Sampling Distribution and Confidence Interval Problems



ASSIGNMENT #3 STAT 601 Review the following:Lecture Handouts : Review the lecture handouts 8 – Sampling Distributions and Confidence Intervals and 9 – Statistical Inference sections 9.1 – 9.4 only. Narrated Powerpoint Lectures under headings 6 – 8.Non-narrated Powerpoint Lectures under headings 6 – 8.RESEARCH PAPER REVIEW QUESTIONSUsing Tables 2 and 3 in the paper “6-month Low Carb Diet” answer the following questions:What is the mean difference from baseline to week 24 for total cholesterol?What is the mean difference from baseline to week 24 for bicarbonate?Give a range of values that is likely to cover the true mean change from baseline for triglycerides, i.e. give the 95% confidence interval for mean triglycerides.Give a range of values that is likely to cover the true mean change from baseline for creatinine, i.e. give the 95% confidence interval for mean creatinine.Did the HDL cholesterol level change significantly from baseline to week 24? Is this result clinically significant?In Table 2, which parameter or variable has the smallest confidence interval?The Atkins Center provided funding and training for the research staff. Would this have an effect on the outcomes of this study? Provide a rationale for your answer.In this study, 51 subjects were enrolled, but only 41 completed. Identify some possible reasons the 10 subjects might have dropped from the study.This was uncontrolled study. Identify ways to improve the control of the study and thus strengthen the study design. Using Tables 3 and 4 in the paper “Scotland Breast Cancer Survival” answer the following questions:Identify the confidence interval (CI) for the women diagnosed with breast cancer in the 1993 group who were negative in all nodes.What was the survival rate (percentage) for women diagnosed with breast cancer in the 1987 group who had unknown pathological tumor size?What tumor size was related to the best survival percentage?What was the difference in survival rate from the 1987 group to the 1993 group for the Health Board of Residence of Argyll and Clyde? How did this figure compare with the other Health Boards of Residence?In comparing the 1987 and 1993 groups, which group was more likely to receive adjuvant therapies? Give your answers as percentages.This study was conducted in Scotland. Can the findings be generalized to the United States?ADDITIONAL PROBLEMSNorth Carolina Birthweight StudyData description:The North Carolina State Center for Health Statistics and Howard W. Odum Institute for Research in Social Science at the University of North Carolina at Chapel Hill make publicly available birth and infant death data for all children born in the state of North Carolina. These data can be accessed at: data contained in NCbirth.JMP represent a random sample of n = 800 births in North Carolina in 2001. The variables and their coding are described in the table below.In addition, the following variables were created and added to the file NCbirth.JMP:White? Coded as White or Non-White (dichotomous version of RACEMOM)Hispanic?Coded as Non-Hisp or Hisp (dichotomous version of HISPMOM)a) Use these data to complete the table below using birthweight as the response (10 pts.)VariablenMeanSD95% CI for ??LCL, UCL)Smoking Status Smoker Non-smoker Minority Status Non-white Whiteb) Interpret the confidence intervals for the mean birthweight of infants born to smokers and non-smokers. Also comment on whether or not these CI’s provide evidence that mothers who smoke during pregnancy have infants with a lower mean birthweight. (4 pts.)c) Interpret the confidence intervals for the mean birthweight of infants born to minority and non-minority mothers. Also comment on whether or not these CI’s provide evidence that the mean birthweights of infants born to these two populations of mothers significantly differ. (4 pts.)d) Using the NC Birthweight Study data estimate the percentage of babies who are born prematurely in the state of North Carolina using a 95% CI confidence interval. Interpret this interval. (4 pts.)e) Now find confidence intervals for the percentage of low birthweight babies born to the population of smoking mothers and to the population of non-smoking mothers. Do these intervals suggest that the percentage of babies born with a low birthweight differs between these two populations of mothers? Explain. (6 pts.)In JMP For part (a) use Fit Y by X with tgrams (birthweight in grams) as Y and Smoke & White? as the X variables. Selecting Means and Std Dev from the Oneway Analysis pull-down menu will give you the required confidence intervals. For part (d) simply get the Distribution of the variable Premie. You can actually get the CI in JMP by selecting Confidence Intervals from the Premie pull-down menu.For part (e) the only way to get the confidence intervals in JMP is to use the process from part (d) above BUT you must first put the smoking status variable, Smoke, in the By box (see next page).2) Type I and II Errors and Drug Safety Testing An interesting paper on the subject can be found here at the link below. The material at the end of the paper gets a bit theoretical but the first few sections are very good. the years, the U.S. Food and Drug Administration (FDA) has worked very hard to avoid making Type I errors. Ho: Drug in question is ineffective or unsafe/dangerousHa: Drug in question is effective and safe.A type I error occurs when the FDA approves a drug that is not both safe and effective. Despite the agency’s efforts, however, bad drugs do on occasion get through to the public. For example, Omniflox, and antibiotic, had to be recalled less than six months