Download 1. Let = number of locations at which the computed confidence

1. Let = number of locations at which the computed confidence interval for that location hits
the true value of the mean yield at its location.
has a binomial(7,0.95)
(a)
P(every one of the seven intervals covers the true mean yield at its location) =
3.
The overall confidence level for the seven simultaneous statements is only 68.9%.
(b)
.
2. (a)
(b)
3. Let
. Now,
if and only if
has a
under the alternative value
297.9853.
(a)
Let
.
.
.
(b)
Let
.
has a
under the alternative value
.
.
(c) The power to detect the alternative mean value 295 is greater than for the alternative mean
value 299 since it is easier to detect a value of µ which is further away from the null mean value
300.
4. (a) The boxplot suggests that the distribution of the data values is fairly symmetric. No
outliers are identified by the plot. The value 110 litres/minute for the mean maximum voluntary
ventilation value considered in part (c) is close to the center of the distribution of the data values
as measured by the median.
(b) Let µ be the mean maximum voluntary ventilation value for this population of healthy
college seniors.
111.6 litres/minute.
litres/minute.
litres/minute.
.
For 95% confidence,
.
A 95% confidence interval for µ is given by
.
We are 95% confident that the mean maximum voluntary ventilation value for this population of
healthy college seniors is between 85.25 litres/minute and 137.95 litres/minute.
(c)
litres/minute
litres/minute.
Since the value 110 is inside the 95% confidence interval in (b), these data are not statistically
significant at the 5% level. At the 5% level, one would accept . At the 5% level, these data are
consistent with the mean maximum voluntary ventilation value being 110 litres/minute.
(d) The analysis assumes the observations are independent and normally distributed. Since our
sample is a simple random sample, the observations are independent if the population size is
much larger than the sample size. Since there are no outliers and
, even if the
distribution of the maximum voluntary ventilation values is even somewhat skewed in the
population, and hence not normally distributed, our confidence level will still be reasonably
accurate. In fact, the normal quantile-quantile plot (or equivalently, normal probability plot) does
not suggest that there is a significant problem with the normal assumption that would affect the
.
analysis. It could easily arise even with independent and normally distributed data with
5. Let µ be the mean diameter (in mm) of a skin test reaction for the population of adolescents
who might participate in an immunologic study of this kind.
mm.
mm.
mm.
.
(a)
mm
Since
mm.
we reject
in favour of
at the 1% level.
These data provide evidence at the 1% level that the mean diameter of the skin test reaction to
the antigen is less than 30 mm.
(b)
for 98% confidence.
mm
A 98% confidence interval for µ is given by:
.
We are 98% confident that the mean diameter of the skin test reaction for this population of
adolescents is between 17 mm and 25 mm.
(c) The key assumption here for the analysis to be valid is that the diameters from different
subjects be independent. If different subjects respond independently to the skin test, this would
be the case. Since
, the normal assumption is not necessary and the stated confidence
level and significance levels will be reasonably accurate even if the distribution of diameter of
the skin test reaction is strongly skewed in the population.
6. It was expected that the blood stream concentration of a particular antibiotic one hour after its
administration could vary substantially from individual to individual. The study was therefore
designed as a matched pairs study where the same individual was exposed to both antibiotics.
This allows a comparison between the two antibiotics while holding subject constant.
Person
1
2
3
4
5
6
Penicillin
42
34
57
40
28
48
Amoxicillin Difference (d=Pen-Amox)
36
6
44
-10
61
-4
35
5
35
-7
50
-2
Let
be the mean difference in blood stream concentration between penicillin and amoxicillin
one hour after administration.
g/ml
g/ml
g/ml.
(a)
Large values of
provide evidence against
in the direction of
.
Using table C.3,
. These data are consistent with the mean blood stream
concentrations one hour after administration being the same for the two antibiotics. In particular,
these data are not even statistically significant at the 20% level.
(b)
for 95% confidence.
A 95% confidence interval for
is given by:
.
With 95% confidence, the mean blood stream concentration one hour after administration of
pencillicin could be as much as 8.74 µg/ml less than amoxicillin or as much as 4.74 µg/ml more
than amoxicillin according to our data.
