Biostatistics Exam 2
What value is p when the probability distribution is symmetric?
0.5
Evaluate P(A U Ac)
1
Evaluate: P(A U Ac)
1
Evaluate: e^0
1
Evaluate: log10
1
What happens when p is very small in regards to binomial distributions?
1. 1-p is close to 1 2. np(1-p) is approximately equal to np
What are a few assumptions of a Poisson variable?
1. A very large number of events are theoretically possible in a given interval 2. Events are independent both within and between intervals 3. Probability of even occurrence is proportional to interval length
What are some characteristic assumptions/ properties of the Poisson distribution?
1. As 1-p approaches 1, np(1-p) approaches np 2. Mean and variance can be represented by a single parameter 3. The mean can be expressed as a functino of n and p
Real life examples of normally distributed data:
1. Blood pressure 2. Serum cholesterol level 3. Height and weight
Characteristics of distributions with smaller degrees of freedom
1. Distributions more spread out 2. As df increases, the t distribution approaches the standard normal
Limitations of a point estimation:
1. Does not provide information about inherent variability of the estimator 2. We do not know how close mean is to mu in any given situation3. Provides no information about the size of the sample
What are a few assumptions of a Bernoulli random variable?
1. Fixed number of trials, each of which yields only two outcomes 2. Outcomes of the n trials are independent 3. Probability for success is a constant for each trial
Some real life scenarios that the Poisson distribution can be used for:
1. Model the number of ambulances needed in a city in a given night 2. The number of particles emitted from a specified amount of radioactive material 3. The number of bacterial colonies growing on a Petri dish
What is a difference between standard normal distributions and T-Distributions?
1. T-distributions have somewhat thicker tails than the normal distribution 2. Extreme values are more likely to occur with T dristribution
The distribution of sample means computed for samples of size n has three important properties:
1. The mean of the sampling distribution is identical to the population mean mu 2. The standard deviation of the distribution of sample means is equal to sigma/ squareroot(n) - this quantity is known as the standard error of the mean 3. Provided that n is large enough, the shape of the sampling distribution is approximately normal
What are the three underlying assumptions of the Poisson distribution?
1. The probability that a single event occurs within an interval is proportional to the length of the interval 2. Within a single interval, an infinite number of occurrences of the event are theoretically possible. We are not restricted to a fixed number of trials 3. The events occur independently both within the same interval and between consecutive intervals
What are the three assumptions of binomial distribution?
1. There are a fixed number of trials n, each of which results in one of two mutually exclusive outcomes 2. The outcomes of the n trials are independent 3. The probability of success p is constant for each trial
What features do the T-distribution have in common with the standard normal distribution?
1. Unimodal, symmetric 2. Mean of 0, with area under the curve 1
What are the characteristics of the normal curve for a normal distribution?
1. Unimodal, symmetric about its mean (mu) 2. Mean, median, and mode of the distribution are all identical 3. Standard deviation (sigma) specifies the amount of dispersion around the mean
What is the simplest possible symbology for "the event of either A or B or both A and B"?
A U B
Degrees of Freedom
A characteristic of the t-distribution Measure the amount of information available in the data that can be used to estimate sigma^2
Poisson distribution
A discrete probability distribution describing the likelihood of a particular number of independent events within a particular interval
Random variable
A variable that can assume a number of different values such that any particular outcome is determined by chance
Continuous random variable
A variable that can take on any value within a specified interval or continuum
When assessing the area under the standard normal curve between two bounds (between two z-values), we compute the area under one tail and sum it with the area under the other tail, and subtract from one. What property allows us to do this?
Additive rule
When assessing the two-tailed probability, we compute the area under one tail and sum it with the area under the other tail. What property allows us to do this?
Additive rule
Why does the T-distribution have DOF, when the standard normal distribution does not?
To account for the information available in estimating the population standard deviation
Variable
Any characteristic that can be measured or categorized
What population does the central limit theorem apply to?
Any population with a finite standard deviation, regardless of the shape of the underlying distribution
Probability distribution
Applies theory of probability to describe the behavior of the random variable
Suppose a vector x containing 10 non-identical items. Suppose then that I ask every student in the class to compile the single line of code sample(x,1) twice in a row in R. How many students will get the same two sub-samplings in their two compilings of sample(x,1)?
Approximately 10% of students (the code instructs R to take 1 sample randomly)
Which is true regarding the sampling distribution of the mean?
As long as n>1, standard error of the mean is always smaller than ơ
Why does the df increase as the t-distribution approaches the standard normal?
Because as the sample size increases, s becomes a more reliable estimate of mu If n is very large, knowing the value of s is nearly equivalent to knowing sigma
Why are the degrees of freedom n-1 rather than n?
Because we lose 1 df by estimating the sample mean
Discrete random variable
Can assume only a fixed or countable number of outcomes
What does the equation n! / x!(n-x!) represent?
