Biostatistics Past Paper PDF 2024/25

Biostatistics minimum requirement questions When writing formulas always interpret the variables even if it 9. What is the definition of classical probability? is not explicitly required in the question If there are N mutually exclusive and equally lik...

Biostatistics minimum requirement questions When writing formulas always interpret the variables even if it 9. What is the definition of classical probability? is not explicitly required in the question If there are N mutually exclusive and equally likely outcomes of an event, and k of these posses a trait, E, 1. What are the permutations of n different elements taken the probability of E is equal to k/N. n at a time and what is their number? Permutations of n different elements taken n at a time are 10. What kind of values can (mathematical) probability all the possible linear arrengements (orders) of all the assume? What is the probability of a certain and an elements. Their number is: impossible event? 𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2)...2 ⋅ 1 Probability is a number between 0 and 1, more rigorously probability can assume any value in the closed interval of 2. Define the combinations of n different elements taken k [0,1]. (A closed interval includes its endpoints, in the at a time in words. above case numbers 0 and 1.) All the possible selections (subsets) of size k of n different Probability of a certain event = 1 elements. Probablity of an impossible event = 0 𝑛 11. Define the complement event of event A! 3. What is the meaning of the ( ) binomial coefficient? 𝑘 The complement of A is the event which occurs when A Define it with a formula and with reference to doesn't occur and the sum of the probabilities of A and its combinatorics (counting techniques). complement event is 1. 𝑛 𝑛! 12. Describe the relationship between the probabilities of ( )= 𝑘 𝑘! (𝑛 − 𝑘)! event A and its complement event B! 𝑛 P(A)+P(B)=1 The ( ) binomial coefficient gives the number of ways k 𝑘 elements can be chosen from n different elements without 13. Define the sum of events A and B! regard to order. The sum of A and B is the event which occurs when either A or B or both of them occur. 4. Define nominal scale and give an example for it. The nominal scale is a list of mutually exclusive 14. Define the product of events A and B! categories to which observations can be classified but The product of A and B is the event which occurs when cannot be ranked. E.g., the sex of a patient can be male both A and B occur. or female. 15. What is the probability of the sum of events A and B? 5. Define ordinal scale and give an example for it. P(A+B)=P(A)+P(B)-P(AB), where The ordinal scale is a list of categories (similar to nominal A+B is the sum of events A and B, scale), in which categories can be ranked according to AB is the product of events A and B. their names or numbers assigned to them. E.g., the efficiency of a drug treatment can be poor, average, good; 16. When are events A and B exclusive? the stage (i.e., extent of progression) of a malignant If AB=0. disease can be classified into stages I-IV. 17. What is the meaning of P(AB)? 6. Define interval scale and give an example for it. P(AB) is the conditional probability of A given B, i.e. the The interval scale is a quantitative scale type in which the probability of occurrence of A if only those cases are numbers assigned to observations have real quantitative considered when B occurs. meaning. Differences between observations measured on an interval scale also express quantitative 18. When are events A and B independent of each other? relationships. However, their ratios are not meaningful A and B are independent if event B has no effect on the due to the lack of an objective zero point on the scale. probability of A and vice versa, i.e. P(AB)=P(A)P(B) or E.g., temperature measured on the Celsius scale: 20 C P(AB)=P(A) or P(BA)=P(B). is not two times higher temperature than 10 C. 19. Using the terms of set theory define 7. Define ratio scale and give an example for it. a. the product ot events A and B The ratio scale is a quantitative scale type in which the b. the sum of events A and B. numbers assigned to observations have real quantitative meaning. Both differences between and ratios of a. AB – the intersection of events A and B (AB) observations measured on a ratio scale express b. A+B – the union of events A and B (AB) quantitative relationships due to the presence of an objective zero point on the scale. E.g. measurement of 20. What is the definition of a random variable? height or blood glucose level and temperature measured If the values assumed by a variable are determined by on the Kelvin scale. 400 K is a temperature two times chance factors, i.e. they cannot be exactly predicted in higher than 200 K. advance, the variable is called a random variable. 8. How are the relative frequency and probability of an event 21. What is a continuous random variable? related to each other? A random variable is continuous if it can assume any The probability of an event is the number around which value within a specified interval of values. the relative frequency (k/n) oscillates (n – the total number of experiments; k – the number of experiments in which the event occurred). If the number of experiments is very large (n→), the variation of relative frequency becomes negligible. This number is called the probability of the event. Biostatistics minimum requirement questions and answers (2024/25) 1 22. What is the definition of the cumulative frequency 31. Define the standard deviation (SD) and the standard error distribution function or cumulative relative frequency of a of the mean (SEM) of a sample with formulas! sample? Cumulative (frequency) distribution function (cdf) of a ∑𝑛 (𝑥𝑖 − 𝑥̅ )2 ∑𝑛 (𝑥𝑖 − 𝑥̅ )2 sample at x gives the fraction of elements in the sample 𝑆𝐷 = √ 𝑖=1 , 𝑆𝐸𝑀 = √ 𝑖=1 which are smaller than or equal to x. 