Statistics and Probability

Please show your calculations.  If there are in your work, the calculations can show me where the errors were made, and I can point you in the right direction. The easiest way to show your work would be to use an Excel spreadsheet.  Submit it with your assignment.  If you prefer to show your work in other ways, you may. Table: Hypothetical Data for Kil-EZ Carcinogenicity Test Number of test rats Number of rats with tumors Rat test dose in mg Kil-EZ per kg rat body weight (mg/kg/day) Probability of a rat’s developing tumors Standard Deviation Lower 95% confidence level Upper 95% confidence level 50 1 0 50 5 20 50 10 40 50 20 80 50 7 * 160 * Acute effects were observed in the 160 mg/kg/day group, leading to significant premature deaths Part A 1. Fill in the fourth column of the Table – the probability of tumor, which is the number of rats with tumors divided by the number of rats in the sample 2. Explain why the probability is lower at a dose of 160 mg/kg.day than at a dose of 80 mg/kg.day 3. Plot the points on a graph with probability on the vertical axis (y-axis) and dose on the horizontal axis (x-axis). 4. Draw a straight line through your points to the origin (x = 0, y = 0) and calculate its slope the way you did in high school algebra. That is the cancer potency in kg-day/mg. (If you don’t remember how to calculate the slope of a straight line, ask)   Should you include the observed probabilities of tumor at the zero dose and the highest dose? Why or why not? 5. Using equation (1), calculate the dose that would correspond to cancer probabilities of 1E-6 and 1E-4. (That is the range EPA considers in setting regulatory standards.) Part B 6. Using the equation for standard deviation (s) in the module, calculate the standard deviation at each dose level and enter these values in the fifth column.  The standard deviation needs to be calculated separately for each dose level. 7. Calculate the 95% confidence intervals for each of the relevant probabilities in your Table, and insert into the sixth and seventh columns of the Table. 8. Using the upper 95% confidence level for the 20 mg/kg-day dose, calculate the cancer potency by drawing a straight line from the 20 mg/kg-day dose level to the origin. 9. Do the same using the lower 95% confidence level. 10. Calculate the range of doses that would correspond to a one in a million risk (p = 1E-6) using your answers to Questions 8 and 9. 11. What conclusions do you draw from your answer? Assignment Part C 12. If you were the EPA administrator, would you ban Kil-EZ to avoid human cancers? Explain your answer.

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