circles was never
completed. These circles are known as the Bluestone Circle and the Bluestone
Horseshoe. The stones in these two formations were transported to
the site from the Prescelly Mountains in Pembrokeshire, southwest Wales.
Excavation at the center of the monument revealed an antler, an antler tine,
and an animal bone. Each of these artifacts was submitted for dating. It was
determined that this sample of three artifacts had mean age 2193.3 B.C, with
a standard deviation of 104.1 years. Assume that the ages are normally distributed
with no obvious outliers. Use an significance level to test
the claim that the population mean age of the Bluestone formations is different
from Corbin’s declared mean age of the ditch, that is, 2950 B.C.
Finally, three additional antler samples were uncovered at the Y and Z
holes.These holes are part of a formation of concentric circles 11 meters and
3.7 meters, respectively, outside of the Sarsen Circle.The sample mean age of
these antlers is 1671.7 B.C. with a standard deviation of 99.7 years.Assume that
the ages are normally distributed with no obvious outliers. Use an
significance level to test whether the population mean age of the Y and Z
holes is different from Corbin’s stated mean age of the ditch—that is, 2950 B.C.
From your analysis, does it appear that the mean ages of the artifacts
for the ditch, the ditch terminals, the Bluestones, and the Y and Z holes
dated by Dillion are consistent with Corbin’s claimed mean age of 2950 B.C.
for the construction of the ditch? Can you use the results from your hypothesis
tests to infer the likely construction order of the various Stonehenge
Using Dillion’s data, construct a 95% confidence interval for the population
mean ages of the various sites. Do these confidence intervals support
Corbin’s claim? Can you use these confidence intervals to infer the likely
construction order of the various Stonehenge structures? Explain.
Which statistical technique, hypothesis testing or confidence intervals,
is more useful in assessing the age and likely construction order of the
Stonehenge structures? Explain.
Discuss the limitations and assumptions of your analysis. Is there any
additional information that you would like to have before publishing your
findings? Would another statistical procedure be more useful in analyzing
these data? If so, which one? Explain. Write a report to Corinn Dillion detailing
Source: This fictional account is based upon information obtained from Archaeometry and
Stonehenge (http://www.eng_h.gov.uk/stoneh). The means and standard deviations used
throughout this case study were constructed by calculating the statistics from the midpoint of
the calibrated date range supplied for each artifact.
a = 0.05
a = 0.05
a = 0.05
a = 0.05
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What Does It Really Weigh?
Many of the consumer products that we purchase have labels that describe
the net weight of the contents. For example, the net weight of a candy bar
might be listed as 4 ounces. Choose any consumer product that reports the
net weight of the contents on the packaging.
(a) Obtain a random sample of size 8 or more of the consumer product.We
will treat the random purchases as a simple random sample.Weigh the
(b) If your sample size is less than 30, verify that the population from which
the sampling was drawn is normal and that the sample does not contain
(c) As the consumer, you are concerned only with situations in which you
are getting “ripped off.” Determine the null and alternative hypotheses
from the point of view of the consumer.
(d) Test the claim that the consumer is getting “ripped off” at the
level of significance. Are you getting “ripped off”? What makes you say
(e) Suppose you are the quality-control manager. How would you structure
the alternative hypothesis? Test this claim at the level of
significance. Is there anything wrong with the manufacturing process?
What makes you say so?
a = 0.05
a = 0.05
SullStatCH09_Fpp516-591 11/20/02 11:22 AM Page 590
Eyeglasses are part medical device and part fashion
statement, a marriage that has always made them a
tough buy. Aside from the thousands of different
frames the consumer has to choose from, there are
various lens materials and coatings that can add to the
durability, and the cost, of a pair of eyeglasses. One
manufacturer even goes so far as to claim that its lenses
are “the most scratch-resistant plastic lenses ever
made.”With a claim like that, we had to test the lenses
One test involved tumbling the lenses in a drum
containing scrub pads of grit of varying size and hardness.
Afterward, readings of the lenses’ haze were
taken on a spectrometer to determine how scratched
they had become. To evaluate their scratch resistance,
we measured the difference between the haze reading
before and after tumbling.
The graphic illustrates the difference between an
uncoated lens (on the left) and the manufacturer’s
“scratch-resistant” lens (on the right).
The following table contains the haze measurements
both before and after the scratch resistance test
for this manufacturer. Haze difference is measured by
subtracting the before score from the after score; in
other words haze difference is computed as
Before After Difference
0.18 0.72 0.54
0.16 0.85 0.69
0.20 0.71 0.51
0.17 0.42 0.25
0.21 0.76 0.55
0.21 0.51 0.30
(a) Suppose it is known that the closest competitor to
the manufacturer’s lens has a mean haze difference
of 1.0. Do the data support the manufacturer’s
scratch resistance claim?
(b) Write the null and alternative hypotheses, letting
represent the mean haze difference for the
(c) We used Minitab (release 13.1) to perform a onesample
t-test.The results are shown below.
Using the Minitab output, answer the following
1. What is the value of the test statistic?
2. What is the P-value of the test?
3. What is the conclusion of this test? Write a paragraph
for the readers of Consumer Reports magazine
that explains your findings.
Note to Readers: In many cases, our test protocol and analytical
methods are more complicated than described in these examples.
The data and discussions have been modified to make the material
more appropriate for the audience.
© 2001 by Consumers Union of U.S., Inc.,Yonkers, NY 10703-1057, a nonprofit organization. Reprinted with permission from the
June, 2001 issue of CONSUMER REPORTS® for educational purposes only. No commercial use or photocopying permitted.To
learn more about Consumers Union, log onto www.ConsumerReports.org .
One-Sample T: Difference
Test of mu 1 vs mu 1
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- Step 1. Set up hypotheses and determine level of significance
Null hypothesis: H0: μ = 2950 B.C
Alternative hypothesis: H1: μ > 2950 B.C &a