(c) We assume that the differences are independent. This will be the case if the subjects respond
independently. Independence of the two responses within subject is not necessary for this
analysis. We also assume that the differences are normally distributed. It is important that this be
based on past experience since these data alone possess very little information regarding this
issue. Moreover, since is very small, we cannot rely on the robustness properties of the onesample -procedures associated with larger sample sizes. The normal quantile-quantile plot (or
equivalently, normal probability plot) does not indicate a departure from normality.
7. Let
the mean protoporphyrin level for the population of adult male alcoholics with ring
sideroblasts in the bone marrow that were sampled.
Let
the standard deviation of the protoporphyrin level for the population of adult male
alcoholics with ring sideroblasts in the bone marrow.
Let
the mean protoporphyrin level for the population of apparently healthy nonalchoholic
adult males that were sampled.
Let
the standard deviation of the protoporphyrin level for the population of apparently
healthy nonalchoholic adult males that were sampled.
The ratio of the largest sample variance to the smallest sample variance
which is
much greater than a typical cut-off value of 3 (or the more conservative cut-off value 2) often
used in practice to rule out the safe application of pooled procedures that assume equal
population standard deviations when sample sizes are not nearly equal as is the case here.
If the protoporphyrin level in each population is normally distributed (not indicated here!), the
two critical values for a test of
:
against
:
with
are
and
.
Since
we would as expected reject . This test is neither necessary nor recommended
in this situation. It is highly non-robust to departures from normality.
(a)
Satterthwaite’s degrees of freedom are:
Large values of provide evidence against
in the direction of
.
Using Table C.3,
. These data provide extremely strong evidence that the
mean protoporphyrin level is higher in the represented alcoholic population than in the
nonalcoholic population. In particular, these data are statistically significant at the 0.5% level.
(b) To get the critical value
for 99% confidence using table C.3 we can round
down to 50.
The 99% confidence interval for
is given by:
390).
With 99% confidence, the mean protoporphyrin level is between 200 and 390 units greater in the
represented alcoholic population than in the nonalcoholic population.
Using a computer, the value of
in this case.
without rounding is
which yields the same interval
8. (a) This is a completely randomized design.
(b) The distribution of the data values for the sensory deprivation treatment group is shifted
towards lower values and has greater spread than that of the control group. In particular, at least
75% of the data values in the control group are above the third quartile of the sensory deprivation
treatment group. There is a single outlier that was identified for the control group.
(c)
Let
the mean alpha-wave frequency for a randomly selected individual when exposed to
the sensory deprivation treatment.
Let
the mean alpha-wave frequency for a randomly selected individual when treated as a
control subject.
= 0.284
Large values of
0.53
provide evidence against
Using table C.3,
in the direction of
.
.
Hence, these data provide very strong evidence that the mean alpha-wave frequency under the
deprived sensory treatment is different from that of the control treatment. In particular, these data
are statistically significant at the 0.5% level.
(d)
for 95% confidence.
The 95% confidence interval for
is given by:
.
We are 95% confident that the mean alpha-wave frequency under the deprived sensory treatment
is between 0.30 and 1.30 units lower than under the control treatment.
(e) It seems reasonable to expect that the two samples are independent. Provided exposure to a
treatment for a subject occurred independently of other subjects subjected to the same treatment,
responses within a specific treatment should also be independent.
The normal quantile-quantile (or equivalently normal probability plot) for the deprived sensory
treatment group does not indicate a departure from normality. There is a value in the control
group that is somewhat outlying. With any outlier it is important to consider its potential cause.
Is it a recording error or an experimental error? Is it indicative of a different mechanism that is at
play? How one proceeds depends on the reason for an outlier. Here, it is probably chance
variation associated with the underlying distribution among subjects subjected to the control
treatment. Even if this is not exactly normal for this group, since the sample sizes are equal, and
, we can expect our pooled procedures to be quite robust to
moreover ,
departures from equal variances, and moreover, normality in the absence of strong skewness.
Our plots do not indicate departures from the assumptions that would affect the validity of the
analysis.
Normal quantile-quantile plot for the sensory deprivation group:
Normal quantile-quantile plot for the control group: