1. (20 points) Use the following data set to perform the t-test and determine p-values: The data studies the effect of dietary calcium on blood pressure. In the experiment, one group of black male adults received calcium supplements for 12 weeks. A control group of black male adults was given placebos for 12 weeks. Earlier tests indicated that calcium could be more effective in the black population. The data are seated systolic blood pressure at the beginning and end of the 12 weeks measured in mm Hg. The data include the decrease after 12 weeks (a negative value indicates an increase). If you choose to use the table for calculating the p value range, please also report what you calculated for t (or for F)!!!!!! Data from Lyle et al., JAMA 257:1772-1776 (1987) Begin 107 110 123 129 112 111 107 112 136 102 Calcium Group End 100 114 105 112 115 116 106 102 125 104 Decrease 7 -4 18 17 -3 -5 1 10 11 -2 Placebo Group Begin End Decrease 123 124 -1 109 97 12 112 113 -1 102 105 -3 98 95 3 114 119 -5 119 114 5 112 114 -2 110 121 -11 117 118 -1 130 133 -3 a. (4) Look at the calcium data as paired. Perform a test to determine the p-value (two-sided) for the decrease. State the null and alternative hypothesis. Calculate a 95% confidence interval for the decrease. Show the steps you used to reach the conclusions. If you use statistical software, show which equations must be used. b. (4) Perform the F test to compare variances on the decrease columns. You will be comparing the variance of the decrease column for calcium to the decrease column for placebo. State the pvalue and the conclusion of the test. c. (4) Treat the two groups (calcium and placebo) as independent samples, equal variances. Look only at the decrease columns (you are now comparing the calcium decrease column to the placebo decrease column). Perform a test to determine the p-value for the decrease. State the null and alternative hypothesis. Calculate a 95% confidence interval for the decrease. d. (4) Treat the two groups as independent samples, unequal variances. Look only at the decrease columns (calcium vs. placebo). Perform a test to determine the p-value for the decrease. State the null and alternative hypothesis. Calculate a 95% confidence interval for the decrease. e. (4) Do a two-sided t-test to compare the sample groups. Look at the Begin column for calcium and the Begin column for placebo. Report the p-value. Based on that p-value, do you have any reason to believe that the sample groups were significantly different headed into the experiment? 2. (14 points) Use the following data table comparing a control group to a group with paclitaxel coated angioplasty balloons implanted in leg arteries: paclitaxel-coated Variables control group balloons group (n = 54) (n = 48) Baseline Rutherford stage 3.1 0.8 3.4 0.8 Degree of stenosis 91 7 89 8 Target lesion length (cm) 7.4 6.7 7.5 6.2 Target lesion diameter (cm) 4.7 0.6 5.0 0.7 No. of runoff vesicles 2.1 0.8 1.8 0.8 NOTE. Values are mean SD. Data taken from Tepe et al. (2008) N Eng J Med. 358(7):689-699. a. (4) Perform the F test on all five variables listed (comparing control group to balloons group). State the p-value and conclusions that should include whether to use the equal variance approach or the unequal variance approach. b. (4) Perform hypothesis tests on all five variables (again comparing one group to the other). State the null hypothesis for each. List the p-value and a specific conclusion. c. (4) Calculate by hand the Satterthwaite's degrees of freedom for target lesion diameter (the fourth variable). Show your steps. Your answer should be the exact number that would be used in a table to look up the p-value in the subsequent hypothesis test. d. (2) Construct a 99% confidence interval for degree of stenosis, balloons group. 3. (10 points) If you were told that a sample of 26 breast-fed infants had a mean hemoglobin level of xbar = 13.9 grams/100 ml with a standard deviation of s = 1.7 g/100ml, give 95% confidence intervals for the mean hemoglobin level of breast-fed infants as indicated below. a. (5) Calculate the 95% confidence interval assuming that the standard deviation () of the breast-fed infants population (normal distribution) is = 2.1g/100ml. b. (5) Calculate the 95% confidence interval assuming that the standard deviation () of the breast-fed infants population is unknown. 4. (10 points) In 1943, Salvadore Luria and Max Delbrck did some pioneering research on bacterial genetics--specifically on the resistance of bacteria to viral attack. The research (years ahead of its time) is particularly relevant now because antibiotic resistance models may prove to be similar. Antibiotic resistance is proving to be one of the greatest threats to modern public health. Suppose 69 colonies appeared on 60 plates. The mean number of colonies on a plate would be 69/60 or 1.15. a. (5) Assuming a Poisson distribution, calculate probability that there will be 0 colonies on a plate. Repeat the calculation for one colony, two colonies, three colonies and four colonies. b. (2) What is mean and the standard deviation for the number of colonies on a plate? c. (3) What is the probability that there will be four or more colonies on a plate? 5. (15 points) Use the direct method of standardization to compute the crude mortality rate and ageadjusted mortality rate for North Carolina in 2000. Use the U.S. population (also in table) as your standardizing population. Age (years) 0-34 35-54 55-64 65+ Total NC population NC deaths 3,984,283 3,740 2,372,270 8,094 723,712 8,035 969,048 51,863 8,049,313 71,732 NC Age-specific rate (per 10,000) US population 139,328,990 82,826,479 24,274,684 34,991,753 281,421,906 US expected deaths per age group a. (5) Fill in the missing values for the age-specific rates in North Carolina, and the expected deaths in the U.S. population using the North Carolina rates. You do not have to list in the tableyou can just report the numbers. b. (5) What is the crude death rate for North Carolina in 2000 (report as per 10,000)? c. (5) What is the age-adjusted death rate for North Carolina in 2000? 6. (9 points) Assume that serum cholesterol levels are normally distributed. We are given that the mean serum cholesterol level of U.S. males aged 20-74 is 211 mg/dL, and that the standard deviation () of that population is 46 mg/dL. You randomly select 16 individuals from that population. a. (3) Would you use the Z table or t table? Why? b. (3) What is the probability that an individual of the 16 selected has a cholesterol level greater than 280 mg/dL? c. (3) What is the probability that a sample mean from a group of 16 has a cholesterol level greater than 280 mg/dL? 7. (4 points) The data set poliomyelitis.dta (or the Excel version - both in Resources under Exam 01) was created by accessing the World Health Organization public database. There are four variables: country lists the African nation, value represents the number of cases of poliomyelitis for the year, latestyear represents the year for the value field, and region represents the region of Africa by United Nations designation. a. Each line represents a unique country. Countries are a subset of regions. Perform a summary statistics operation on value by region. Which region has the fewest countries? Region: _______________; # of countries: _____________________ b. Which region has the highest mean number of cases per country? Region:__________: Mean polio cases per country: __________; Std deviation:_______ c. What is the mean number of cases per country (all of Africa, not by district)? ____________. What is the median number of cases per country (all of Africa again)?_______________. d. Would you say the mean or median is more representative of a typical African country? e. Which country is the biggest outlier? country value latestyear region Cape Verd 0 2006 East Afr Liberia 0 2006 East Afr Togo 0 2006 East Afr Mauritania 0 2006 East Afr Niger 11 2006 East Afr Guinea 0 2006 East Afr Nigeria 1099 2006 East Afr Guinea-Bis 0 2006 East Afr Mali 0 2006 East Afr Gambia 0 2006 East Afr Benin 0 2006 East Afr Ghana 0 2006 East Afr Senegal 0 2006 East Afr Burkina Fa 0 2006 East Afr Cte d'Ivoi 0 2006 East Afr Sierra Leo 0 2006 East Afr Mauritius 0 2006 West Afr Rwanda 0 2006 West Afr Kenya 2 2006 West Afr United Repu 0 2006 West Afr Madagasca 0 2006 West Afr Seychelles 0 2006 West Afr Eritrea 0 2006 West Afr Malawi 0 2006 West Afr Algeria 0 2006 West Afr Zambia 0 2006 West Afr Mozambiqu 0 2006 West Afr Uganda 0 2006 West Afr Ethiopia 17 2006 West Afr Comoros 0 2006 West Afr Burundi 0 2006 West Afr Zimbabwe 0 2006 West Afr Swaziland 0 2006 Southern Botswana 0 2006 Southern South Afric 0 2006 Southern Lesotho 0 2006 Southern Namibia 19 2006 Southern Congo 0 2006 Middle A Gabon 0 2006 Middle A Central Afr 0 2006 Middle A Chad 1 2006 Middle A Democratic 12 2006 Middle A Cameroon 2 2006 Middle A Equatorial 0 2006 Middle A Angola 2 2006 Middle A Sao Tome a 0 2006 Middle A