𝑛−1 𝑛(𝑛 − 1) 23. What is the definition of the cumulative distribution xi: the elements of the sample, 𝑥 ̅is the mean of the function of a random variable? sample, n is the number of elements in the sample. The cumulative distribution function (cdf) of a random variable at x represents the probability that the random 32. What is the difference between the standard deviation variable assumes a value smaller than or equal to x. (SD) and standard error of the mean (SEM) of a sample? Write your answer in words, do not use formulas. 24. What is the probability that a continuous random variable The SD of a sample gives an unbiased estimation of the assumes a value in the interval between a and b ? population SD, whereas the SEM is the SD of the sample The probability that a continuous random variable mean, i.e. it describes how accurately the sample mean assumes a value in the (a,b) interval is equal to the area approaches the population mean. If the number of under the curve of the probability density function elements of the sample increases, the SD approaches the between a and b. square root of the population variance, the SEM approaches 0. 25. Define the mean of a discrete random variable! 𝑛 33. Define the coefficient of variation (CV) in words and with a formula. 𝑀(𝑥) = ∑ 𝑥𝑖 𝑝𝑖 The coefficient of variation (CV) is the standard deviation 𝑖=1 (SD) expressed as the percentage of the mean (𝑥 ̅): 𝑆𝐷 where xi is the ith value of the random variable, and pi is 𝐶𝑉 = 100 𝑥̅ the probability that the random variable assumes the value of xi. 34. Define the mean of a sample with a formula and interpret the variables! 26. Define the variance of a random variable with a formula. Variance of a random variable: 𝑛 𝑛 1 ̅= 𝑥 ∑ 𝑥𝑖 𝑆2 = 𝑀((𝑥 − 𝜇)2 ) = ∑ 𝑝𝑖 (𝑥𝑖 − 𝜇)2 𝑛 𝑖=1 𝑖=1 where x and xi are the values of the random variable where xi designates the elements of the sample and n is which it can assume with a probability of pi,  is the mean the number of elements in the sample. of the random variable, n is the number of possible values of the random variable. 35. What is an ordered array? An ordered array is a listing of the values of a sample from 27. Define the variance of a random sample with a formula. the smallest to the largest values. Variance of random sample: ∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅)2 36. Define the median of a sample! 𝑆2 = The median of a sample is the value which divides it into 𝑛−1 ̅ is the mean of where xi are the elements of the sample, 𝑥 two equal parts such that the number of values equal to the sample and n is the number of elements in the or greater than the median is equal to the number of sample. values equal to or less than the median. If the number of elements is odd, the median will be the middle value in 28. Define the variance of a random variable in words. the ordered array. If the number of elements is even, the The variance of a random variable is the expected value median will be the average of the two middle values in the of the squared deviation of the random variable from its ordered array. mean. 37. Define the i-th percentile of a sample! 29. Define the variance of a random sample in words. The i-th percentile of a sample is the smallest value that Variance of a random sample: a statistic estimating the is equal to or greater than i% of the observations. variance of a random variable or population from which the random sample has been taken. 38. Define the first, second and third quartile (Q1,Q2,Q3) of a sample. 30. What is the central limit theorem? The first, second and third quartile of a sample are the Given a population of an arbitrary distribution with a mean smallest values which are equal to or greater than 25%, 50% and 75%, respectively, of the observations. of  and an SD of , the means of the samples of size n OR taken from the population will be approximately normally Q1 is the 25th percentile, Q2 is the 50th percentile (or the distributed if n is large. The expected value of the sample median), Q3 is the 75th percentile. mean will be , and the SD of the sample mean (i.e., the standard error of the mean) will be 𝜎 ⁄√𝑛. 39. Define the mode of a sample! The mode of a sample is the value which occurs the most frequently. Biostatistics minimum requirement questions and answers (2024/25) 2 40. How can a histogram be constructed? Parameter  of a Poisson distribution is equal to the mean The class intervals (or the values of the variable) are and the variance of the distribution. displayed on the horizontal axis. Above each class interval (or variable value) a bar is erected so that the 47. When does a random variable follow a standard normal height corresponds to the frequency or the relative distribution? frequency of the respective class interval (or variable If it follows a normal distribution and the mean and value). standard deviation are 0 and 1, respectively. 41. Calculate the mean, median and mode of the data set 48. Calculate the standardized z value corresponding to a given below. (A different data set may be given on the value of 135 of a normally distributed random variable exam or in written tests.) with a mean value of 120 and a standard deviation of 10. Data set: 5, 8, 9, 5, 9, 1, 6, 5 Calculate the probability that the above random variable will assume a value less than 135. (Numbers different ∑𝑛𝑖=1 𝑥𝑖 5+8+9+5+9+1+6+5 from the ones given above may be present on the exam ̅= 𝑥 = =6 𝑛 8 or in written tests.) or 𝑥 − 𝜇 135 − 120 ∑𝑘𝑖=1 𝑓𝑖𝑥𝑖 1⋅1+3⋅5+1⋅6+1⋅8+2⋅9 𝑧= = = 1.5 ̅= 𝑥 = =6 𝜎 10 𝑛 8 where  is the mean and  is the standard deviation. where 𝑓𝑖 is the frequency of elements 𝑥𝑖 According to the table of the standard normal distribution: ordered array: 1,5,5,5,6,8,9,9 P(x