Download Chapter 10 Review Name: Period

Chapter 10 Review
Name: _____________________________
Period: ______________
1. The pesticide diazinon is in common use to treat infestations of the German cockroach, Blattella germanica. A
study investigated the persistence of this pesticide on various types of surfaces. Researchers applied a 0.5%
emulsion of diazinon to glass and plasterboard. After 14 days, they randomly assigned 72 cockroaches to two
groups of 36, placed one group on each surface, and recorded the number that died within 48 hours. On glass,
18 cockroaches died, while on plasterboard, 25 died.
(a) Construct and interpret a 90% confidence interval for the difference in the proportion of cockroaches that
die on each surface.
(b) Based only on this interval, do you think that the difference in proportion of cockroaches that died on each
surface is significant? Justify your answer.
2. A study of iron deficiency among infants compared blood hemoglobin levels of a random sample of
one-year-old infants who had been breast-fed to a random sample of one-year old infants who had been fed with
standard infant formula. Here are the results.
n
s
Breast-fed infants
23
13.3
1.7
Formula-fed infants
19
12.4
1.8
We wish to test the hypothesis
against
, where and are the population mean
blood hemoglobin levels for breast-fed and formula-fed infants, respectively.
(a) What additional information would you need to confirm that the conditions for this test have been met?
(b) Assuming the conditions have been met, run the significance test with a = .01.
(c) Given your conclusion in part B, which type of error, Type I or Type II, is it possible to make? Describe
that error in the context of this study.
Daphnia pulicaria is a water flea—a small crustacean that lives in lakes and is a major food supply for
many species of fish. When fish are present in the lake water, they release chemicals called kairomones that
induce water fleas to grow long tail spines that make them more difficult for the fish to eat. One study of this
phenomenon compared the relative length of tail spines in Daphnia pulicaria when kairomones were present to
when they were not. Below are data on the relative tail spine lengths, measured as a percentage of the entire
length of the water flea.
3.
Fish kairomone present
Fish kairomone absent
Relative tail spine length
n
x̅
s
214
37.26
4.68
152
30.67
4.19
(a) Do the data provide convincing evidence that the mean relative tail spine length of Daphnia is longer in the
presence of fish kairomones?
Assume the conditions for inference have been met.
(b) What additional information would you need to confirm that the conditions for this test have been met?
4. An association of Christmas tree growers in Indiana sponsored a sample survey of 500 randomly
selected Indiana households to help improve the marketing of Christmas trees. One question the researchers
asked was, “Did you have a Christmas tree this year?” Respondents who had a tree during the holiday season
were asked whether the tree was natural or artificial. Respondents were also asked if they lived in an urban
area or in a rural area. The tree growers want to know if there is a difference in preference for natural trees
versus artificial trees between urban and rural households. Among the 160 who lived in rural areas, 64 had a
natural tree. Among the 261 who lived in an urban area, 89 had a natural tree.
(a) Construct and interpret a 95% confidence interval for the difference in the proportion of rural and urban
Indiana residents who had a natural Christmas tree this year.
(b) Suppose the tree growers asked 10 different questions about Christmas tree preferences in an effort to find
differences between urban and rural residents. Explain why it would be unwise to perform 10 different
2-sample significance tests (a = 0.05) on their data and conclude that any significant result establishes a
difference between the two groups.
Chapter 10 Review
Answer Section
OTHER
1. ANS:
A.
State: We wish to estimate, with 90% confidence, the difference
, where
and
are, respectively, the proportion of all cockroaches similar to those in the study that die on
plasterboard and glass. Plan: We should use a 2-sample z-interval for
. Conditions:
Random: Cockroaches were randomly assigned to one of the two treatment groups, plasterboard or
glass. 10%: Since no sampling took place, the 10% restriction does not apply. Normal/Large
Sample: Number of successes and failures in the two groups are 25, 11, 18, and 18, all of which are
at least 10. Do: The critical z for 90% confidence is 1.645,
The interval is ,
and
.
or
.
Conclude: We are 90% confident that the interval from 0.008 to 0.380 captures the true difference in
the proportion of cockroaches that die on plasterboard versus the proportion that die on glass. B.
Since 0 is not within the 90% confidence interval of plausible values for the difference in proportion
of cockroach deaths in the two groups, this interval provides evidence that there is a significant
difference. A two-tailed test of significance would reject the null hypothesis that
at the a
= 0.10 level.
PTS: 1
2. ANS:
A. The subjects were randomly assigned to the two treatment groups, so independence and random
conditions have been met.
We should be concerned about the somewhat small sample sizes: we
need to know that there were no outliers in the samples and we should plot the data to determine
whether there is strong skew present.
B.
Using Table B and conservative df = 18, the two tailed
P-value is between 0.1 and 0.2. Using the calculator and df = 37.60, P-value = 0.1065.
C. If there is no difference between the mean blood hemoglobin levels of breast-fed and
formula-fed infants, the probability of obtaining a sample mean difference this far or farther from 0
is 0.1065. We cannot reject H0. We do not have convincing evidence that there is a difference in
the mean blood hemoglobin levels of the infants in the two groups. D. It’s possible we made a
Type II error, which would be failing to conclude that there is a difference in mean blood
hemoglobin levels between the two groups when there is a difference.
PTS: 1
3. ANS:
A. State: We are testing the hypotheses
, where
is the mean
length of Daphnia tail spines when fish kairomones are present and
is the mean length of
Daphnia tail spines when fish kairomones are absent. We will use a significance level of a = 0.05.
Plan: The procedure is a two-sample t-test for the difference of means. The problem states that the
conditions for this procedure have been met. Do:
. This very
large t-statistic yields an extremely small P-value (by calculator: P-value ).
Conclude: Since the
P-value is much less than any conventional value for a, we reject H0: we have strong evidence that
the mean length of Daphnia pulicaria tail spines in the presence of fish kairomones is longer than
when the kairomones are absent.
B. We need to know more about the design of the experiment. If this was a controlled
experiment, were individual water fleas randomly assigned to the two treatments? If water fleas were
merely sampled from environments that did or did not have fish kairomones, can we view the
samples as independent SRSs from the two populations? The sample sizes are very large, so we are
not concerned about the shapes of the population distributions
PTS: 1
4. ANS:
State: We wish to estimate, with 95% confidence, the difference
, where
and
are,
respectively, the proportion of rural and urban residents who had a natural Christmas tree this year.
Plan: We should use a 2-sample z-interval for
. Conditions: Random: The study used 500
randomly-selected Indiana households. We can view this as SRSs from rural and urban
populations. 10%: There are certainly more than 10x500=5000 households in Indiana. Normal:
Number of successes and failures in the two groups are 64, 96, 89, and 172, all of which are at least
10.
Do: The critical z for 95% confidence is 1.96 and
interval is given by
.
, or
The
.
Conclude: We are 95% confident that the interval from –0.035 to 0.155 captures the true difference
in the proportion of rural versus urban residents of Indiana who had a natural Christmas tree this
year. B. At the _ = 0.05 level, we expect to make a Type I error and reject a true H0 about once in
every 20 tests we perform. If we perform 10 such tests, the chances are quite good that we will
make at least one Type I error. (In fact, the probability is
).
PTS: 1