SPH244_2023_Homework_6

docx

School

University of California, Davis *

*We aren’t endorsed by this school

Course

244

Subject

Statistics

Date

Feb 20, 2024

Type

docx

Pages

Uploaded by SuperWillpowerButterfly25

SPH 244 Fall 2023 Homework 6 You must submit your homework online via Canvas in a single Word document or PDF. Late homework assignments will not be accepted. General guidelines for homework:  You are allowed to work in study groups on homework, but you must write your own programs and your own analysis for your answers and summaries.  For the data analysis portion of the homework, please include: o A brief statement (a few sentences or a paragraph) interpreting your findings. o All tables and figures (i.e., graphs/plots) requested, suitably labeled as would appear in a formal report or manuscript. For example, do not use SAS variable names. Instead, use the appropriate descriptions and labels (i.e. proc format, format statements and label statements). o The SAS or R code that generated the output must be included at the end of your document in its entirety as a code appendix.  For each data analysis step, follow best coding practices for reproducible research, e.g., clear and detailed comments, white spaces, indenting, descriptive variable names, and easy to follow logical order. 1. Glaucoma is a disease of the eye, characterized by elevated intraocular pressure. Intraocular pressure is calibrated to measure pressure continuously in millimeters of mercury (mmHg). Intraocular pressure in the general population has a symmetric distribution with a mean of 16 mmHg and standard deviation of 5 mmHg. You may use a table, an online calculator, SAS, or R to find the probabilities and/or to help you sketch regions for some parts of this problem. a. What is the name of the probability distribution that is most appropriate for this scenario? (Bernoulli, Binomial, Poisson, or Normal)? What is/are the assumed value(s) of the parameter(s) based on the information provided? b. What proportion of the population has an intraocular pressure between 6mmHg and 26mmHg? Sketch a picture and shade the region of interest on the scale of X (mmHg) and be sure to label values of X (mmHg) on the x-axis for the mean and the values of X (mmHg) that correspond to +/-1, +/-2, and +/-3 standard deviations above/below the mean. You should be able to do this in your head. c. What proportion of the population has an intraocular pressure between 6mmHg and 26mmHg? Sketch a picture and shade the region of interest on the scale of Z (no units) and be sure to label values of Z on the x-axis for the mean and the values of Z that correspond to +/-1, +/-2, and +/-3 standard deviations above/below the mean. You should be able to do this in your head. d. How are the probabilities and the pictures that you sketched in parts (b) and (c) related? e. What proportion of the population has an interocular pressure under 6 mmHg? Sketch a picture

and shade the region of interest (either using X or Z scale) and be sure to label values on the x- axis. f. What proportion of the population has an interocular pressure over 23 mmHg? Sketch a picture and shade the region of interest (either using X or Z scale) and be sure to label values on the x- axis. g. Find the intraocular pressure that separates the top 10% from the rest of the population. Sketch a picture and shade the region of interest (either using X or Z scale) and be sure to label values on the x-axis. h. An ophthalmologist is trying out a new tonometer (the instrument that measures intraocular pressure) in people she is pretty sure are clinically normal. The first 6 people all have pressures that read over 23 mmHg, which surprises the doctor. If these people really are just a sample from a population with healthy eyes, what is the chance that all six have interocular pressures over 23 mmHg?

2. Suppose you have a random sample of 16 subjects from an optometry practice that generally serves patients who are older than the population in Question 1 above. The mean of your sample is 21mmHg and the standard deviation of your sample is 4mmHg. You may use a table, an online calculator, SAS, or R to find the probabilities and/or to help you sketch regions for some parts of this problem. a. What symbols/notation do we use for the mean of your sample and for the standard deviation of your sample and what are their values? b. What proportion of your random sample of 16 subjects have pressure between 6mmHg and 26mmHg? Sketch a picture and shade the region of interest on the scale of X (mmHg) and be sure to label values of X (mmHg) on the x-axis for the mean and the values of X (mmHg) that correspond to +/-1, +/-2, and +/-3 standard deviations above/below the mean. c. How does this probability compare to the probability in Question 1b? Does this make sense, why? d. Calculate a 95% confidence interval for the sample mean. Show your formula and calculation for full credit. e. Based on your 95% confidence interval, do you think the mean of this sample of patients has the same mean as the overall population, why or why not? f. Bonus question . Sketch three distributions on top of each other on the scale of X (the population, the sample, and the sample mean, but not Z) and make sure that they are roughly drawn to the correct scale. Label the center of each distribution on the x-axis and also label 2 standard deviations (or standard errors) above and below the center of each distribution . Recall that each of these is a probability distribution so the area under each curve must be equal to 1 and in order to get that to look approximately correct, the peak of each curve must be different, because the spreads are different.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version