Stat 250 1. A hospital decides to bring in a new surgical resident with a known high success rate for total knee replacements. The new resident has a current success rate of 0.96. Suppose on his first day, the resident is supposed to complete 4 total knee replacement operations in a row. Assume his probability of success does not change. a. What is the expected number of successes for the intern? b. What is the variance of the distribution? c. What is the probability that at least 3 replacements will be successful? d. What is the probability that all replacements will fail? 2. Suppose a new screening tool is developed for prostate cancer in men with much higher sensitivity and specificity as the PSA test. Let X be the distribution of scores resulting from that test in men aged 45-65. Suppose after a log transformation, X~N(-0.25, 0.76). What is the range of scores that results in the top and bottom 2.5% of scores for this test? 3. Suppose body mass index (BMI) varies approximately to the normal distribution in a population of boys aged 2-20. A national survey analyzed the BMI for American adolescents in this age range and found the =17.8 and the =1.9. A) What is the 25th percentile of this distribution? B) What is the z-score corresponding to finding a boy with at least a BMI 19.27? C) What is the probability of finding a boy with at least this BMI? D) what proportion of boys will have their BMI to be between 15 and 18? 4. Researchers are interested in conducted an analysis about the effectiveness of a new investigational drug aimed at increasing overall survival among HIV positive patients coinfected with Hepatitis C. Current national data claims that 5 year survival for patients with HIV and Hepatitis C is 32%. After treatment analysis, the research team estimates that 5 year survival among those treated with the new medication is 37%. A) Does the national data report contain a parameter or an estimator? B) Does the study report a parameter or an estimator? 5. Researchers are interested in estimating the prevalence of hypertension in a population of adults aged 40-65 for a new grant. Suppose the population prevalence is 37.5%. The research team has enrolled 167 participants, and 79 have been confirmed hypertensive. A) Can we approximate with the normal distribution in this hypothetical example? Why or why not? B) Create a standardized z-score for the estimate calculated by the research team. C) What is the probability of observing at least 79 to be confirmed hypertensive from this sample? Stat 1. An investigator wants to determine if a new program aimed at reducing the overall number of pregnancies in a highly populated area is effective (reducing average births below 6 per household). In order to carry out the investigation, the investigator conducts a SRS of 189 participants in the program and finds the average number of births is 4.7 with a sample standard deviation of 0.9 births. Construct a 95% confidence interval for the average number of births per household in this study. Provide a correct interpretation of your interval. 2. Recently there has been quite a bit of controversy about the lack of definitive knowledge about the adverse side effects associated with hormone based birth control usage. Suppose a researcher is interested in the effects of birth control in women aged 18-25 relevant to blood pressure. The researcher assumes that women in this demographic that are taking hormone based birth control are at risk for increased blood pressure. Assume the following table contains the summary statistics of two SRSs, one from a cohort of women taking hormone based birth control, and one from a cohort of women birth control free. Study Arm Birth Control No Birth Control Average Systolic Blood Pressure (mmHG) 124 116 Sample Standard Deviation Sample Size 6.8 4.3 246 267 A) Carry out the appropriate statistical procedure to determine if there is a significant difference in the systolic blood pressure between the two groups of women. Write out your null and alternative hypotheses and interpret your results. (Use an alpha level of 0.05) B) Construct a 95% confidence interval about your point estimate for the average difference in systolic blood pressure between the two groups of females. Provide an interpretation for your interval. 3. Food deserts have become a particularly hot topic in the public health community as an explanation for a variety of negative health outcomes in disadvantaged areas. A researcher is interested in comparing the average shelf space provided by supermarkets, convenience stores, and corner stores in two different communities in order to address the growing concern for food deserts. Assume the following table reports the summary statistics for shelf space dedicated to fresh fruits and vegetables (meters) from SRSs of a variety of different stores that supply groceries from each community. Community Average Shelf Space for Fresh Fruits and Sample Standard Deviation Sample Size \"Food Desert\" Other Community Vegetables (m) 13.7 89.4 2.4 14.6 12 14 A) Carry out the appropriate statistical test in order to determine if there is a significant difference in mean shelf space between the two communities at the alpha level of 0.05. Assume all conditions for the test are met. Do not assume equal variances. Write out your null and alternative hypotheses, and correctly interpret your pvalue. B) Construct a 95% confidence interval for your estimate for the average difference in shelf space provided for fresh fruits and vegetables. Provide an interpretation for your interval. 4. Current data provided by the 2010 census indicates that the prevalence of breast cancer in the US is still on the rise, while the mortality rates have slightly declined over time. A researcher at a local health department conducts an observational study to determine the prevalence in the particular region. The researcher gathers 2,168 participants and confirms 246 cases of breast cancer in her sample. A) Assuming the census data indicates a national prevalence of 23.5%, conduct a formal statistical test to determine if the prevalence calculated in the researcher's study differs significantly from the national prevalence level at the alpha value of 0.05. B) Interpret your results from the hypothesis test conducted in part A)