Combination of n objects chosen at a time; it represents the number of ways in which x objects can be selected from a total of n objects without regard to order
What is the complement of True-Negative test?
False- Positive test
A ____________ displays each observed outcome and the number of times it appears in the set of data.
Frequency table
What is the frequentist definition of probability theory?
If an experiment is repeated n times under essentially identical conditions, and if the event A occurs m times, then as n grows large, the ration m/n approaches a fixed limit that is the probability of A
What is the term and symbol for "the event of both A and B"?
Intersection; A ∩ B
Why do people tend to say that a sample size of 30 is a good target for small studies?
The T-distribution approximates the standard normal to within 5% for DOF near 29
What happens as n increases for sampling distributions of the mean?
The amount of sampling variation decreases
What happens when the standard normal distribution is symmetric about z=0?
The area under the curve to the right of z is equal to the area to the left of -z.
What happens if n is large enough in sampling distributions of the mean?
The distribution of the sample means is approximately normal
What is the term and symbol for event "either A or B, or both A and B"
union; A U B
What are the values of mu and sigma when a single curve is tabulated for a normal distribution?
0
What is the variance of X in regards to binomial random variables and Bernoulli trials?
np(1-p)
What qualities does a normal distribution share with a binomial distribution?
p is constant but n approaches infinity
Suppose event A occurs m times in n repetitions of an experiment. Using the frequentist definition of probability of an event A, what is the probability of the complimentary event, Ac?
(n-m) / n
Evaluate: log0.1
-1
Evaluate the expression: P(A U Ac) + P(A ∩ Ac) + (P(A ∩ Ac) * P(A U Ac))
0
Evaluate: P(A ∩ Ac)
0
Evaluate: log1
0
Evaluate: lne^2
2
What percentage of the standard normal distribution lies within ± 1 standard deviation from the mean?
68.2%
Consider a 95% CI of 102 ≤ mean ≤ 118, obtained through a sample of 100 items. Suppose instead that a sample of 25 had generated the same sample parameters (i.e. same mean and standard deviation). What would the 95% CI then be?
94 ≤ mean ≤ 126
What percentage of the standard normal distribution lies within ± 2 standard deviations from the mean?
95.4%
When is the Bayes Theorem used?
Diagnostic testing or screening
Which direction is the distribution skewed when p > 0.5?
Left
Normal distribution (a.k.a. Gaussian distribution or bell-shaped curve)
Most common continuous distribution is constant
What variables are used for Bernoulli's random variables?
Mutually exclusive variables
What is the equation for the odds ratio?
OR = (P(disease | exposed) / (1-P(disease | exposed))) / (P(disease | unexposed) / (1-P(disease | unexposed)))
Prior probability
Our initial belief about the probability of an outcome
What are independent events?
Outcome of one event has no effect on the occurreence or nonoccurrence of the other
What is the equation for the multiplicative rule when the events are said to be independent?
P (A ∩ B) = P(A) P(B)
What is the equation for the additive rule?
P(A U B) = P(A) + P(B)
Suppose A and B are independent. Which of the following is true?
P(A | B) = p(A)
What is the multiplicative rule of probability in equation form?
P(A ∩ B) = P(A) * P(B | A)
Equation for Multiplicative rule of probability
P(A ∩ B) = P(A)P(B | A) or P(B ∩ A) = P(B)P(A | B)
If events A and B are not mutually exclusive, the additive rule no longer applies. What, then, is the way to describe the union of A and B?
P(A) + P(B) - P(A ∩ B)
What is the equation for the frequentist definition of probability theory?
P(A) = (m/n)
Equation to find the probability of Ac using frequentist equation
P(Ac)= 1-(m/n)
What is the formula for a conditional probability?
P(B | A) = P(A ∩ B)/P(A)
Bayes Theorem Formula
P(E1 | H) = (P(E1) P(H | E1)) / ((P(E1) P(H|E1) + P(E2)P(H | E2) + P(E3)P(H | E3))
What is the Poisson distribution equation?
P(X = x) = e^(-lamda)lamda^(x) / x!
Interval estimation
Preferred over point estimation Provides a range of reasonable values that are intended to contain the parameter of interest - the population mean mu, in this case - with a certain degree of confidence
What is the equation for the relative risk?
RR = P(disease | exposed) / P(disease | unexposed)
Which direction is the distribution skewed when p < 0.5?
Right
What feature does the T-distribution NOT have in common with the standard normal distribution?
Same tails (neither thicker, nor thinner)
Why is it impossible to tabulate the area associated with each and every normal curve?
Since a normal distribution could have an infinite number of possible values for its mean and standard deviation
What does the standard deviation of a normal distribution specify?
Specifies the amount of dispersion around the mean
What does the T-Distribution reflect?
The extra variability introduced by the estimate s
What happens when the farther the underlying population departs from being normally distributed?
