Exam 1

Question 1

Y

Answer 1

Data

Question 2

Yi

Answer 2

An observation of the data

Question 3

Y bar

Answer 3

Mean of the data

Question 4

s²

Answer 4

Variance of the data

Question 5

s

Answer 5

Standard deviation of the data

Question 6

µ

Answer 6

Mean

Question 7

σ²

Answer 7

Variance

Question 8

σ

Answer 8

Standard deviation

Question 9

Answer 9

Arithmetic Mean

The average

Most common

Unbiased estimate of µ if assumptions on the earlier slide are met

Question 10

Geometric Mean

Answer 10

Used when the values are multiplied

Used in population ecology

Question 11

Harmonic Mean

Answer 11

• Greater weight to extreme small values

Used for rates, and in population genetics to estimate effective population size

Question 12

Symmetrical Distributions

Answer 12

If the distributions are perfectly symmetrical then the arithmetic mean, median, and mode are equal.

Question 13

Variance

Answer 13

The variance of the population (σ2) can be estimated from data

Remember that variance is in units2

Question 14

Standard Deviation

Answer 14

The square root of the variance

On average, s does not change when you increase sample size

Question 15

Standard Error of the Mean

Answer 15

The standard deviation of the estimated population mean

Decreases when you increase sample size

Question 16

Coefficient of Variation

Answer 16

• The sample standard deviation divided by the sample mean

Often multiplied by 100 to represent a percent

Question 17

Classical definition

Answer 17

P=0, outcome will never happen

P=1, event will always happen

Note: typically cannot measure

Question 18

Frequentist definition

Answer 18

P=0, outcome didn’t occur in any trial

P=1, outcome occurred in every trial

Note: can measure

Question 19

Experiment

Answer 19

¡A set of trials

¡E.g. all the crocodiles in a nest

Question 20

Trial

Answer 20

¡Each replicate event

¡E.g. a particular crocodile

Question 21

Sample space { }

Answer 21

¡The set of all possible outcomes

¡E.g. hatched and didn’t hatch

Question 22

Outcome ( )

Answer 22

¡A possible result of a event

¡E.g. hatched

Question 23

Event

Answer 23

¡A process with a beginning and end

¡E.g. hatching of an crocodile

Question 24

Defining Sample Space

Answer 24

nOutcomes are mutually exclusive

nOutcomes are exhaustive

Question 25

Axiom 1

Answer 25

The sum of all the probabilities of outcomes within a single sample space = 1.

Question 26

nComplex events

Answer 26

¡Composites of simple events in the sample space ¡ ¡Rolling 3 OR rolling 4 ¡First crocodile hatching OR the second crocodile hatching

Question 27

nShared events

Answer 27

¡Multiple simultaneous occurrences of simple events in the sample space ¡Rolling 1 on the first roll AND 1 on the second roll ¡First AND second crocodile hatch

Question 28

Axiom 2

Answer 28

The probability of a complex event equals the sum of the probabilities of the outcomes that make up that subset.

Question 29

Conditional Probabilities

Answer 29

Used to calculate probabilities when you know that an outcome has occurred or will occur Probability of a given b

Question 30

Bayes’ Theorem

Answer 30

The conditional probability rewritten in terms of the inverse conditional probability.

Question 31

Frequency Distribution

Answer 31

nBased on number of observations within a given range ¡Non-negative integers

Question 32

Probability Distributions

Answer 32

nBased on probabilities and often represented by the density ¡Density is calculated using the area of each bar ¡Thus, densities can be greater than 1

Question 33

Discrete

Answer 33

nTake on finite numbers nE.g., integers

Question 34

Continuous

Answer 34

nTake on any value within a range nE.g., diameter of your head

Question 35

Discrete Distributions

Answer 35

nBernoulli ¡Uncommon because of simplicity nBinomial ¡Common in biological data nPoisson ¡More appropriate when a successful outcome is rare

Question 36

Bernoulli Distribution

Answer 36

nA single event nOnly two outcomes ¡E.g., 0 and 1 nP(1)=p and P(0)=1-p

Question 37

Binomial Distribution

Answer 37

nMultiple Bernoulli trials nProbability of successful outcomes ¡E.g. 5 of 10 eggs hatch nNeed to know ¡Probability of success (p) ¡Number of trials (n) ¡Number of successful trials (X)

Question 38

Poisson Distributions

Answer 38

nUsed when ¡Number of trials is unclear ¡p is very small ¡Sampling fixed area of space or time interval nNeed to know ¡Number of observed occurrences (x) ¡Rate parameter (λ)

Question 39

Rate Parameter

Answer 39

Estimated from experiments or prior data Represents the average of the Poisson Distribution, and also the variance

Question 40

Expected Value of the Distribution

Answer 40

Is the mean or average

Question 41

Variance of the Distribution

Answer 41

The spread of the data Calculated as the squared deviations from the mean

Question 42

Continuous Distributions

Answer 42

Uniform Distributions Used to represent truly random events Normal Distributions Extremely common and important in biology (as well as many other fields) There are lots of other continuous distributions F, t, Beta, Gamma, Log-normal, exponential, chi-squared, etc.

Question 43

Problem with Continuous Distributions

Answer 43

Probability for any given value is zero There are an infinite number of different possibilities

Question 44

Solution

Answer 44

Calculus and axioms Axiom 1 state the sum of all probabilities must equal 1.0 So, using integral calculus, we can find the function that satisfies axiom 1

Question 45

Parameters Min (a) and max (b)

Answer 45

Uniform Distribution

Question 46

Probability Density Function (PDF)

Answer 46

Is the function that gives the probability for x

Question 47

Cumulative Density Function (CDF)

Answer 47

Is a function that gives the area under the curve (i.e. the cumulative probability)

Question 49

Parameters Mean (µ) and Standard deviation (σ)

Answer 49

Normal Distribution

Question 50

Works for any type of distribution Sum or average random samples from a distribution Result is a distribution that is normal!

Answer 50

Central Limit Theorem

Question 52

Lambda and Max of x axis

Answer 52

Poisson Dist.

Question 53

of trials and probability of success

Answer 53

Binomial Dist

Question 54

Min and Max

Answer 54

Uniform Dist.