Biostatistics Minimum Requirement Questions

?
  • Created by: kpaul1234
  • Created on: 13-07-22 12:25
What are the permutations of n different elements taken n at a time and what is their number?
Permutations of n different elements taken n at a time are all the possible linear arrangements (orders) of all the elements. Their number is:
n!=n(n-1)(n-2)...2⋅1.
1 of 72
Define the combinations of n different elements taken k at a time in words.
All the possible selections (subsets) of size k of n
different elements.
2 of 72
What is the meaning of the (n/k) binomial coefficient?
Define it with a formula and with reference to
combinatorics (counting techniques).
(n/k)=n!/[k!(n-k)!]
The (n/k) binomial coefficient gives the number of ways k elements can be chosen from n different elements without regard to order.
3 of 72
Define nominal scale and give an example for it
Nominal scale is a list of mutually exclusive categories to which observations can be classified. E.g. the sex of a patient can be male or female.
4 of 72
Define ordinal scale and give an example for it
Ordinal scale is a list of categories, in which categories can be ranked according to their names or numbers assigned to them. E.g. the
efficiency of a drug treatment can be poor, average, good.
5 of 72
Define interval scale and give an example for it
The interval scale is a quantitative scale type, in which the numbers assigned to observations have real quantitative meaning. Their ratios aren't meaningful to the lack of an objective 0 point on the scale.
E.g. temp measured on the Celsius scale.
6 of 72
Define ratio scale and give an example for it.
Ratio scale is a quantitative scale type in which the
numbers assigned to observations have real quantitative meaning. They express quantitative relationships due to presence of an objective 0 point on the scale. E.g. measurement of height or blood gluc.
7 of 72
How are the relative frequency and probability of an event related to each other?
The probability of an event is the number around which the relative frequency (k/n) oscillates (n – the total number of experiments; k – the number of experiments in which the event occurred).
8 of 72
What is the definition of classical probability?
If there are N mutually exclusive and equally likely
outcomes of an event, and k of these posses a trait, E, the probability of E is equal to k/N.
9 of 72
What kind of values can (mathematical) probability
assume? What is the probability of a certain and an
impossible event?
Probability is a no. between 0&1, more
rigorously probability can assume any value in the
closed interval of [0,1]. (A closed interval includes its endpoints).
Probability of a certain event=1
Probablity of an impossible event=0.
10 of 72
Describe the relationship between the probabilities of event A and its complement event B!
P(A)+P(B)=1.
11 of 72
Define the sum of events A and B!
The sum of A and B is the event which occurs when either A or B or both of them occur.
12 of 72
What is the probability of the sum of events A and B?
P(A+B)=P(A)+P(B)-P(AB), where
A+B is the sum of events A and B,
AB is the product of events A and B.
13 of 72
Define the product of events A and B!
The product of A and B is the event which occurs when both A and B occur.
14 of 72
Define the complement event of event A!
The complement of A is the event which occurs when A doesn't occur and the sum of the probabilities of A and its complement event is 1.
15 of 72
When are events A and B exclusive?
If AB=0.
16 of 72
When are events A and B independent of each other?
A and B are independent if event B has no effect on the probability of event A and vice versa, i.e.
P(AB)=P(A)⋅P(B) or P(A|B)=P(A) or P(B|A)=P(B).
17 of 72
Using the terms of set theory define
a. the product ot events A and B
b. the sum of events A and B.
a. AB – the intersection of events A and B (AnB)
b. A+B – the union of events A and B (AuB).
18 of 72
What is the meaning of P(A|B)?
P(A|B) is the conditional probability of A given B, i.e. the probability of occurrence of event A if only those cases are considered when event B occurs.
19 of 72
What is the definition of a random variable?
If the values assumed by a variable are determined by chance factors, i.e. they cannot be exactly predicted in advance, the variable is called a random variable.
20 of 72
What is a continuous random variable?
A random variable is continuous if it can assume any value within a specified interval of values.
21 of 72
What is the definition of the cumulative relative
frequency of a sample?
The cumulative relative frequency of a sample at x gives the fraction of elements in the sample which are smaller than or equal to x.
22 of 72
What is the definition of the cumulative distribution function of a random variable?
The cumulative distribution function (cdf) of a random variable at x gives the probability that the random variable assumes a value smaller than or equal to x.
23 of 72
What is the probability that a continuous random
variable assumes a value in the interval between a and b?
The probability that a continuous random variable
assumes a value in the (a,b) interval is equal to the area under the curve of the probability density function between a and b.
24 of 72
Define the mean of a discrete random variable!
M(x)=nΣi=1 xipi
where xi is the ith value of the random variable, and pi is the probability that the random variable assumes the value of xi.
25 of 72
Define the variance of a discrete random variable with a formula.
Variance of a discrete random variable:
S^2=M((x-μ)^2)=nΣi=1 pi(xi-μ)^2
