Attached are 8 Biostatistics problems. Please provide worked out solutions. Thank you

1. The following is a table relating a group of 1000 patients' true breast cancer statuses and their corresponding test results after receiving a mammogram. Test Result Cancer Status Breast Cancer No Breast Cancer Total Positive 223 13 236 Negative 80 684 764 Total 303 697 1000 A) What is the probability of a positive test, given that the patient has breast cancer? (2 points) B) What is the formal term used to describe this probability measure? (1 point) C) What is the probability of a negative test, given that the patient is breast cancer free? (2 points) D) What is the formal term used to describe this probability measure? (1 point) 2. 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? 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? 4. HIV/AIDS patients are regularly monitored for their CD4 counts to make sure that antiretroviral therapies are effective. Suppose the distribution of CD4 counts in a population is approximately normal with =237 and =43. A) HIV patients are defined as moving into the AIDS stage of their disease course after their CD4 counts are less than 200. What proportion of patients in this population is suffering from AIDS? B) HIV positive patients maintaining a CD4 count over 350 can frequently carry on regular lives with complications mostly limited to medication side effects. What proportion of patients in this population is considered healthy? 5. Researchers are interested in the popularity of the new HPV vaccine among a population of university aged women. National data released last year indicates that the vaccine is increasing in popularity, and 24% of young women and girls aged 11-25 have now been estimated to have received at least one of the three shots associated with the vaccine. Suppose researchers randomly sample 7 girls in the population, and 3 report having at least one of the injections. A) Can we carry out our calculation for the probability of seeing such an occurrence with an approximation of normality? B) Calculate the probability of observing such a result. C) If we were to repeat the sample, how many women in this age range would we expect to be vaccinated? 6. Researchers estimate that the prevalence of HPV among young women aged 18-30 is on the rise. Suppose the national statistics report that 39% of women in this age range are now HPV positive. A research team is investigation the relationship between HPV infection and cervical cancer, and samples 167 women to participate in their study. Assume that they find 78 women to be confirmed HPV positive by confirmed PAP smear. A) Calculate the probability of observing such a number of women HPV positive based on the research team's sample. B) What is the mean and standard deviation for the distribution, under the assumptions made in part A? C) What is the probability that 89 women would be observed to be HPV positive given the same conditions? 7. Suppose a database contains population based statistics for a group of hypertensive potential participants in a new clinical trial. The systolic blood pressure of the population varies according to a normal distribution with mean 141 and a standard deviation of 8.4 mmHG. A) From a sample of 37 patients, what is the probability of finding a patient with a systolic blood pressure between 120-125? B) What is the blood pressure associated with the highest 5% in the population? 5% correlates to score of 0.2912