Biostats Homework 2

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School

Indiana University, Bloomington *

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381

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Statistics

Date

Apr 3, 2024

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docx

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Homework #2 Questions Homework #2 Questions Homework #2 Questions SPH-Q 381.SP24 Introduction to Biostatistics—Spring 2024 Sections 12247 and 13871 Dr. Dave Fluharty Homework #2 8 Points 12 Questions points are indicated by the question. February 19, 2024 No late assignments accepted except in exceptional circumstances. You may use your book, notes, videos, internet, or other resources. You may discuss the questions with other students, HOWEVER, THE ASSIGNMENT YOU SUBMITT MUST BE DONE INDIVIDUALLY. For example, you might meet with a study group to discuss what questions mean and how they might be approached (including hints for JMP), but then separate and do the problems yourself. Contact Dr. Dave if this is unclear. We will review the homework questions in the first class after the homework is uploaded. Students may also ask Dr. Dave questions about this homework during Drop-In Office Hours or individual meetings. INSTRUCTIONS: 1. BE SURE TO READ THE ENTIRE QUESTION BEFORE ANSWERING IT (in the past students have lost points on this). 2. JMP is not used in this assignment. Most of our homeworks will have extensive use of JMP. 3. Write out your calculations. Do not just write down the answer unless it is just looking up a value. First write the formula you used and then show the calculations. 4. Submit the assignment in Microsoft Word or PDF format. a. Do not submit in other formats which may not be readable for grading. b. Please put your answers in a different color than the questions (this makes grading much easier). c. Be sure the answers you upload are complete. You are responsible for ensuring you uploaded the entire assignment. 5. Please read the questions carefully. In the past students have lost points for not answering the question that asked or not answering all of the question. If you have any questions—particularly if you have difficulty working with JMP--please contact Dr. Dave!! PURPOSE OF THIS HOMEWORK: To ensure you understand the basic concepts of probability. GENERAL HINTS: Be sure you see the question. For Questions 6 and 7, the questions you need to answer are in the box. JMP HINTS: JMP is not used in this assignment. Page 1 of 7

Homework #2 Questions Homework #2 Questions Homework #2 Questions HOMEWORK PROBLOEMS: 1. (1 point) Relative Risk (RR) Question: a. Write out the Relative Risk formula. Relative Risk = P( Outcome exposed to risk) / P(Outcome unexposed to risk) b. What is the relative risk of contracting a disease if exposed to a substance believed to be harmful given the sample data below? RR = 400 x 40400/4400 x 400 = 16160000/1760000  9.182 is the RR  400 people who were exposed to the substance contract the disease while 4000 who were exposed to the substance do Not contract the disease. Hint: Think about the total number who were exposed when doing the calculation. You need to add two numbers.  and 400 people who were NOT exposed to the substance contract the disease and 40,000 who were NOT exposed do NOT contract the disease. 2. (1 point) Odds Ratio (OR) Question: a. Write out the Odds Ratio formula. OR= P(outcome|exposed)/ 1-P(Outcome|Exposed)]/P(outcome|Unexposed)/1-P(outcome|unexposed)] b. What is the Odds Ratio for the data in the prior question? Data repeated for your convenience. 400 4400 4000 4400 ÷ 400 40400 40000 40400 ≈ 10 ∨ ¿  400 people who were exposed to the substance contract the disease while 4000 who were exposed to the substance do Not contract the disease. Hint: Think about the total number who were exposed when doing the calculation. You need to add two numbers.  and 400 people who were NOT exposed to the substance contract the disease and 40,000 who were NOT exposed do NOT contract the disease. Page 2 of 7

Homework #2 Questions Homework #2 Questions Homework #2 Questions 3. (0.5 point) The following is problem 5 of chapter 5: Let A represent the event that a particular individual is exposed to high levels of carbon monoxide and B the event that he or she is exposed to high levels of nitrogen dioxide. Write out the answers to the following in words. a) What is the event A ∩ B? This event represents individuals who are exposed to both high levels of carbon monoxide and high levels of nitrogen dioxide. In other words, they are exposed to both harmful substances. b) What is the event A ꓴ B? The event A ∪ B represents the union of events A and B, which means the event where a particular individual is exposed to high levels of either carbon monoxide or nitrogen dioxide or both. c) What is the complement of A? This event represents the complement of A, which includes all individuals who are not exposed to high levels of carbon monoxide. This encompasses everyone who was either not exposed to the substance at all or was exposed to levels below the threshold considered "high." d) Are events A and B mutually exclusive? No, Events A and B are not mutually exclusive because it is possible for an individual to be exposed to both high levels of carbon monoxide and high levels of nitrogen dioxide simultaneously. This means they can occur together, 4. (1 point) For Mexican American infants born in Arizona in 1986 and 1987, the probability that a child’s gestational age is less than 37 weeks is 0.142 and the probability that his or her birth weight is less than 2500 grams is 0.051 [19]. Furthermore, the probability that these two events occur simultaneously is 0.031. a) Let A be the event that an infant’s gestational age is less than 37 weeks and B the event that his or her birth weight is less than 2500 grams. Construct a Venn diagram to illustrate the relationship between events A and B. Annotate your diagram with the three probabilities listed above. Page 3 of 7 B A 0.031 0.051 0.142

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Homework #2 Questions Homework #2 Questions Homework #2 Questions b) Are A and B independent? Why? They are not independent due to plugging in the numbers to this equation, making P(A ∩ B) ≠ P(A) x P(B) P(A) x P(B) = (0.142)(0.051) = 0.007242 c) For a randomly selected Mexican American newborn, what is the probability that A or B or both occur? Hint: Use the formula that accounts for double counting. P(A or B or both) = P(A) + P(B) + P(A ∩B)  0.142 + 0.051+ 0.031  0.224 d) What is the probability that event A occurs given that event B has occurred? Hint: Conditional probability formula. P(A ∩B)/P(B) 0.031/0.051  0.6078 5. (0.5) a) Given the following probabilities, what is P(A | B)? show your calculation. b) What is the name of this type of probability? P(A∩B) = .7 P(B) = .9 P(A|B) = P(A∩B)/P(B)  0.7/0.9  0.778 This is called conditional probability Hint: PGM p. 116 6. (1 point) This is problem #7 from Chapter 3. The question at the bottom of the box. Page 4 of 7

Homework #2 Questions Homework #2 Questions Homework #2 Questions I disagree with this statement because an increase in the number of deaths over time does not necessarily mean the population is becoming less healthy. It’s essential to consider these factors and look at other health indicators as well. For a comprehensive understanding of population health, one should consider various factors including disease prevalence, life expectancy, quality of life, etc. 7. (1 point0 This is problem #8 from Chapter 3. The questions are at the bottom of the box. Show your calculations. Hint: Remember your glossary and index may be helpful. A) Crude birth rate  (Live births/ population) x1000  70,704/6,863,560 x 1000 = 10.30 % B) Crude death rate  (deaths/population) x 1000  58,844/6,863,560 x 1000 = 8.58 % C) Infant mortality rate  (deaths of infants/live births) x 1000  263/70,704 x1000 = 3.72 % 8. (0.5) This question is based on problem #4 in Chapter 3 a. What is an adjusted rate? Given an example of an adjusted rate. An adjusted rate is a corrected version of a crude rate that considers the influence of one or more factors that might skew the data. Essentially, it removes the bias caused by these factors and allows for a more Page 5 of 7

Homework #2 Questions Homework #2 Questions Homework #2 Questions accurate comparison between different groups or populations. An example of an adjusted rate would be comparing the crude death rates of two countries with vastly different age distributions. An older population naturally has a higher death rate. b. When should adjusted rates be used? They should be used when Comparing groups with different characteristics that might influence the outcome (e.g., age, sex, ethnicity), and when evaluating trends over time when the population composition changes. c. Give an example of a crude rate. When should a crude rate be used? A crude rate is a simple calculation that uses the raw data without any adjustments. It expresses the frequency of an event (birth, death) per unit of population in a specific time period. An example of one would be the total number of live births per 1,000 people in a population per year. You would use a crude rate when you need a quick and straightforward estimate of an event's frequency. d. Give an example of a specific rate. When should a specific rate be used? An example of a specific rate is when we can calculate the specific death rate by age group to see how mortality risk changes with age. You use specific rates when you want to analyze variations within a population based on specific characteristics (age, sex, occupation). 9. (0.3 points) How does the direct method of standardization differ from the indirect method? When would you use the indirect method? Both direct and indirect standardization methods compare rates across populations while accounting for compositional differences. Direct methods utilize a reference population's age-specific rates, while indirect methods apply standard rates for individual subgroups. Choose the direct method for large populations with stable rates and the indirect method for smaller populations or unstable rates, considering their specific strengths and limitations. 10. (0.4 points) Please name the rule of probability of which each of the following is an example. Write out the name—do not just give a number that may have been in a list: a. The probability of rain is between 0.0 and 1.0, inclusive. The rule of probability that each outcome must have a probability between 0 and 1, inclusive, is called the Law of Probability. b. The members of a fraternity that have cars have cars that are red or blue or white. P(red) = 0.2, P(white) = 0.5, P(blue) = 0.3. Each person who has a car has one and only one car. Page 6 of 7

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Homework #2 Questions Homework #2 Questions Homework #2 Questions The rule of probability that the sum of the probabilities of all mutually exclusive outcomes must equal 1 is called the Law of Total Probability. 11. (0.3 points) Calculate the following AND name the probability rule: a. The probability of IU Willing the game on Saturday is 0.3. If no ties are allowed, what is the probability the opponent will win? The Rule is Complementary probability. Here’s the calculation: Calculation: P(opponent wins) = 1 - P(IU wins) = 1 - 0.3 = 0.7 b. If the probabilities of the color of cars owned by members of a fraternity that have cars are P(red) = 0.2, P(white) = 0.5, P(blue) = 0.3. What is the P (Red U White)? The Rule is the Addition rule (for mutually exclusive events). Here’s the calculation: Calculation: P(Red U White) = P(Red) + P(White) = 0.2 + 0.5 = 0.7 c. If the P(Dr. Dave Teaches SPH-Q 381 next term) = 0.9 and the P(US Senator Todd Young will run for Senate in 2028) = 0.5, what is the probability (Dr. Dave will teach SPH=Q 381 next term U US Senator Todd Young will run for Senate in 2028). The rule is the addition rule for the union for two events. So, the probability of either event occurring would be calculated: P(A ∪ B)=P(A)+P(B)−P(A∩B) Substituting the given values: P(A ∪ B)=0.9+0.5−0=1.4 However, a probability cannot exceed 1. This suggests that the events are not independent, and there is some overlap between them. 12. (0.5 points) Which definition of probability do you prefer and why? (No answer is incorrect as long as you have a reasonable reason). The classical definition of probability, also known as the "theoretical" probability, is based on the assumption that all outcomes in a sample space are equally likely. It is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes in the sample space. This definition is often preferred for its simplicity and ease of calculation when dealing with simple random processes where it is clear what the equally likely outcomes are. For example, when flipping a fair coin, there are two equally likely outcomes: heads or tails. The classical probability of getting heads is therefore 1/2, as there is one favorable outcome (heads) out of two possible outcomes (heads and tails). Page 7 of 7

Biostats Homework 2

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