where x&xi are the values of the random variable, which it can assume w/ a probability of pi, μ is the mean, n is the no. of possible values of the random variable.
26 of 72
Define the variance of a sample with a formula
Variance of a sample:
S^2=(nΣi=1(xi-x̄)^2)/(n-1)
27 of 72
Define the variance of a random variable.
Variance of a random variable is the expected value of the squared deviations of the random variable from its mean.
28 of 72
Define the variance of a sample in words
Variance of a sample: a statistic estimating the variance of a random variable or population from which the sample has been taken.
29 of 72
What is the central limit theorem?
Given is a population of an arbitrary distribution w/ a mean of μ&an SD of σ. The means of samples of size n taken from this population will be approx normally distributed, if n is large. The mean of the distribution of the sample means will be=μ&SD=σ√n.
30 of 72
Define the standard deviation (SD) and the standard error of the mean (SEM) of a sample with formulas!
SD=√(nΣi=1(xi-x̄)^2)/(n-1), SEM=√(nΣi=1(xi-x̄)^2)/n(n-1)
xi: the elements of the sample, x̄ is the mean of the
sample, n is the number of elements in the sample.
31 of 72
What is the difference between the standard deviation (SD) and standard error of the mean (SEM) of a sample? Write your answer in words, do not use formulas.
The SD of a sample gives an unbiased estimation of the population SD, whereas the SEM is the SD of the distribution of the sample means. If the no. of elements of the sample increases, SD approaches the √ of the population variance, the SEM approaches 0.
32 of 72
Define the coefficient of variation (CV) in words and with a formula.
The coefficient of variation (CV) is the standard
deviation (SD) expressed as the percentage of the mean (x̄):
CV=100 SD/x
33 of 72
Define the mean of a sample with a formula and
interpret the variables!
x̄=1/n nΣi=1 xi
where xi designates the elements of the sample and n is the number of elements in the sample.
34 of 72
What is an ordered array?
An ordered array is a listing of the values of a sample from the smallest to the largest values.
35 of 72
Define the median of a sample!
The median of a sample is the value which divides it into 2 equal parts such that the no. of values = or > the median is = the no. of values = or < the median.
36 of 72
Define the i-th percentile of a sample!
The i-th percentile of a sample is the smallest sample value that is equal to or greater than i% of the observations.
37 of 72
Define the first, second and third quartile (Q1,Q2,Q3) of a sample.
The 1st, 2nd&3rd quartile of a sample are the smallest sample values which are equal to or greater than 25%, 50%&75%, respectively, of the observations.
OR
Q1 is the 25th percentile, Q2 is the 50th percentile (or the median), Q3 is the 75th percentile.
38 of 72
Define the mode of a sample!
The mode of a sample is the value which occurs the most frequently.
39 of 72
How can a histogram be constructed?
The class intervals are displayed on the horizontal axis.
Above each class interval a bar is erected so that the height corresponds to the frequency or the relative frequency of the respective class interval.
40 of 72
Calculate the mean, median and mode of the data set given below. (A different data set may be given on the exam or in written tests.)
Data set: 5, 8, 9, 5, 9, 1, 6, 5
41 of 72
Calculate the standard deviation and standard error of the mean of the sample given below. (A different data set may be given on the exam or in written tests.)
Sample: 3, 7, 5
42 of 72
Give the k th element of a binomial distribution with parameter p (the probability of the first possible outcome of a trial) if the total number of trials is n (k=0,1,2,3...,n), and define the probability it means!
Pn,k is the probability that a given event occurs k times in n independent trials w/ 2 possible outcomes w/ probabilities p&1-p:
Pn,k=n!/((n-k)!k!) p^k (1-p)^(n-k)
where n is the no. of trials, k is the no. of
occurrences given by a probability p.
43 of 72
Give the kth element of a Poisson distribution with
parameter λ and define the probability it means!
Pk=λ^k /k! e^-λ
Pk is the probability that the given event occurs k
number of times in an infinite number of independent trials.
44 of 72
Give the sum of the elements of a Poisson distribution with parameter λ!
infinityΣk=0 λ^k /k! e^-λ=1
45 of 72
Which two properties of the Poisson distribution are equal to λ, the parameter of the distribution?
Parameter λ of a Poisson distribution is equal to the mean and the variance of the distribution.
46 of 72
When does a random variable follow a standard normal distribution?
If it follows a normal distribution and the mean and standard deviation are 0 and 1, respectively.
47 of 72
Calculate the standardized z value corresponding to a value of 135 of a random variable distributed according to a normal distribution w/ a mean value of 120&a SD of 10. Calculate the probability that the above random variable will assume a value <135.
(Numbers different from the ones given above may be present on the exam or in written tests.)
48 of 72
What is the probability that a normally distributed
random variable falls in the regions ± 1σ, ± 2σ and ± 3σ (μ is the mean and σ is the standard deviation of the normal distribution)?
A normally distributed random variable falls within 1σ, 2σ and 3σ around the mean with probabilities of 68%, 95% and 99.7%, respectively.
49 of 72
What is a type I error in a statistical test?
A type I error is committed when a true null hypothesis is rejected.
50 of 72
What is a type II error in a statistical test?
A type II error is committed when a false null hypotheses is not rejected.
51 of 72
What is the relationship between the probability of
