Statistics 2MA3 - Assignment #3

2001-03-26

Due 2001-04-06 17:00

1. 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?
2. 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%?
3. 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?
4. 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?
5. 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.
6. Repeat the previous analysis using the sign test to test the hypothesis that the median difference is zero. State your assumptions and your conclusions.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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

12. 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.
13. 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.
14. 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.
15. 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.
16. 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.

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Statistics 2MA3