Statistics 2MA3 - Assignment #3
2001-03-26
Due 2001-04-06 17:00
Do these exercises by hand with a pocket calculator (where
feasible), then check your work with R..
- Use the Hospital Stay data in Table 2.11 on p. 39 of Rosner.
(HOSPITAL.DAT, HOSPITAL.DOC on the data disk.) Plot a histogram of
first temperature after admission. Compute the mean temperature.
Assuming that the standard deviation of temperature is 1 degree,
find a 2-sided 95% confidence interval for the mean temperature
and find a p-value to test the hypothesis that the mean
temperature is 98.6 `F against the two-sided alternative. Test the
hypothesis that the standard deviation is 1 degree, with a 2-sided
5% test. Without assuming a value for standard deviation, find a
2-sided 95% confidence interval for the mean temperature and find
a p -value to test the hypothesis that the mean temperature
is 98.6 `F against the two-sided alternative. State your
conclusions. Which of the above calculations may be invalid?
- Normal body temperature is 37 `C. You are concerned if the
mean temperature in the group of treated patients is 38 `C or
higher. From past experience, you know that measurements have a
standard deviation of 0.5 degrees. How many observations would be
required to test this hypothesis at the 5% level, and ensure that
the Type II Error Rate is no more than 1%?
- In an opinion survey, how many independent subjects are
required to ensure that an estimated percent is within 2
percentage points of the true percent 19 times out of 20?
- How many independent normal observations would be required to
ensure that the upper limit of a 90% confidence interval for the
variance is no more than 3 times the lower limit?
- Using the pharmacology data described in Table 8.17 on p. 318
of Rosner, compute a p-value to test the hypothesis that
the mean body temperature is unaffected by taking aspirin. Is a
one-sided or two-sided test appropriate? Do what you can to test
any assumptions you make. State your conclusions.
- Repeat the previous analysis using the sign test to test the
hypothesis that the median difference is zero. State your
assumptions and your conclusions.
- Analyze the Obstetrics data in Table 8.16 on p. 317 of Rosner.
Give an appropriate graphical display of the data. State any
assumptions you make and test any assumptions you can test.
- You have analyzed the Obstetrics data in Table 8.16 on p. 317
of Rosner as a two-sample t-test. Repeat the analysis, this time
as an analysis of variance for a one-factor design. Show that the
F statistic in the anova table is the square of the two-sample t
statistic and has the same p-value. Show that the mean
squared error (also called the mean squared residual or residual
variance) is the same as the pooled variance estimate in the
t-test. The graphical displays, assumptions and conclusions are
exactly the same for both analyses.
- Continuing with the Obstetrics data, give a 95% confidence
interval for the conditional variance of birth weight, given
treatment, that is, the residual mean squared error after fitting
treatment as a factor.
- Analyze the Pulmonary Disease data in Table 12.23 on p. 568 of
Rosner. Answer problem 12.6 with a comparative box plot and
an ANOVA table. Give a 95% confidence interval for the residual
variance s2.
- Analyze the following data which give plasma epinephrine
concentrations for two different subjects under (1) isoflurane,
(2) halothane and (3) cyclopropane anesthesia. Present your
results in an ANOVA table. State your assumptions and your
conclusions. Give a 95% confidence interval for the residual
variance s2. If this study
were to be done again, what changes to the design would you
recommend?
Subject:
|
1
|
1
|
1
|
1
|
1
|
1
|
2
|
2
|
2
|
2
|
2
|
2
|
Anesthesia:
|
1
|
1
|
2
|
2
|
3
|
3
|
1
|
1
|
2
|
2
|
3
|
3
|
Epinephrine:
|
0.28
|
0.36
|
0.30
|
0.88
|
1.07
|
1.53
|
0.51
|
0.32
|
0.39
|
0.39
|
1.35
|
0.49
|
- Use the Hospital Stay data in Table 2.11 on p. 39 of Rosner.
(HOSPITAL.DAT, HOSPITAL.DOC on the data disk.) Remove the 7th
subject from the analysis. Plot duration of stay (dependent
variable) against age (independent variable). Fit a straight line
to the data and add it to the graph. Summarize the fit in an ANOVA
table and state your assumptions and conclusions.
- Use the Hospital Stay data in Table 2.11 on p. 39 of Rosner.
(HOSPITAL.DAT, HOSPITAL.DOC on the data disk.) Leave in the 7th
subject, which has an unusually long duration of stay, but try to
reduce its impact by taking a log transformation. Give a pairs
plot and a correlation matrix for the following variables:
log-transformed duration of stay, age, first temperature and first
white blood cell count. Fit the model log(duration) ~
age+temp1+wbc1. Summarize the fit in an ANOVA table. Plot the
observed values against the fitted values and add a diagonal line
to the plot. Plot the residuals against the fitted values. State
your assumptions and conclusions.
- Continuing with the Hospital Stay data, fit the model
log(duration) ~ temp1+age+wbc1 and give the ANOVA table.
Discuss how it differs from the previous fit.
- The following data come from a paper "Changes in growth
hormone status related to body weight in cattle," with x = body
weight (kg) and y = metabolic clearance rate.
x
|
110
|
110
|
110
|
230
|
230
|
230
|
360
|
360
|
360
|
360
|
505
|
505
|
505
|
505
|
y
|
235
|
198
|
173
|
174
|
149
|
124
|
115
|
130
|
102
|
95
|
122
|
112
|
98
|
96
|
Fit a straight line to the data by least squares. Can
metabolic clearance rate be predicted as a linear function of body
weight? Present your results in an ANOVA table with tests for
non-linearity and for the slope of the regression line. Plot
appropriate graphs. State your assumptions and your conclusions.
What metabolic clearance rate would you predict for a body weight
of 300 kg? Give a 95% confidence interval for the residual
variance.
- Analyze the Sexually Transmitted Disease data from Table 10.25
on p. 415 of Rosner as a 3 x 3 contingency table and give a
p-value. State your conclusions.