committing a type I error and the level of significance?
The level of significance is equal to the probability of committing a type I error if the null hypothesis is true.
52 of 72
What is the p value in hypothesis testing?
The p value is the probability of obtaining a value of the test statistic as extreme or more extreme than the one actually computed provided the null hypothesis is true.
53 of 72
What is a two-sided statistical test?
The rejection area is split into two parts in a two-sided statistical test, i.e. the null hypothesis is rejected when the value of the statistic is significantly larger or smaller than according to the null hypothesis.
54 of 72
Write the formula for the statistical test used for single population mean hypothesis testing when the SD of the population is known and interpret the variables!
z=(x̄-μ)/(σ/√n)
where: x̄=the mean of the sample, μ=the population mean, σ=the standard deviation of the population, n=the number of elements in the sample.
55 of 72
What kind of hypothesis testing can the F test be used for?
It can be used to compare the variances of two random variables following normal distributions.
56 of 72
What kind of quantities can be compared with a twosample independent groups t-test?
It can be used to compare the means of two
independent random variables with normal distributions if the standard deviation of the random variables is not significantly different according to an F test.
57 of 72
When does a statistic give an unbiased estimation of a parameter?
When the expected value of the statistic is equal to the parameter.
58 of 72
When is a sample representative?
When we use random sampling, i.e. each element of the population has equal probability of being sampled.
59 of 72
What are the most important attributes of the quality of an estimate? Define them briefly.
1. accuracy or unbiasedness: An estimate is unbiased or accurate if its expected value is equal to the mean of the estimated parameter.
2. precision or reproducibility: An estimate is precise or reproducible if its standard deviation is small.
60 of 72
Define the null hypothesis for a two-sample, two-sided (two-tailed) t-test.
The means of the two populations (μ1 and μ2) under investigation are identical, i.e μ1=μ2.
61 of 72
Write the null hypothesis for an F test.
The variances of the two populations under investigation are equal, i.e.σ1^2=σ2^2
62 of 72
What is the definition of specificity of a clinical diagnostic test?
Specificity is the probability of obtaining a negative test result in a patient without the examined disease condition, i.e. the reliability of the test in correctly identifying those patients who do not have the condition: P(T- | D-).
63 of 72
What is the definition of sensitivity of a clinical diagnostic test?
Sensitivity is the probability of obtaining a positive test result in a patient who has the examined disease condition, i.e. the reliability of the test in correctly detecting those patients who have the condition: P(T+ | D+).
64 of 72
What is the definition of positive predictive value?
Positive predictive value is the probability that a subject with a positive diagnostic test result has the disease condition: P(T+ | D+)
65 of 72
What is the definition of negative predictive value?
Negative predictive value is the probability that a subject with a negative test result does not have the disease condition: P(T- | D-).
66 of 72
Draw the 2x2 contingency table characterizing the
sensitivity and specificity of a clinical diagnostic test.
Test result
positive | negative
Disease | present | true positive | false negative
Disease | absent | false positive | true negative
67 of 72
What is the definition of odds?
Odds is the ratio of the probability of the occurrence of an event to the probability that it will not occur. If p is the probability of occurrence of an event, then
odds=p/(1-p)
68 of 72
Define odds ratio from a epidemiological point of view
Odds ratio is the ratio of the odds that a risk factor is present in the diseased population to the odds that it is present in the healthy population.
69 of 72
What is relative risk in epidemiological studies?
Relative risk (RR) is the ratio of the relative frequency (or probability) of a disease in the risk factor-exposed population to the relative frequency (or probability) of the disease in the risk factor-free population.
70 of 72
Calculate the expected frequency in the upper left cell of the contingency table given below assuming the independence of the two random variables.
Categories of the 1st variables
A1 | A2 | Marginal Frequencies
Categories of | B1 | Nexpected=? | | NB1
2nd variable | B2 | | | NB2
Marginal frequencies | NA1 | NA2 | N
Nexpected=P(A1)P(B1)N=NA1/N NB1/N N = (NA1NB1) / N
71 of 72
Specify the x^2 statistic for a test of independence.
x^2=Σfor all cells (Nobserved-Nexpected)^2 / Nexpected
72 of 72

Other cards in this set

Card 2

Front

Define the combinations of n different elements taken k at a time in words.

Back

All the possible selections (subsets) of size k of n
different elements.

Card 3

Front

What is the meaning of the (n/k) binomial coefficient?
Define it with a formula and with reference to
combinatorics (counting techniques).

Back

Preview of the front of card 3

Card 4

Front

Define nominal scale and give an example for it

Back

Preview of the front of card 4

Card 5

Front

Define ordinal scale and give an example for it

Back

Preview of the front of card 5
View more cards

Comments

No comments have yet been made

Similar Mathematics resources:

Coding
See all Mathematics resources »See all Biostatistics resources »