# Regression analysis

1. A hospital administrator is reviewing the relationship between the length of in-patient stays, x, in days and the total cost of care, y, in dollars. She collected the following sample data:

X= 4 6 1 8 10

Y= 1105 1545 345 2125 2785

(a) Construct the regression model in the form , which is the best fit for this bivariate data. [COMMENTS & HINTS: Determine the least squares line. Round off the constant and the coefficient to the nearest whole dollar.] Document or explain your work in some reasonable manner. Use available technology to expedite calculations.]

(b) Using the aforementioned regression model, predict the total cost of care, rounding to the nearest whole dollar, based upon a five (5) day in-patient stay.

(c) Using the aforementioned regression model, predict the number of days for an in-patient stay associated with a total cost of care approximately $566.

(d) Determine the Pearson product moment correlation coefficient, r. [COMMENTS & HINTS: Document or explain your work in some reasonable manner. Round off to the nearest ten-thousandth if necessary.]

(e) What percentage of the variation is due to the linear relationship between the independent variable, x, and the dependent variable, y. [COMMENTS & HINTS: Round off to the nearest ten-thousandth, then express as a percentage to two decimal places. In other words, compute the coefficient of determination .]

2. The office manager at a busy medical doctor's office suspects that one out of every eight patient appointments gets rescheduled or canceled. To test her claim, she randomly selects 160 appointments over the past year; of these, ten were rescheduled or canceled. Is her claim reasonable at the 5% level of significance, or is the true population proportion different from what she claims? [COMMENTS & HINTS: What population parameter is being tested? Use both hypothesis testing and confidence interval methods to confirm your conclusion. For bonus credit, determine the corresponding p-value.]

3. The populations, x, in thousands for ten predominantly rural U.S. counties are as follows, along with the corresponding number, y, of general physicians (e.g., practitioners of family or internal medicine).

X= 36 54 75 32 64 60 42 80 45 55

Y= 20 27 35 20 30 26 25 40 24 22

(a) Construct the regression model in the form , which is the best fit for this bivariate data. [COMMENTS & HINTS: Determine the least squares line. Round off the constant and the coefficient to the nearest whole dollar.] Document or explain your work in some reasonable manner. Use available technology to expedite calculations.]

(b) Using the aforementioned regression model, approximately how many general physicians would be predicted for a predominantly rural county with a population of 50,000?

(c) Using the aforementioned regression model, anticipate the population, expressed properly and rounded off to the nearest whole person, for a predominantly rural county with 32 general physicians.

(d) Determine the Pearson product moment correlation coefficient, r. [COMMENTS & HINTS: Document or explain your work in some reasonable manner. Round off to the nearest ten-thousandth if necessary.]

(e) What percentage of the variation is due to the linear relationship between the independent variable, x, and the dependent variable, y. [COMMENTS & HINTS: Round off to the nearest ten-thousandth, then express as a percentage to two decimal places. In other words, compute the coefficient of determination .]

4. A cold capsule is intended to contain 10 units of aspirin. A simple random sample of 100 capsules yielded a sample mean of 10.25 units. The population standard deviation is adequately estimated from prior research as 1.25 units. Is the true population mean of 10 units accurate, according to the available evidence and at the 1% level of significance?

5. According to prior research, the amount of time a medical doctor spends with patients has a standard deviation of 7.2 minutes. Further study is needed to estimate the true mean time medical doctors spend with their patients. If a researcher wants to be 96% confident that the estimate is within 2.4 minutes of the population parameter, what is the minimum sample size necessary for such findings? [COMMENTS & HINTS: Round up!]

6. In an investigation study, researchers were treating benign tumors with repeated dosages of vitamin E. According to a pilot study, 30 out of 36 patients responded favorably to theses treatments. How many more patients are needed to be 99% confident that the estimate will be within 0.125 of the true population proportion for those who respond favorably to the treatments? [COMMENTS & HINTS: Round up!]