The larger the value of n that is necessary to ensure the normality of the sampling distribution
What is an identifying characteristic of the Poisson distribution?
The mean is equal to the variance
What does n! allow us to calculate?
The number of ways in which the n individuals can be ordered; note that we have n choices for the first position, n-1 choices for the second position, and so on
Exhaustive
The probabilites of mutually exclusive events sum to 1
Student's T-Distribution or T-Distribution
The probability distribution for small sample sizes taken from a normal population or a large random sample is taken from any population, t = (x-bar - μ) / (s / square root n) with df = n-1, the standardized statistic's distribution
Law of total probability
The probability of an event is the sum of its probability across every possible condition
Posterior probability
The probability that a hypothesis is true after consideration of the evidence
Conditional probability
The probability that an event B will occur given that we already know the outcome of another event A (Does the prior occurrence of A cause the probabiity of B to change?)
Statistical inference
The process of drawing conclusions about an entire population based on the information in a sample
Confidence interval
The range of values within which a population parameter is estimated to lie
Standard error of the mean
The standard deviation of the distribution of sample means is equal to sigma/ squareroot(n)
Central limit theorem
The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution.
What does the central limit theorem allow us to quantify?
The uncertainty inherent in statistical inference without having to make a great many assumptions that cannot be verified
What is the interpretation of the symbol -z(α/2)?
The value of the Z standardization that cuts off an area of (α/2) in the left tail
What happens when we select larger and larger random samples in regards to mu?
The variability of the sample mean - out estimator of the population mean mu - becomes smaller
How are standard normal distributions and T-Distributions similar?
They are both unimodal and symmetric around its mean of 0
What is the relationship between the mean and the variance of a Poisson distribution?
They are identical and can be represented as a single parameter (lamda)
What is the complement of False-Negative test?
True-Positive test
Mutually exclusive or disjoint events
Two events that cannot occur simultaneously
Point estimation
Using the sample data to calculate a single number to estimate the parameter of interest
Suppose I want to know the z-value for a series of probabilities and I enter a code into R as follows: qnorm(c(0.1,0.5,0.75,0.9,0.95)). I could have run this code multiple times, i.e. qnorm(0.1), then qnorm(0.5), then qnorm(0.75), etcetera, but clearly I didn't. What technique did I use to execute my target calculation in a single line of code?
Vectorization
What is the proper interpretation of the 95% CI?
We are 95% confident that the mean of the distribution lies within the CI
What do we consider when trying to construct a one-sided confidence interval?
We consider the area in one tail or the standard normal distribution only
When is the binomial distribution impractical to use as a basis for calculations?
When n becomes too large
When is the standard error of the mean always smaller than the standard deviation of the population?
When n is greater than 1
When is the quantity np(1-P) the largest?
When p is equal to 0.5
When are one-sided confidence intervals used?
When we are concerned with either an upper limit for the population mean mu or a lower limit for mu, but NOT both
Binomial distribution
a frequency distribution of the possible number of successful outcomes in a given number of trials in each of which there is the same probability of success
Null event
an event that will never happen
What is the term and symbol for event "not A"
complement; Ac
What is a valid and useful code for calculating the binomial probability of observing 3 occurrences in 8 observations of something that might occur with probability 18%?
dbinom(5,8,prob=0.82)
What happens to the value of np(1-p) as p approaches 0 or 1 in regards to binomial random variables and Bernoulli trials?
it decreases
What qualities does a normal distribution share with a Poisson distribution?
lamda approaches infinity
What two parameters of a normal distribution define a normal curve?
mu and sigma
What happens when p is very large or very small in regards to binomial random variables and Bernoulli trials?
nearly all the outcomes take the same value and the variability among outcomes is small
What is the mean of a binomial random variable equal to?
np
What is the mean value of X equal two in regards to binomial random variables and Bernoulli trials?
np
What is the variance equation of a binomial random variable?
np(1-p)
For a fixed n, what value p maximizes the variance of a binomial distribution (n*p*(1-p))?
p=0.5
When two events are mutually exclusive, the additive rule of probability states:
that the probability that either of the two events will occur is equal to the sum of the probabilities of the individual events
Population mean
the average value assumed by a random variable
Population variance
the dispersion of the value relative to the population mean
What is obtained by multiplying the number of independent Bernoulli trials by the probability of success at each trial?
the mean value of a binomial random variable X - or the average number of "successes" in repeated samples of size n
Probability density
the probability distribution of continuous random variable
Describe the notation P(B | A)
the probability of the event B given the event A has already occurred
Multiplicative rule of probability definition
the probability that two events A and B will both occur is equal to the probability of A multiplied by the probability of B given that A has already occurred
Population standard deviation
the square root of the population variance
When two events are not mutually excusive the probability that either of the events will occur is equal to:
the sum of the individual probabilities minus the probability of their intersection P(A U B) = P(A) + P(B) - P(A ∩ B)