after its approval due to reports of severe adverse reactions, which included a number of deaths. Similarly, Fenoterol, an inhaled drug intended to relieve asthma attacks, was found to increase the risk of death rather than decrease it. a) Is there any way for the FDA to completely eliminate the occurrence of type II errors? Explain. (2 pts.)b) Name two ways the probability of making a type II error can be decreased. (2 pts.) 3) Power and Sample SizeData from the Framingham Study ()allow us to compare the distributions of initial serum cholesterol levels for two populations of males: those who go on to develop coronary heart disease and those who do not. The mean serum cholesterol level of the population of men who do not develop heart disease is ????????mg/dl and the ???????mg/dl. Suppose you wish to conduct a study to see if men who go on to develop coronary heart disease (CHD) have a greater mean serum cholesterol level, i.e. you have the following hypotheses in mind when conducting your study using the Framingham data:Ho: ? < 219 mg/dl Ha: ? > 219 mg/dlYou wish to conduct your test using an ??????? level of significance. To be clear, in this study men had their serum cholesterol levels at the start of this study and then they were followed prospectively. Some of these men went on to develop coronary heart disease and some did not. We are comparing the initial or baseline cholesterol levels of these two groups of men. We want to know of those that developed CHD had a mean cholesterol level above 219 mg/dl at baseline.a) State in words what a type I error is in this particular situation. (1 pt.)b) State in words what a type II error is in this particular situation. (1 pt.)c) What is the probability of making a type I error? (1 pt.)d) Use the JMP DOE > Sample Size and Power calculator to find the power and the probability of making a type II error when n = 25 and the population mean making the alternative true is 244 mg/dl. (See JMP tutorial on how to use Power Calculator) (2 pts.)e) Construct and turn in a copy of a plot of the power vs. the sample size when the population mean which makes the alternative true is equal to 244 mg/dl. Use it to estimate the sample size required to achieve a power of 90%. (3 pts.)4)? In an effort to detect hypertension in young children, blood-pressure measurements were taken on 30 children aged 5-6 years living in a specific community.? For these children the mean diastolic blood pressure was found to be 56.2 mm Hg with standard deviation 7.9 mm Hg.? From a nationwide study, we know that the mean diastolic blood pressure is 64.2 mm Hg for 5- to 6-year-old children.? a) Is there evidence that the mean diastolic blood pressure for children in this community is different from the nationwide average of children of the same age group? Use a 95% confidence interval for the mean to answer this question. (4 pts.)5) In the 1980s it was generally believed that autism affected about 5% of the nation’s children. Some people believe that the increase in the number of chemicals in the environment has led to an increase in the incidence of autism. A recent study examined 384 children and found that 46 of them showed signs of some form autism a) Is this strong evidence that the level of autism has increased? Conduct an appropriate test. (5 pts.)b) Give a 95% CI for the proportion of children in the population who show signs of some form of autism and interpret this interval. (4 pts.)6) Suppose we wanted to estimate the proportion of children in U.S. who exhibit signs of autism using a 95% CI with a margin of error no larger than 3%, i.e. E = .03.a) What sample size is required if researchers are willing to assume that true proportion is at most 8%? (3 pts.)b) What sample size is required if researchers are not willing to assume anything about this proportion? (3 pts.)7) Suppose we wanted to estimate the mean systolic blood pressure of individuals more than 30% above their ideal body weight. How many such individuals would we have to sample to estimate this mean with a 95% confidence interval and margin of error no larger than 5 mmHg? Use your own prior knowledge of the range of systolic blood pressures to find an estimate of and use that in determining the sample size. (4 pts.)8) The Bayley Scales of Infant Development yield scores on two indices – the Psychomotor Development Index (PDI) and the Mental Development Index (MDI) – which can be used to assess a child’s level of function in each of these areas at approximately one year of age. Among healthy infants, both indices have a mean value of 100. As part of study assessing the development and neurologic status of children who have undergone reparative heart surgery during the first three months of life, the Bayley Scales were administered to a sample n = 144 of one-year-old infants born with congenital heart disease. These data are contained in the data file PDI-MDI.JMP contains the variables:PDI = psychomotor development indexMDI = mental development indexa) Is there evidence that the mean PDI score for children born with congenital heart disease who undergo reparative heart surgery during the first three months of life have a mean score less 100, which is the mean for healthy infants? Summarize your findings and check assumptions. (4 pts.)b) Is there evidence that the mean MDI score for children born with congenital heart disease who undergo reparative heart surgery during the first three months of life have a mean score different from 100, which is the mean for healthy infants? Summarize your findings your findings and check assumptions. (4 pts.) ................
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